Journal of Topology 1 (2008) 159–186
c 2007 London Mathematical Society doi:10.1112/jtopol/jtm011
Axioms for higher ...

Author:
Igusa K.

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Journal of Topology 1 (2008) 159–186

c 2007 London Mathematical Society doi:10.1112/jtopol/jtm011

Axioms for higher torsion invariants of smooth bundles Kiyoshi Igusa Abstract This paper attempts to explain the relationship between various characteristic classes for smooth manifold bundles which are known as ‘higher torsion’ classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz–Reidemeister torsion and higher Miller–Morita–Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples. We also show how any higher torsion invariant, that is, any characteristic class satisfying the two axioms, can be computed for a smooth bundle with a ﬁberwise Morse function with distinct critical values. Finally, we explain the statements of the conjectured formulas relating higher analytic torsion classes, higher Franz–Reidemeister torsion and Dwyer–Weiss–Williams smooth torsion.

Contents 1. Introduction . . . . 2. Preliminaries . . . . . . . . . 3. Axioms 4. Statement . . . . 5. Extension to relative case . 6. Stability of higher torsion . 7. Computation of higher torsion 8. Proof of the main theorem 9. Existence of higher torsion References . . . . .

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159 162 163 165 167 172 173 177 182 186

1. Introduction Higher analogues of Reidemeister torsion and Ray–Singer analytic torsion were developed by J. Wagoner, J.R. Klein, the author, M. Bismut, J. Lott, W. Dwyer, M. Weiss, E.B. Williams, S. Goette and many others [3, 4, 11, 13, 14, 17, 21, 20, 26]. This paper develops higher torsion from an axiomatic viewpoint. There are three main objectives to this approach: (1) Make higher torsion easier to understand. (2) Simplify the computation of these invariants. (3) Explain the theorems relating higher Franz–Reidemeister torsion, Miller–Morita– Mumford (tautological) classes, Dwyer–Weiss–Williams higher torsion and higher analytic torsion classes. Received 22 February 2007; published online 25 October 2007. 2000 Mathematics Subject Classiﬁcation 55R40 (primary), 57R50, 19J10 (secondary). I am in great debt to John R. Klein and E. Bruce Williams for their help in completing the ﬁnal crucial steps in the proof of the main theorem. I also beneﬁtted greatly from conversations with Sebastian Goette, Xiaonan Ma, Wojciech Dorabiala and Gordana Todorov. Also, I would like to thank Bernard Badzioch for his inspired presentation of this work during the Arbeitsgemeinshaft at Oberwolfach in April, 2006. Finally, I should not forget to thank the organizers Thomas Schick, Ulrich Bunke and Sebastian Goette of the conference on higher torsion in G¨ o ttingen in September 2003 at which the ﬁrst version of these results were developed and announced. Research for this paper was supported by NSF grants DMS 02-04386 and DMS 03-09480.

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The following theorems are examples of results which will make much more sense from the axiomatic viewpoint. Theorem 1.1 (Hain–I–Penner [17, 19]). The higher Franz–Reidemeister torsion invariants for the Torelli group τ2k (Tg ) ∈ H 4k (Tg ; R) are proportional to the Miller–Morita–Mumford classes. This had been conjectured by J.R. Klein [22]. The precise proportionality constant was computed in [17]. We will see that this theorem is an example of the uniqueness theorem for even higher torsion invariants. The next theorem is Theorem 0.2 in [13]. See [3], [13], [14] for more details. Theorem 1.2 [14]. Suppose that p : M → B is a smooth bundle with closed oriented manifold ﬁber X, F is a Hermitian coeﬃcient system on M and H ∗ (X; F) admits a π1 B invariant metric. Suppose further that there exists a ﬁberwise Morse function on M . Then the Chern normalizations (from [3]) of the higher analytic torsion classes T2k (E) ∈ H 4k (B; R) and the higher Franz–Reidemeister torsion class are deﬁned and agree up to a correction term which is a multiple of the transfer to B of the Chern character of the vertical tangent bundle of M : ch E T2k (E) = τ2k (E) + ζ (−2k)rk(F)trB (ch4k (T v M )).

This theorem, together with the uniqueness theorem for odd higher torsion invariants proved below, suggests that nonequivariant higher analytic torsion classes are odd torsion invariants. Recently, Sebastian Goette has claimed that he can prove his theorem in general, that is, without the existence of a ﬁberwise Morse function. Finally, in the case where we compare diﬀerent smooth structures on the same topological manifold bundle, both Dwyer–Weiss–Williams higher torsion of [11] and the higher Franz– Reidemeister torsion of [17] are deﬁned and must be proportional. Details will be given in forthcoming joint work with Sebastian Goette. In this paper, we deﬁne a higher torsion invariant to be a characteristic class τ (E) ∈ H 4k (B; R) of ‘unipotent’ smooth bundles E → B (deﬁned in Section 2) which satisﬁes two axioms (Section 3). We show in Section 4 that each such invariant is the sum of even and odd parts τ = τ ev + τ od . The main theorem (Theorem 4.4) is shown below. Theorem 1.3. Nontrivial even and odd torsion invariants τ ev , τ od exist in degree 4k for all k > 0 and they are uniquely determined up to scalar multiples. The uniqueness statement is simply a reﬂection of the fact that, given the value of a higher torsion invariant on the universal oriented S 1 bundle over CP ∞ and the associated S 2 bundle, the higher torsion invariant can be computed or shown to be uniquely determined in all other cases. Since the cohomology of CP ∞ is one dimensional in degree 4k, these universal examples are given by two scalars s1 and s2 (Subsection 4.2). The ﬁrst step is to extend the deﬁnition to the case where the ﬁber of E → B has a boundary giving a subbundle ∂ v E which we call the vertical boundary of E. The higher torsion in this case is deﬁned in Subsection 5.1 to be half the torsion of the double of E plus half the torsion of ∂ v E. In Subsection 5.2, we extend to the relative case (E, ∂0 E) when the vertical boundary is a union of two subbundles ∂ v E = ∂0 E ∪ ∂1 E each of which is unipotent. The relative higher

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torsion is simply deﬁned to be τ (E, ∂0 E) = τ (E) − τ (∂0 E). In Section 7, we use the transfer axiom to compute the higher torsion for any oriented linear disk or sphere bundle in terms of s1 and s2 . We extend this to ﬁberwise products of linear disk bundles, such as those that arise from ﬁberwise Morse functions (E, ∂0 E) → (I, 0). If the critical points have distinct critical values, then the additivity axiom allows us to express the higher torsion of E as a sum of higher torsions of disks which we have already computed. In Subsection 7.3, we make crucial use of Hatcher’s construction of disk bundles with exotic smooth structures. Since these admit ﬁberwise Morse functions, their higher torsion can be computed. By a theorem of B¨ okstedt, these examples rationally generate all possible smooth disk bundles in the stable range. In Section 6, we show that higher torsion is stable. This implies that the higher torsion is determined (by the two scalars s1 , s2 ) for all disk bundles. By additivity, this also implies that the higher torsion is determined on all h-cobordism bundles. At this point, we are ready to prove the main theorem. We deﬁne the diﬀerence torsion to be τ δ = τ − aM2k − bτ2k , where M2k is the higher Miller–Morita–Mumford class and τ2k is the higher Franz–Reidemeister torsion and the real numbers a, b are uniquely determined by the requirement that τ δ = 0 on all oriented linear sphere bundles. In that case, the computation of Hatcher’s example shows that τ δ = 0 on all disk bundles and the h-cobordism calculation implies that τ δ (E, ∂0 E) depends only on the ﬁber homotopy type of the pair (E, ∂0 E). In Section 8, we show that this ﬁber homotopy invariant must be trivial. The main theorem follows. In Section 9, we show that there are two linearly independent higher torsion invariants of degree 4k given by the higher Miller–Morita–Mumford (MMM) classes E ((2k)!ch4k (T v E)) ∈ H 4k (B; Z) M2k (E) = trB

and the higher Franz–Reidemeister (FR) torsion invariants τ2k (E) ∈ H 4k (B; R). Using basic properties of higher FR-torsion proved in [17] and [19], in particular the framing principle and the transfer theorem, it is easy to show that τ2k satisﬁes the axioms. Basis properties of the transfer map imply that M2k also satisﬁes these axioms. It is easy to see that M2k is an even higher torsion invariant, that is, it is zero when the ﬁber is a closed odd dimensional manifold. However, τ2k has both even and odd components. The uniqueness theorem implies the known theorem [17] that τ2k and M2k are proportional whenever the ﬁber is a closed oriented even dimensional manifold, for example, an oriented surface. The proportionality constant is determined by computing one example, for example, the universal S 2 bundle over CP ∞ and we get: τ2k (E) =

(−1)k ζ(2k + 1) M2k (E). 2(2k)!

This is not a new proof of this formula since we use this formula in the proof of the uniqueness theorem. However, the point is that it is now very transparent. Finally in Subsection 9.3, we characterize the tangential and exotic components of higher torsion given by τ = τT + τx, where τ T is an even higher torsion invariant and τ x is proportional to higher FR-torsion. We will see that DWW smooth torsion is exotic by deﬁnition. This explains why we believe that DWW smooth torsion is proportional to higher FR torsion. Bismut and Lott [4] showed that nonequivariant higher analytic torsion classes are trivial for bundles with closed even dimensional ﬁbers. Thus, we believe that they are odd higher torsion

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invariants. S. Goette’s theorem above says that, assuming the existence of a ﬁberwise Morse function, T2k is proportional to the odd part of τ2k . And the proportionality constant is 1 if they are normalized in the same way. Finally, we want to point out that X. Ma [23] has shown that higher analytic torsion satisﬁes the transfer axiom and W. Dorabiala and M. Johnson [10] have shown that DWW homotopy torsion satisﬁes the product formula (Corollary 5.10) which is a variation of the transfer axiom. Becker and Schultz [1] have shown that the transfer in stable homotopy, especially for smooth bundles, can be characterized by axioms very similar to ours. The main diﬀerence is that they use a product formula instead of the formula analogous to our transfer axiom. This paper combines two earlier works: my lecture notes [18] and the preliminary manuscript ‘Axioms for higher torsion II.’ 2. Preliminaries We consider smooth ﬁber bundles p

→ B, F →E− where E and B are compact smooth manifolds and p is a smooth submersion. We usually assume that the ﬁber F is a closed oriented manifold of dimension n. But, we also consider the case when F is oriented with boundary ∂F . This gives a subbundle ∂F → ∂ v E → B of E. We call ∂ v E the vertical boundary of E. (The boundary of E is the union of ∂ v E and p−1 (∂B).) We assume that B is connected. We assume that the action of π1 B on F preserves the orientation of F . We will assume that the bundle E → B is unipotent in the sense that the rational homology of its ﬁber F is unipotent as a π1 B-module. In other words, H∗ (F ; Q) has a ﬁltration by π1 B-submodules so that the subquotients have trivial π1 B actions. In particular, π1 B does not permute the components of F . Note that unipotent π1 B-modules form a Serre category. In fact, it is the Serre category generated by the trivial modules. A unipotent module is the same as a nilpotent module. However, ‘unipotent’ is used in the case of a representation of a group over a ﬁeld. A unipotent ﬁbration is not the same as a nilpotent ﬁbration. Recall that a ﬁbration E → B is nilpotent if there is a sequence of ﬁbrations p1

p2

p3

B = E0 ←− E1 ←− E2 ←− · · · converging to E (so that E is the inverse limit of the Ei ), so that each pi is a principal ﬁbration with ﬁber an Eilenberg–MacLane space K(πi , ni ) with ni → ∞ as i → ∞. (See, for example, [15].) This implies that the action of π1 B on the homotopy groups of the ﬁber is nilpotent. Quasi-nilpotent means that the action of π1 B on the homology of the ﬁber is nilpotent. A unipotent ﬁbration could thus correctly be called a rationally quasi-nilpotent ﬁbration. The action of π1 B on H∗ (X; Q) is nilpotent which is the same as saying that it is a unipotent representation over the ﬁeld Q. (See, for example, [9].) Many theorems about nilpotent and quasi-nilpotent ﬁbrations apply to unipotent ﬁbrations. For example, in [25] it is noted that we only need a ﬁnite index subgroup of π1 B to act nilpotently on the homology of the ﬁber. Similarly, Sebastian Goette pointed out to me that any characteristic class of unipotent bundles such as the higher torsion discussed in this paper extends uniquely to bundles for which a ﬁnite index subgroup of π1 B acts unipotently on the ﬁber. Proposition 2.1. If E → B is a unipotent bundle with oriented ﬁber F having boundary ∂F , then H∗ (∂F ; Q) and H∗ (F, ∂F ; Q) are also unipotent π1 B-modules. In particular, the vertical boundary ∂ v E → B is a unipotent bundle.

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Proof. By Poincar´e duality, H∗ (F ; Q) ∼ = H ∗ (F ; ∂F ; Q) is a unipotent π1 B-module. Its dual H∗ (F, ∂F ; Q) must also be unipotent. Since unipotent modules form a Serre category, the long exact homology sequence of (F, ∂F ) implies that H∗ (∂F ; Q) is also unipotent. Let T v E denote the vertical tangent bundle of E. This is the subbundle of the tangent bundle T E of E consisting of all tangent vectors which go to zero in B, that is, T v E is the kernel of T p : T E → T B. The Euler class e(E) ∈ H n (E; Z) of the bundle E is deﬁned to be the usual Euler class of T v E. The transfer E : H ∗ (E; Z) → H ∗ (B; Z) trB

(2.1)

is given by E trB (x) = p∗ (x ∪ e(E)),

where p∗ : H ∗+n (E; Z) → H ∗ (B; Z) is the push-down operator given over R by integrating along ﬁbers. Over Z, it is given as the composition of two maps H k +n (E; Z) → H k (B; H n (F ; Z)) → H k (B; Z), where the ﬁrst map comes from the Serre spectral sequence of the bundle and the second map is induced by the coeﬃcient map H n (F ; Z) → Z given by evaluation on the orientation class of the ﬁber. For details, see [24] or [19]. If the orientation of the ﬁber F is reversed, both e(E) and p∗ change sign. Thus, the transfer is independent of the choice of orientation of F . For the basic properties of the transfer, see [2]. The main property that we need is that, for closed ﬁbers F , E E trB = (−1)n trB . E So, rationally, trB = 0 if n = dim F is odd.

3. Axioms We deﬁne a higher torsion invariant (in degree 4k) to be a real characteristic class τ (E) ∈ H 4k (B; R) for unipotent smooth bundles E → B with closed oriented ﬁbers satisfying the additivity and transfer axioms described below. When we say that τ is a ‘characteristic class’ we mean it is a natural cohomology class, that is, τ (f ∗ E) = f ∗ (τ (E)) ∈ H 4k (B ; R) if f ∗ E is the pull-back of E along f : B → B. Naturality implies that τ is zero for trivial bundles: τ (B × F ) = 0. 3.1. Additivity If E = E1 ∪ E2 where E1 , E2 are unipotent bundles over B with the same vertical boundary E1 ∩ E2 = ∂ v E1 = ∂ v E2 , then the Additivity Axiom says that τ (E) = 12 τ (DE1 ) + 12 τ (DE2 ), where DEi is the ﬁberwise double of Ei . This wording of the Additivity Axiom comes from Ulrich Bunke.

(3.1)

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3.2. Transfer Suppose that p : E → B is a unipotent bundle with closed ﬁber F and q : S n (ξ) → E is the S bundle associated to an SO(n + 1) bundle ξ over E. Then S n (ξ) is a unipotent bundle over both E and B. The Transfer Axiom says that the higher torsion invariants of these bundles τB (S n (ξ)) ∈ H 4k (B; R) and τE (S n (ξ)) ∈ H 4k (E; R) are related by the formula: n

E τB (S n (ξ)) = χ(S n )τ (E) + trB (τE (S n (ξ))).

(3.2)

Note that χ(S n ) = 2 or 0 depending on whether n is even or odd respectively. 3.3. Examples As stated in the introduction, two examples of higher torsion invariants are the higher Miller–Morita–Mumford classes M2k (E) and the higher Franz–Reidemeister torsion invariants τ2k (E). The MMM classes, for closed ﬁber F , are given by E M2k (E) = trB ((2k)!ch4k (T v E)),

where ch4k (T v E) = 12 ch4k (T v E ⊗C). Although this is an integral cohomology class (for k > 0), we consider it as a real characteristic class. This invariant is deﬁned for any smooth bundle E is zero. with closed oriented ﬁber F . If n = dim F is odd, then twice the transfer map trB Proposition 3.1. M2k (E) = 0 for closed odd dimensional ﬁbers F . Theorem 3.2. M2k is a higher torsion invariant for every k 1. The higher FR torsion invariants ([17, 19]) τ2k (E, ∂0 E) ∈ H 4k (B; R) are deﬁned for any relatively unipotent bundle pair (E, ∂0 E) → B. By this we mean that the vertical boundary ∂ v E is a union of two subbundles ∂0 E, ∂1 E with the same vertical boundary ∂0 E ∩ ∂1 E = ∂ v ∂0 E = ∂ v ∂1 E and that the rational homology of the ﬁber pair (F, ∂0 F ) of the bundle pair (E, ∂0 E) is a unipotent π1 B-module. Theorem 3.3. The higher FR torsion invariants τ2k are higher torsion invariants for unipotent bundles with closed manifold ﬁbers. These two theorems will be proved later. J.-M. Bismut and J. Lott [4] constructed even diﬀerential forms on B called analytic torsion forms. In some cases, these are closed and topological (that is, independent of the metric and horizontal distribution used to deﬁned the form). For example, if F is a closed oriented manifold and π1 B acts trivially on H∗ (F ; Q), then they obtain a (nonequivariant) analytic torsion class BL T2k (E) ∈ H 4k (B; R).

They showed, that is, the following Proposition. BL (E) = 0 for closed even dimensional ﬁbers F . Proposition 3.4. T2k

We also have the following theorems of X. Ma and U. Bunke. Theorem 3.5 [23].

T2k satisﬁes the transfer axiom.

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Theorem 3.6 [6]. Let E → B be the S 2n −1 bundle associated to an U (n) bundle ξ over B. Then (4k + 1)! B unke ζ(2k + 1)ch4k (ξ). (E) = 4k T2k 2 (2k)! Remark 3.7. Bunke uses a diﬀerent normalization of the analytic torsion. We should multiply by (2πi)−2k to get the Bismut–Lott normalization: BL (E) = (−1)k (2π)−2k T2k

(4k + 1)! ζ(2k + 1)ch4k (ξ). 24k (2k)!

As we noted in the introduction, S. Goette has extended the deﬁnition of the higher analytic torsion class to the case when π1 B acts orthogonally on H∗ (F ; Q), that is, preserving some metric. However, we need to extend it to the unipotent case. 4. Statement We give the statement of the main theorem. We begin with the following elementary observations. Lemma 4.1. For each k, the set of all higher torsion invariants τ of degree 4k is a vector space over R. Proof. The axioms are homogeneous linear equations in τ . Lemma 4.2. If τ is a higher torsion invariant, then so is (−1)n τ , where n = dim F considered as a function of the bundle E → B. Proof. We need to show that (−1)n τ satisﬁes the axioms: (−1)n τ (E) = (−1)n 12 τ (DE1 ) + (−1)n 12 τ (DE2 ) E (−1)m +n τB (S m (ξ)) = (−1)n χ(S m )τ (E) + (−1)m trB (τE (S m (ξ))).

The additivity axiom (the ﬁrst equation) is the same as before. The transfer axiom (the second equation) is the same as before if both n = dim F and m are even. If one or both are odd, then E =0 the terms on the right with the wrong sign are zero since χ(S m ) = 0 for odd m and trB for odd n. 4.1. Even and odd higher torsion These lemmas imply that higher torsion invariants can always be expressed as a sum of odd and even parts: τ = τ ev + τ od , where τ ev =

τ + (−1)n τ , 2

τ od =

τ − (−1)n τ . 2

Here τ ev is even and τ od is odd in the sense of the following deﬁnition. Definition 4.3. A higher torsion invariant τ is called even (respectively odd) if τ (E) = 0 for all unipotent bundles with closed odd (respectively even) dimensional ﬁbers. The main theorem of this paper is the following.

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Theorem 4.4 (Main Theorem). Nontrivial even and odd torsion invariants exist in degree 4k for all k > 0 and they are unique up to a scalar factor. Corollary 4.5. Every even torsion invariant is a scalar multiple of M2k and every odd torsion invariant is a scalar multiple of the odd part of the higher FR-torsion τ2k . 4.2. The scalars s1 , s2 The main theorem is that a higher torsion invariant is determined by two scalars in every degree. These scalars are given as follows. Let λ be the universal U (1) = SO(2) bundle over CP ∞ . Let S 1 (λ) → CP ∞ be the circle bundle associated to λ and S 2 (λ) → CP ∞ the associated S 2 bundle (the ﬁberwise suspension of S 1 (λ)). Since the cohomology of CP ∞ is a polynomial algebra in c1 (λ), H 4k (CP ∞ ; R) ∼ = R is generated by ch4k (λ) = c1 (λ)2k /(2k)!. Given any higher torsion invariant τ , there exist scalars s1 , s2 ∈ R, so that (1) τ (S 1 (λ) = τ od (S 1 (λ) = 2s1 ch4k (λ) (2) τ (S 2 (λ) = τ ev (S 2 (λ) = 2s2 ch4k (λ) The main theorem says that τ is uniquely determined by s1 and s2 . For example, we have the following calculations which will be explained later. Proposition 4.6. M2k (S 2 (λ)) = 2(2k)!ch4k (λ). Thus, s2 = (2k)! for the higher MMM classes M2k (and s1 = 0). Proposition 4.7. τ2k (S n (λ)) = (−1)k +n ζ(2k + 1)ch4k (λ). So, sn = 12 (−1)k +n ζ(2k + 1) for the higher FR torsion invariants τ2k . 4.3. Consequences of uniqueness The uniqueness of even torsion gives us the following. Corollary 4.8 [19]. If E → B is a unipotent bundle with closed even dimensional ﬁbers, then (−1)k ζ(2k + 1) τ2k (E) = M2k (E). 2(2k)! Theorem 1.1 in the introduction is a special case of this corollary. The uniqueness of odd torsion can now be expressed as follows. Corollary 4.9. Any odd torsion invariant is a scalar multiple of the odd part of higher Franz–Reidemeister torsion which is given by od τ2k = τ2k −

(−1)k ζ(2k + 1) M2k . 2(2k)!

Theorem 1.2 in the introduction says that analytic torsion classes are odd torsion invariants on certain bundles. We expect that the same formula should hold in general. The uniqueness theorem also tells us something about higher Franz–Reidemeister torsion. Corollary 4.10.

The higher FR torsion invariant τ2k has coeﬃcients in ζ(2k + 1)Q.

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Proof. Let π : R → ζ(2k + 1)Q be any Q linear retraction. Then, by linearity of the axioms, π∗ τ2k is a higher torsion invariant with coeﬃcients in ζ(2k+1)Q. Proposition 4.7 gives the same value of s1 , s2 for π∗ τ2k and τ2k . Therefore, the uniqueness theorem tells us that π∗ τ2k = τ2k .

The rest of this paper is devoted to the proof of the main theorem. We show that, given the values of the scalars s1 , s2 above, any higher torsion invariant can be computed in suﬃciently many cases to determine it completely. 5. Extension to relative case In order to compute the higher torsion invariant τ (E), we need to cut E into simpler pieces and compute the relative torsion of each piece. To do this, we need to extend τ ﬁrst to the case when F has a boundary and then to the case of a unipotent bundle pair (F, ∂0 F ) → (E, ∂0 E) → B. 5.1. Higher torsion in the boundary case Suppose that E → B is a unipotent smooth bundle with ﬁber F an oriented compact manifold with boundary. By Proposition 2.1, the vertical boundary ∂ v E is also unipotent. And it follows from the Mayer–Vietoris sequence that the ﬁberwise double DE is also unipotent. A higher torsion invariant τ can now be extended to the boundary case by the formula τ (E) := 12 τ (DE) + 12 τ (∂ v E). We will show that this extension of τ satisﬁes boundary analogues of the additivity and transfer axioms. These axioms refer to bundles whose ﬁbers have corners. If E is such a bundle, we deﬁne τ (E) to be equal to τ (E ), where E is E with the vertical corners rounded oﬀ in a standard way to be explained at the end of this section. We need the following lemmas. Lemma 5.1. Suppose that Ei are smooth unipotent bundles over B with the same vertical boundary. Then τ (E1 ∪ E2 ) + τ (E3 ∪ E4 ) = τ (E1 ∪ E3 ) + τ (E2 ∪ E4 ). Proof. Both sides are equal to

1 2

τ (DEi ) by the additivity axiom.

Lemma 5.2. τ (∂ v E) = τ (∂ v (E × D2 )) assuming E is unipotent. Proof. Since ∂ v (E × D2 ) = ∂ v E × D2 ∪ E × S 1 we have, by the additivity axiom, that τ (∂ v (E × D2 )) = 12 τ (∂ v E × S 2 ) + 12 τ (DE × S 1 ). But τ (∂ v E × S 2 ) = 2τ (∂ v E) and τ (DE × S 1 ) = 0 by the transfer axiom. Lemma 5.3 (additivity of transfer). If E = E1 ∪ E2 is a union of two smooth bundles along their common vertical boundary ∂ v E1 = ∂ v E2 = E1 ∩ E2 , then v

E1 E2 ∂ E trB (x) = trB (x|E1 ) + trB (x|E2 ) − trB

E1

(x|∂ v E1 ).

Proposition 5.4 (additivity for boundary case). If (E1 , ∂0 ), (E2 , ∂0 ) are unipotent bundle pairs over B with E1 ∩ E2 = ∂0 E1 = ∂0 E2 , then τ (E1 ∪ E2 ) = τ (E1 ) + τ (E2 ) − τ (E1 ∩ E2 ).

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Proof. We expand each term using the deﬁning equation: τ (Ei ) := 12 τ (DEi ) + 12 τ (∂ v Ei ) τ (E1 ∪ E2 ) := 12 τ (∂ v (E1 ∪ E2 )) + 12 τ (D(E1 ∪ E2 )) τ (E1 ∩ E2 ) := 12 τ (D(E1 ∩ E2 )) + 12 τ (∂ v (E1 ∩ E2 )). The terms in the last two equations are arranged so that the vertical sums match the following two examples of Lemma 5.1: τ (∂ v E1 ) + τ (∂ v E2 ) = τ (∂ v (E1 ∪ E2 )) + τ (D(E1 ∩ E2 )) τ (DE1 ) + τ (DE2 ) = τ (D(E1 ∪ E2 )) + τ (D(I × (E1 ∩ E2 ))), where the last term is τ (D(I × (E1 ∩ E2 ))) = τ (∂ v (D2 × (E1 ∩ E2 ))) = τ (∂ v (E1 ∩ E2 )) by Lemma 5.2. The proposition follows. Proposition 5.5 (transfer for boundary case). If X → D → E is an oriented linear disk or sphere bundle, then E (τE (D)). τB (D) = χ(X)τ (E) + trB

Proof. We consider ﬁrst the case when D = D(ξ) is an oriented linear Dn -bundle and F is E (τE (D(ξ))). This is just half of the sum of closed, that is, we will show: τB (D(ξ)) = τ (E) + trB the following two examples of the original transfer axiom. E τB (S n (ξ)) = χ(S n )τ (E) + trB (τE (S n (ξ))) E τB (S n −1 (ξ)) = χ(S n −1 )τ (E) + trB (τE (S n −1 (ξ)))

The transfer axiom takes care of the case when D is a sphere bundle and F is closed. The remaining case when ∂F is nonempty is given by the following lemma. Lemma 5.6. With the ﬁber X of q : D → E ﬁxed, the transfer formula for F closed implies the transfer formula for F with boundary. Proof. Write DE = E ∪ E as the union of two copies of E along its vertical boundary. Let D, D be two copies of D with D ∩ D = q −1 (∂ v E). Then the transfer formula τB (D) = E χ(X)τ (E)+trB (τE (D)) is half the sum of the following two transfer formulas with closed ﬁbers DF, ∂F respectively.

E ∪E τB (D ∪ D ) = χ(X)τ (E ∪ E ) + trB (τE ∪E (D ∪ D ))

E ∩E (τE ∩E (D ∩ D )) τB (D ∩ D ) = χ(X)τ (E ∩ E ) + trB

The additivity of transfer (Lemma 5.3) is used here. 5.2. Relative torsion Suppose that (F, ∂0 F ) → (E, ∂0 E) → B is a unipotent smooth bundle pair. By this, we mean that the vertical boundary ∂ v E is the union of two subbundles ∂ v E = ∂0 E ∪ ∂1 E which meet along their common vertical boundary: ∂0 E ∩ ∂1 E = ∂ v ∂0 E = ∂ v ∂1 E and that both E and ∂0 E are unipotent. This implies the weaker condition that the pair (E, ∂0 E) is relatively unipotent, that is, that H∗ (F, ∂0 F ) is a unipotent π1 B-module. We use the abbreviation (E, ∂0 ) for (E, ∂0 E).

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Suppose that τ is a higher torsion invariant which has been extended to the boundary case as above. Then for any unipotent smooth bundle pair (E, ∂0 ) → B, we deﬁne the relative torsion by τ (E, ∂0 ) := τ (E) − τ (∂0 E). Proposition 5.7 (additivity in the relative case). Suppose that E → B is a smooth bundle which can be written as a union of two subbundles E = E1 ∪ E2 which meet along a subbundle of their respective vertical boundaries: E1 ∩ E2 = ∂0 E2 ⊆ ∂ v E1 . Let ∂ v E1 = ∂0 E ∪ ∂1 E be a decomposition ∂ v E1 , so that ∂0 E2 ⊆ ∂1 E1 and (Ei , ∂0 ) → B, i = 1, 2 are unipotent smooth bundle pairs. Then (E, ∂0 E1 ) → B is unipotent and τ (E1 ∪ E2 , ∂0 E1 ) = τ (E1 , ∂0 ) + τ (E2 , ∂0 ).

∂ 0 E2

∂ 1 E1 E2 E1

← ∂ 0 E1

Proof. By Proposition 5.7, both sides of the equation are equal to τ (E1 ) + τ (E2 ) − τ (E1 ∩ E2 ) − τ (∂0 E1 ). Here is another variation of the additivity axiom which is also trivial to prove. Proposition 5.8 (horizontal additivity). Suppose that (E, ∂0 ) → B is a union of two unipotent bundle pairs (Ei , ∂0 Ei ) in the sense that E = E1 ∪ E2 and ∂0 E = ∂0 E1 ∪ ∂0 E2 with E1 ∩ E2 ⊆ ∂1 E1 ∩ ∂1 E2 . Let E0 = E1 ∩ E2 and ∂0 E0 = E0 ∩ ∂0 E and suppose (E0 , ∂0 ) is a unipotent bundle pair. Then (E, ∂0 ) is unipotent and τ (E, ∂0 ) = τ (E1 , ∂0 ) + τ (E2 , ∂0 ) − τ (E0 , ∂0 ).

E1

E0

∂ 0 E1

E2 ∂ 0 E2

∂ 0 E0 To state the transfer axiom in the relative case, we need the relative transfer: (E ,∂ 0 )

trB

: H ∗ (E; Z) → H ∗ (B; Z),

given by (E ,∂ 0 )

trB

(x) = p∗ (x ∪ e(E, ∂0 )),

where p∗ : H ∗+n (E, ∂ v E; Z) → H ∗ (B; Z)

(5.1)

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is the push-down operator (given over R by integrating along ﬁbers) and e(E, ∂0 ) ∈ H n (E, ∂ v E; Z) is the relative Euler class given by pulling back the Thom class of the vertical tangent bundle T v E along any vertical tangent vector ﬁeld which is nonzero along the vertical boundary ∂ v E and which points inward along ∂0 E and outward along ∂1 E. The relative transfer satisﬁes the following two equations for any x ∈ H ∗ (E; Z). (E ,∂ 0 )

trB

∂0 E E (x) = trB (x) − trB (x|∂0 E)

(E ,∂ 1 )

trB

(5.2)

(E ,∂ 0 )

= (−1)n trB

(5.3)

And, ﬁnally, trB 0 ◦p∗ : H ∗ (B) → H ∗ (B) is multiplication by the relative Euler characteristic of the ﬁber pair (F, ∂0 F ): (E ,∂ )

χ(F, ∂0 ) := χ(F ) − χ(∂0 F ). p

→ B and (X, ∂0 ) → Proposition 5.9 (transfer in the relative case). Let (F, ∂0 ) → (E, ∂0 ) − q (D, ∂0 ) − → E be unipotent smooth bundle pairs, so that the second is an oriented linear S n or Dn bundle with ∂0 X = S n −1 , Dn −1 or ∅. Then τB (D, ∂0 D ∪ q −1 ∂0 E) = χ(X, ∂0 )τ (E, ∂0 ) + trB

(E ,∂ 0 )

(τE (D, ∂0 )).

Proof. We already did the case when both ∂0 F and ∂0 X are empty. The case when ∂0 X is empty follows easily from the formula τB (D, q −1 ∂0 E) = τB (D) − τB (q −1 ∂0 E). The general case follows from the following two examples of the ∂0 X = ∅ case. τB (∂0 D, ∂0 D ∩ q −1 ∂0 E) = χ(∂0 X)τ (E, ∂0 ) + trB

(E ,∂ 0 )

τB (D, q

−1

∂0 E) = χ(X)τ (E, ∂0 ) +

(τE (∂0 D)) (E ,∂ 0 ) trB (τE (D))

Take the second formula minus the ﬁrst to prove the proposition. A useful special case is the case when D → E is the pull-back of a linear bundle over B. In this case, D is the ﬁber product of two bundles. Corollary 5.10 (product formula). Suppose that (F, ∂0 ) → (E, ∂0 ) → B and (X, ∂0 ) → (E , ∂0 ) → B are unipotent smooth bundle pairs, so that the second is an oriented linear S n or Dn bundle with ∂0 X = S n −1 , Dn −1 or ∅. Let D = E ×B E be the ﬁber product of these bundles and let ∂0 D = ∂0 E ×B E ∪ E ×B ∂0 E . Then τ (D, ∂0 ) = χ(X, ∂0 )τ (E, ∂0 ) + χ(F, ∂0 )τ (E , ∂0 ).

Proposition 5.11. For any unipotent bundle pair (E, ∂0 ) → B, we have τ (E, ∂0 ) + (−1)n τ (E, ∂1 ) = 2τ ev (E, ∂0 ), where n = dim F is the ﬁber dimension of E.

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Proof. Using the deﬁnition of relative torsion and additivity resulting from the decomposition ∂ v E = ∂0 E ∪ ∂1 E, we can write each term as a linear combination of higher torsions of the bundles DE, D∂0 E, D∂1 E, ∂ v ∂0 E = ∂ v ∂1 E. For example, 1 1 1 1 τ (E, ∂0 ) = τ (DE) + τ (D∂1 E) − τ (D∂0 E) − τ (∂ v ∂0 E). 2 4 4 2 The proposition follows immediately. 5.3. Further extension Given any higher torsion invariant τ initially deﬁned only for unipotent bundles with closed ﬁbers, we have extended the deﬁnition to all unipotent bundle pairs (E, ∂0 E). If we assume the main theorem, we can extend it further. Theorem 5.12. Any higher torsion invariant can be extended uniquely to all relatively unipotent smooth bundle pairs (E, ∂0 E), so that the relative versions of the axioms (additivity, horizontal additivity and transfer) hold. Proof. Suppose that (E, ∂0 ) → B is a relatively unipotent smooth bundle pair. Then the torsion τ (E, ∂0 ) can be deﬁned as follows. (i) Let D(ν) → E be the normal disk bundle and let ∂0 D = D(ν)|∂0 E. (ii) Embed ∂0 E ﬁberwise into the northern hemisphere of B × S N for N large. (iii) Thicken this embedding to a codimension zero embedding ∂0 D → B × S N . Then the union of B × DN +1 with D along ∂0 D is unipotent. (iv) Deﬁne the higher torsion of (D, ∂0 ) by τ (D, ∂0 ) := τ (B × DN +1 ∪ D). (v) Deﬁne the higher torsion of (E, ∂0 ) by E τ (E, ∂0 ) := τ (D, ∂0 ) − (s1 + s2 )trB (ch4k (ν)),

where s1 , s2 are as given in Subsection 4.2. These formulas show that the extension of τ to the relatively unipotent case is unique if it exists. However, this extension does exist in the cases τ = M2k and τ = τ2k which span all possibilities by the main theorem. Therefore, the extension is unique for all τ . The additivity, horizontal additivity and relative transfer formulas hold for M2k and τ2k when extended to the relatively unipotent case. Therefore, these axioms hold for all τ . 5.4. Rounding oﬀ corners Here is a brief description of how corners can be rounded oﬀ in a canonical way. First, we need a deﬁnition. Let Ck be one of the subsets of Rn given by the conditions x1 0, x2 0, · · · , xk 0 connected in that order by a sequence of ‘and’s and ‘or’s. For k = 0, there is only one such subset: C0 = Rn . But C4 = {x ∈ Rn | (x1 0 or (x2 0 and x3 0)) and x4 0} is one of the many possibilities when k = 4. Suppose that M is a compact topological n-manifold with boundary which has a smooth structure given by some embedding into a smooth open manifold of the same dimension. Then M is a smooth manifold with corners if every point x ∈ M has a neighborhood diﬀeomorphic to a neighborhood of 0 in some Ck . Consequences: (1) The product of manifolds with corners is a manifold with corners. (2) If M n is embedded in the interior of N n , then N × [0, 1] ∪ M × [0, 2] is also a manifold with corners. This is locally equivalent to unions such as E1 ∪ E2 in Proposition 5.7.

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In our model Ck , we deﬁned a positive vector at 0 to be any vector whose ﬁrst k coordinates are all positive. Note that a positive vector always points outward, that is, Ck ⊆ Ck + v for any positive vector v. A positive vector ﬁeld on the boundary of M is any smooth vector ﬁeld deﬁned in some neighborhood of ∂M which maps to a positive vector in the model at every point of ∂M . If we take the smooth ﬂow generated by this vector ﬁeld, we get a homeomorphism F : ∂M × (− , 0] ∼ =U sending (x, 0) to x for all x ∈ ∂M , where U is some neighborhood of ∂M in M . For each point x ∈ ∂M , we can ﬁnd a small open neighborhood Vx of x in ∂M and a homeomorphism gx : Wx → Vx , where Wx is an open set in Rn −1 and a continuous function hx : Wx → (− , 0), so that φx = F ◦ (gx , hx ) : Wx → Vx × (− , 0) → U is a smooth embedding. Since F is smooth in the second coordinate, φx will remain smooth if we alter hx by any operation which is smooth in x. Since ∂M is compact, it is covered by a ﬁnite number of the open sets Vi and we get a ﬁnite number of smooth embeddings φi : Wi → U which meet all of the ﬂow lines given by the vector ﬁeld. Let W be the abstract closed smooth n − 1 manifold given by pasting together the Wi by identifying points which map to the same point in ∂M . Take a smooth partition of unity ψi : Wi → [0, 1] on W . Then the continuous function h : W → (− , 0) given by h(x) = ψi (x)hi (x) gives a smooth embedding φ : W → U by φ(x) = F (g(x), h(x)). Furthermore, the set C = {F (g(x) × [h(x), 0]) | x ∈ W } is homeomorphic to ∂M × I and the closure of the complement of C in M is a smooth manifold with smooth boundary φ(W ). We call this the manifold obtained from M by smoothing the corners. This construction can be carried out for bundles E. We just need to choose the positive vector ﬁeld to be vertical. We can also extend any choice of data on E × 0 ∪ E × 1 to all of E × [0, 1] without any trouble: Extend the vector ﬁeld, then choose more embeddings φi , then patch them together using a partition of unity using the already given partition of unity near E × 0 and E × 1. So, the rounding of corners is canonical. 6. Stability of higher torsion Smooth bundles are stabilized by taking products with disks. The following special case of the product formula says that higher torsion is a stable invariant. Corollary 6.1 (stability of torsion). If (E, ∂0 ) → B is a unipotent smooth bundle pair, then so is (E × Dn , ∂0 E × Dn ) and the relative torsion is the same: τ (E × Dn , ∂0 E × Dn ) = τ (E, ∂0 ). If M is a compact smooth manifold, then we recall that a concordance of M is a diﬀeomorphism of M ×I which is the identity on M ×0∪∂M ×I. Let C(M ) be the space of concordances of M with the C ∞ topology: C(M ) = Diff (M × I rel M × 0 ∪ ∂M × I).

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The classifying space BC(M ) is the space of h-cobordisms of M rel ∂M . Recall that an hcobordism of M rel ∂M is a compact smooth manifold W with boundary ∂W = M × 0 ∪ ∂M × I ∪ M , where M is another compact smooth manifold with the same boundary as M , so that the inclusion M × 0 → W is a homotopy equivalence. A mapping B → BC(M ) is a smooth bundle over B whose ﬁbers are all h-cobordisms of M rel ∂M , so that the subbundle with ﬁber M × 0 ∪ ∂M × I is trivial. We will call such a bundle an h-cobordism bundle over B. There is a suspension map σ : C(M ) → C(M × I) which is highly connected when dim M is large by the concordance stability theorem ([16] or the last chapter of [19]). The limit is the stable concordance space P(M ) = lim C(M × I n ) →

which is well known to be an inﬁnite loop space. Therefore, the set [B, BP(M )] of homotopy classes of maps from B to the classifying space BP(M ) is an additive group. By the concordance stability theorem, this group is isomorphic to [B, BC(M × I n )] for suﬃciently large n. Proposition 6.2. For any h-cobordism bundle E → B, let ∂0 E be the trivial subbundle with ﬁber M × 0 ∪ ∂M × I. Then the higher torsion invariant E → τ (E, ∂0 ) gives an additive map τ : [B, BC(M × I n )] → R. Proof. This is an immediate consequence of the horizontal additivity of τ (Proposition 5.8) since the H-space structure on the h-cobordism space BC(M × I n ) is given by lateral union that is, the sum of two mappings B → C(M × I n ) is given by lateral union of the corresponding h-cobordism bundles followed by rescaling. (The lateral union is an h-cobordism of M × I n −1 × [0, 2]. The last coordinate needs to be rescaled down to [0, 1].) Corollary 6.3. Suppose that Eα is a collection of h-cobordism bundles which spans the Q vector space [B, BC(M × I n )] ⊗ Q. Suppose also that τ (Eα , ∂0 ) = 0 for all α. Then τ (E, ∂0 ) = 0 for any h-cobordism bundle E classiﬁed by a map B → BC(M × I n ). 7. Computation of higher torsion We will now show how the higher torsion invariants of unipotent bundles can be computed in many cases given the values of the parameters s1 , s2 . We recall that these parameters are given by τ (S n (λ)) = 2sn ch4k (λ), where S n (λ) is the S n bundle associated to a complex line bundle λ over B. 7.1. Torsion of disk and sphere bundles Theorem 7.1. The higher torsion of the Dn -bundle Dn (ξ) associated to an SO(n)-bundle ξ over B is given by τ (Dn (ξ)) = (s1 + s2 )ch4k (ξ). Proof. For n = 2, this is by deﬁnition of τ in the boundary case: τ (D2 (ξ)) = 12 τ (S 2 (ξ)) + 12 τ (S 1 (ξ)) = (s2 + s1 )ch4k (ξ).

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The general case follows from the product formula (Corollary 5.10) and the splitting principle. If n = 2m + 1, then we may assume by the splitting principle that ξ is a direct sum of m complex line bundles λi and a trivial real line bundle. Then τ (Dn (ξ)) = τ D2 (λ1 ) ×B · · · ×B D2 (λm ) ×B (B × I) = (s1 + s2 )

ch4k (λi ) = (s1 + s2 )ch4k (ξ)

by the product formula. The even case is similar. From the formula τ (Dn (ξ)) = 12 τ (S n ) + 12 τ (S n −1 (ξ)) = (s1 + s2 )ch4k (ξ), we get the following by induction on n. Corollary 7.2. For n > 0, the higher torsion of the S n -bundle S n (ξ) associated an SO(n + 1)-bundle ξ over B is given by τ (S n (ξ)) = 2sn ch4k (ξ), where sn depend only on the parity of n. Comparing this with Proposition 9.2 and Theorem 9.6 below, we get the following. Lemma 7.3. Let k 1 and let a, b ∈ R be given by a :=

s1 + s2 , (2k)!

b :=

(−1)k +1 2s1 . ζ(2k + 1)

Then τ (E) = aM2k (E) + bτ2k (E) for E any oriented linear sphere and disk bundle over B. Remark 7.4. If the value of τ on a bundle E is determined by the known values of τ on disk and sphere bundles, then this lemma implies that τ (E) = aM2k (E) + bτ2k (E) for that bundle. 7.2. Morse bundles Given a Morse function f : (M, ∂0 M ) → (I, 0), a compact n-manifold M will be decomposed as a union of handles Di × Dn −i attached along S i−1 × Dn −i to the union of lower handles and the base ∂0 M × [0, ]. Each such handle has a critical point at its center with index i. The core of the handle is Di × 0. This is also the union of trajectories of the gradient of f (with respect to some metric on M ) which converge to the critical point. The tangent plane is the negative eigenspace of the second derivative D2 f at the critical point. Suppose that (E, ∂0 ) → B is a smooth bundle pair and f : E → I = [0, 1] is a ﬁberwise Morse function with f −1 (0) = ∂0 E and with distinct critical values. In other words, for each b ∈ B, the restriction fb : (Fb , ∂0 ) → (I, 0) is a Morse function with critical points x1 (b), . . . , xm (b) having critical values fb (x1 ) < fb (x2 ) < · · · < fb (xm ). For 1 j m, let ξj be the negative eigenspace bundle of the ﬁberwise second derivative of f along xj . Let ηj be the complementary positive eigenspace bundle. Then E has a ﬁltration E = Em ⊃ Em −1 ⊃ · · · ⊃ E0 , where E0 ∼ = ∂0 E × I and Ej = Ej −1 ∪ D(ξj ) ×B D(ηj ), where Ej −1 ∩ D(ξj ) ×B D(ηj ) = S(ξj ) ×B D(ηj ). By additivity we get the following.

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Lemma 7.5. If ∂0 E → B is unipotent and the bundles ξj , ηj are oriented, then (E, ∂0 ) → B is a unipotent bundle pair with torsion invariant τ ((D(ξj ), S(ξj )) ×B D(ηj )). τ (E, ∂0 ) = Each summand in the above lemma can be determined using the product formula (Corollary 5.10) as follows. Lemma 7.6. The value of τ on the ﬁber product of an oriented linear disk bundle D(η) and the oriented relative i-disk bundle (D(ξ), S(ξ)) is given by τ ((D(ξ), S(ξ)) ×B D(η)) = (−1)i τ (D(η)) + τ (D(ξ), S i−1 (ξ)) = (−1)i (s1 + s2 )ch4k (η) + (si − si−1 )ch4k (ξ). Remark 7.7. Since si −si−1 = (−1)i (s2 −s1 ), the even and odd parts of the above formula are: τ ev ((D(ξ), S(ξ)) ×B D(η)) = (−1)i s2 (ch4k (η) + ch4k (ξ)) τ od ((D(ξ), S(ξ)) ×B D(η)) = (−1)i s1 (ch4k (η) − ch4k (ξ)). Putting these together, we get the following theorem which is a mild improvement over the obvious. Namely, the negative eigenspace bundles need not be oriented. Of course, the sum ξj ⊕ ηj of negative and positive eigenspace bundles must be oriented since it is the vertical tangent bundle along the jth component of the Morse critical set. Theorem 7.8. Suppose that (E, ∂0 ) → B is a unipotent smooth bundle pair and f : (E, ∂0 E) → (I, 0) is a ﬁberwise Morse function with distinct critical values. Let ξj , ηj be the negative and positive eigenspace bundles associated to the jth critical point whose index we denote by i(j). Then (−1)i(j ) (s1 + s2 )ch4k (ηj ) + (−1)i(j ) (s2 − s1 )ch4k (ξj ). τ (E, ∂0 ) = Proof. Suppose ﬁrst that the bundles ξj , ηj are oriented. Then the two lemmas above apply to prove the theorem. If these bundles are not oriented, then there is a ﬁnite covering of B so that, on the pull-back E, the function E → E → I is a Morse function with B oriented eigenspace bundles ξj , ηj which are pull-backs of ξj , ηj . Thus, the theorem applies to However, the induced map in real cohomology H ∗ (B; R) → H ∗ (B; R) is a monomorphism E. and the two expressions in our theorem are elements of H 4k (B; R) which go to the same element R). So, they must be equal. of H 4k (B; 7.3. Hatcher’s example One crucial example of a Morse bundle to which the above theorem holds is Hatcher’s construction. This constructs an exotic disk bundle over B = S n out of an element of the kernel of the J-homomorphism J : πn −1 O → πns −1 (S 0 ). B¨okstedt [5] interpreted this as a mapping from G/O to the stable concordance space of a point H(∗) = BP(∗). This means that a mapping B → G/O gives an exotic disk bundle over B. Since G/O is the homotopy ﬁber of the map BO → BG, a map B → G/O is unstably the same as an n-plane bundle ξ : B → SO(n) together with a homotopy trivialization of the

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associated sphere bundle, that is, we have a ﬁber homotopy equivalence g : S n −1 (ξ) B × S n −1 . We write this as a family of homotopy equivalences gt : Stn −1 (ξ) → S n −1 , t ∈ B and extend to the disk Dtn (ξ) (the ﬁber over t ∈ B of the disk bundle Dn (ξ)) g t : Dtn (ξ), Stn −1 (ξ) → (Dn , S n −1 ). Assuming that n and m are large enough, we can lift g t up to a family of embeddings gt : Dtn (ξ), Stn −1 (ξ) → (Dn , S n −1 ) × Dm . If we let η be the complementary bundle to ξ, we can extend this to a family of codimension zero embeddings Gt : Dtn (ξ), Stn −1 (ξ) × Dtm (η) → (Dn , S n −1 ) × Dm . n Embed Dn into a larger n-disk D+ by adding an external collar along ∂Dn = S n −1 , that is, n ∼ n n −1 × I). For each t ∈ B, the image of Gt together with the thickened collar D+ = D ∪ (S (S n −1 × I) × Dm forms an n + m disk (if the corners are rounded). ∆t (ξ) := Gt Dtn (ξ) × Dtm (η) ∪ (S n −1 × I) × Dm .

This is the ﬁber over t ∈ B of a smooth disk bundle ∆(ξ) → B which we call a Hatcher disk S n −1 × I Dm

G t (D tn (ξ) × D tm (η)) n D+

Dn

bundle. By Theorem 7.8, the torsion of Hatcher’s bundle is: τ (∆(ξ)) = (−1)n (s1 + s2 )ch4k (η) + (−1)n (s2 − s1 )ch4k (ξ). Since ch4k (η) + ch4k (ξ) = 0 (ξ ⊕ η being trivial), this simpliﬁes to the following. Lemma 7.9. The higher torsion of Hatcher’s example is given by τ (∆(ξ)) = (−1)n +1 2s1 ch4k (ξ) where s1 is deﬁned in Subsection 4.2. As pointed in [17], this calculation gives another proof of B¨ okstedt’s theorem. Theorem 7.10 [5]. Hatcher’s construction gives a rational homotopy equivalence

→ H(∗) = BP(∗). ∆ : G/O − Proof. By Farrell and Hsiang [12], we know that H(∗) is rationally homotopy equivalent to BO. Therefore, it suﬃces to show that ∆ induces a rational isomorphism on homotopy groups. But the rational homotopy groups of these spaces are zero except in degree 4k where they have

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rank 1. Thus, it suﬃces to show that ∆ is rationally nontrivial on π4k . This follows from the od and Lemma 7.9 when applied to a nontrivial existence of nontrivial odd torsion, such as τ2k element of the kernel of J : π3k −1 O → π3k −1 G. Lemma 7.9 also implies the following. Lemma 7.11. For any smooth oriented disk bundle D → B, we have τ (D) = aM2k (D) + bτ2k (D), where a, b are given in Lemma 7.3 Proof. By Corollary 6.1, we can stabilize D without changing the value of the three higher torsion invariants τ (D), τ2k (D), M2k (D). After stabilizing, D becomes an n + 1 disk bundle over B, where n > dim B and we can ﬁnd a smooth section B → ∂ v D. We can thicken this up to get a linear n disk bundle D(ξ) ⊆ ∂ v D. Suppose ﬁrst that ξ is a trivial bundle. Then (D, D(ξ)) is an h-cobordism bundle classiﬁed by an element of [B, BC(Dn )] which is rationally generated by Hatcher’s examples by B¨okstedt’s theorem. So, by Corollary 6.3, the ‘diﬀerence torsion’ τ − aM2k − bτ2k is equal to zero on D by Lemma 7.9 and the choice of a, b. If the linear bundle ξ is nontrivial, we simply take the ﬁber product of D with the disk bundle of the complementary bundle η. The argument above applies to the higher torsion of D ×B D(η) which is related to the torsion of D by: τ (D ×B D(η)) = τ (D) + τ (D(η)). Since the lemma holds for D ×B D(η) and D(η), it holds for D. 8. Proof of the main theorem We have enough calculations to prove the main theorem, namely that τ = aM2k + bτ2k or, equivalently, that the diﬀerence is zero. 8.1. The diﬀerence torsion We deﬁne the diﬀerence torsion by τ δ := τ − aM2k − bτ2k . This is a linear combination of higher torsion invariants and therefore, a higher torsion invariant. Furthermore, this new invariant has the property that s1 = s2 = 0. So, by Lemma 7.11 and Corollary 7.2, we have the following. Lemma 8.1. The diﬀerence torsion is zero on all disk bundles and linear sphere bundles. Lemma 8.2. If q : D → E is an oriented linear disk bundle, then the diﬀerence torsion of any unipotent bundle pair (E, ∂0 ) is equal to that of (D, ∂0 ) as a bundle over B, where ∂0 D = q −1 (∂0 E)), that is, τBδ (D, ∂0 ) = τ δ (E, ∂0 ). Proof. By the relative transfer formula, we have

δ E τE (D) . τBδ (D, ∂0 ) = τ δ (E, ∂0 ) + trB

But τEδ (D) = 0, since D is a linear disk bundle over E. Lemma 8.3. The diﬀerence torsion is zero on any unipotent bundle pair (E, ∂0 ) → B whose ﬁbers (F, ∂0 ) are h-cobordisms.

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Proof. Choose a smooth ﬁbered embedding of ∂0 E into B × DN for some large N . Let ∂0 D be the normal disk bundle of ∂0 E in B × DN . Since E is ﬁber homotopy equivalent to ∂0 E, the linear disk bundle ∂0 D extends to a linear disk bundle D over E. By the above lemma, τ δ (E, ∂0 ) = τ δ (D, ∂0 ). So, it suﬃces to show that τ δ (D, ∂0 ) = 0. However, this follows by additivity since the union of B × DN with D along the inclusion ∂0 D → B × DN is a disk bundle over B: τ δ (D, ∂0 ) = τ δ ((B × DN ) ∪ D) = 0.

Lemma 8.4 (First Main Lemma). of unipotent smooth bundle pairs.

The diﬀerence torsion τ δ is a ﬁber homotopy invariant

Proof. Suppose that (E1 , ∂0 ) and (E2 , ∂0 ) are unipotent smooth bundle pairs over B that are ﬁber homotopy equivalent. Then we want to show that τ δ (E1 , ∂0 ) = τ δ (E2 , ∂0 ). If we replace (E2 , ∂0 ) by a large dimensional disk bundle (D2 , ∂0 ), we can approximate the ﬁber homotopy equivalence by a ﬁberwise smooth embedding g : (E1 , ∂0 ) → (D2 , ∂0 ). Let D1 be the normal disk bundle of E1 in D2 . Then the complement of D1 in D2 is a unipotent ﬁberwise h-cobordism which has trivial τ δ by the previous lemma. Thus, τ δ (D1 , ∂0 ) = τ δ (D2 , ∂0 ) by additivity. By Lemma 8.2, this implies that τ δ (E1 , ∂0 ) = τ δ (E2 , ∂0 ). 8.2. Vanishing of the ﬁber homotopy invariant The ﬁnal step in the proof of Theorem 4.4 is to show the following. Lemma 8.5 (Second Main Lemma). Any higher torsion invariant τ δ which is also a ﬁber homotopy invariant of unipotent smooth bundle pairs must be zero. This is the crucial step in the proof of the main theorem which I completed with the help of John Klein and Bruce Williams. The proof is in three parts. First, we show that τ δ satisﬁes excision. Then we use obstruction theory to reduce to the case when the ﬁbers are rationally trivial. Finally, we use immersion theory to reduce to the h-cobordism case (Lemma 8.3), where it holds by the assumption that τ δ is a ﬁber homotopy invariant. 8.2.1. Excision. Since τ δ is a ﬁber homotopy invariant, it is well deﬁned on any ﬁbration pair (Z, C) → B with ﬁber (X, A) which is unipotent in the sense that H∗ (X; Q) and H∗ (A; Q) are unipotent as π1 B-modules and smoothable in the sense that it is ﬁber homotopy equivalent to a smooth bundle pair (E, ∂0 ) with compact manifold ﬁber (F, ∂0 ). By deﬁnition, the relative torsion is related to the absolute torsion by τ δ (Z, C) = τ δ (Z) − τ δ (C).

(8.1)

Lemma 8.6. Given unipotent smoothable ﬁbrations (Z, C) and Y over B and any continuous mapping f : C → Y over B, the union Y ∪C Z is also unipotent and smoothable with τ δ (Y ∪C Z) = τ δ (Y ) + τ δ (Z, C).

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Proof. Unipotence follows from the Mayer–Vietoris sequence for the ﬁber homology of Y ∪C Z. The formula is additivity of torsion. Smoothing of Y ∪C Z, very similar to the construction in Subsection 5.3, goes as follows. First, choose smoothings (E, ∂0 ) and E for (Z, C) and Y which are both codimension 0 submanifolds of B × Rn . The continuous map f : C → Y gives a map ∂0 E → E over B which is ﬁberwise homotopic to a smooth embedding φ : ∂0 E → E × Dm . Since the normal bundle of φ(∂0 E) in E × Dm is trivial, we have a codimension 0 embedding φ : ∂0 E × Dm +1 → E × Dm . The smoothing of Y ∪C Z is E × Dm +1 ∪φ E × Dm × I with corners rounded. Lemma 8.7. For any unipotent smoothable ﬁbration pair (Z, C) the ﬁberwise smash Z/B C = B ∪C Z and ﬁberwise suspension ΣB (Z/B C) → B, with ﬁbers X/A and Σ(X/A) respectively, are unipotent and smoothable with τ δ (Z/B C) = τ δ (Z, C) = −τ δ (ΣB (Z/B C)). Remark 8.8. By taking an iterated ﬁberwise suspension Σm B (Z/B C) → B, this lemma reduces the proof of Lemma 8.5 to the case where the ﬁber is highly connected and pointed. However, it would be incorrect to take the ﬁberwise limit and assume that ﬁbers are contractible. Proof. The coﬁbration sequence X/A ∨ X/A → X/A → Σ(X/A) and additivity give: τ δ (ΣB (Z/B C)) = τ δ (Z/B C) − 2τ δ (Z/B C) = −τ δ (Z/B C). This is equal to −τ δ (Z, C) by the previous lemma. 8.2.2. Making ﬁbers rationally trivial. Following suggestions of Bruce Williams, we will reduce the proof of Lemma 8.5 to the case of smoothable ﬁbrations with rationally trivial ﬁbers. Such ﬁbrations will necessarily be unipotent. The proof was reorganized following suggestions of the referee. Lemma 8.9. Suppose that X → E → B is a unipotent ﬁbration with section, where B and X have the homotopy type of ﬁnite complexes. Then there is a ﬁnite ﬁltration of unipotent ﬁbrations over B Σm B E = E 0 → E 1 → · · · → Ek starting with an iterated ﬁberwise suspension Σm X → Σm B E → B of E satisfying the following conditions. (i) The map Ek → B is a rational weak homotopy equivalence. (ii) There is a coﬁbration sequence of ﬁbrations over B: αj

B × S n j −→ Ej → Ej +1 . (iii) The total rank of the rational homology of the ﬁber of Ej +1 is one less than that of Ej . Furthermore, if the original ﬁbration E → B is smoothable, then so is each Ej → B. Remark 8.10. By Lemma 8.7, if E → B is smoothable, then so is E0 = Σm B E. Each Ej would also be smoothable by induction on j since Ej +1 = Ej ∪B ×S n j B × Dn j +1 . Since B × (Dn j +1 , S n j ) is a trivial bundle pair, its torsion is zero. So, τ δ (Ek ) = τ δ (Ek −1 ) = · · · = τ δ (E0 ) = (−1)m τ δ (E).

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Therefore, the statement that τ δ (E) = 0 for all smoothable unipotent E is reduced to the case of the bundle Ek → B which is a rational weak equivalence. Proof. By induction on the total rank of the rational homology of the ﬁber X, it suﬃces to construct a mapping α = α0 : B × S n 0 → E0 = Σm B E over B, so that the induced homomorphism (substituting n0 = n + m) α∗ : H n +m (S n +m ; Q) → H n +m (Σm X; Q) ∼ = H n (X; Q) is nonzero. We will construct the mapping α to be pointed on ﬁbers (that is, α preserves the section). Since B is a ﬁnite complex, the existence of α for some m 0 is equivalent to the existence of a section of the bundle (Ωn Q)B E → B, which is associated to E → B with ﬁber Ωn QX = colimΩn +m Σm X. We will use obstruction theory to construct this section. Suppose that sk is a section of the bundle (Ωn Q)B E over the k-skeleton B k of B. Then we have an obstruction class θ(sk ) ∈ Hk +1 B; πns +k X which is the obstruction to extending sk −1 = sk |B k −1 to B k +1 . The key point is that πns +k X is rationally isomorphic to H n +k (X). Therefore, if we take n to be maximal so that H n (X; Q) = 0 (such an n exists since X is equivalent to a ﬁnite complex), these obstruction groups will be rationally trivial for k 1. But we still need to construct s1 and we need to show that rational triviality of the obstruction group is suﬃcient. Since πns (X) ⊗ Q ∼ = H n (X; Q) is unipotent, it contains a nonzero element which if ﬁxed under the action of π1 B. A nonzero multiple of this element lifts to a stable homotopy class β ∈ πns (X). Since X is homotopy ﬁnite, the torsion subgroup T of πnS (X) is ﬁnite. The group π1 B permutes the elements in the coset β + T . Therefore, the sum, say γ ∈ πns (X), of all elements in this coset is ﬁxed under the action of π1 B and is still rationally nontrivial. Note that any nonzero multiple of γ has the same properties. These properties imply that γ deﬁnes a component (Ωn Qγ )B E of (Ωn Q)B E with connected ﬁber Ωn Qγ (X). We let s1 be any section of this subbundle over the 1-skeleton B 1 . Given sk for k 1, we need to kill the obstruction class θ(sk ) by a geometric construction. Let be the order of the ﬁnite group πns +k (X) and let f be the operation on the ﬁbration (Ωn Q)B E given by right multiplication by a stable self-map of S n of degree . This operation sends (Ωn Qγ )B E to (Ωn Qγ )B E. We claim that θ(f ◦ sk ) = 0 and therefore, we get a section sk +1 of (Ωn Qγ )B E over B k +1 . To see that the obstruction vanishes, we look at the operation f on ﬁbers. Composing with the homotopy equivalence Ωn Q0 X Ωn Qγ X given by addition with γ, we get a map

f

Ωn Q0 X − → Ωn Qγ X −→ Ωn Qγ X which sends x to f (x ∨ γ) f (x) ∨ γ. But the map f is multiplication by on πk Ωn Q0 X = πns +k X and therefore is zero on πk both on Ωn Q0 X and on Ωn Qγ X. This implies that θ(f ◦ sk ) = f θ(sk ) = 0 as claimed. 8.2.3. Proof for rationally trivial ﬁbers. The statement and proof of the following lemma which is the last step in the proof of Lemma 8.5 follows several discussions I had with John Klein. Lemma 8.11. τ δ (Z) = 0 for smoothable ﬁbrations with rationally trivial ﬁbers.

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Proof. We use the fact that a ﬁbration Z → B with rationally trivial ﬁber X is necessarily unipotent. This implies the existence of a universal torsion class which, by immersion theory, can be lifted to a space of h-cobordisms where it is trivial by assumption. The ﬁrst step is to construct a universal class. By ﬁberwise suspension using Lemma 8.7 and Remark 8.8, we may assume that the map Z → B has a section and the ﬁber X is simply connected. We choose a smooth bundle E → B ﬁber homotopy equivalent to Z. By embedding E in B × Rn for large n and taking a tubular neighborhood, we may assume that the vertical tangent bundle of E is trivial and the ﬁber is a compact n-manifold M n embedded in Rn so that both M X and ∂M are simply connected. Also, the image of the section B → E will have a neighborhood which is a trivial disk bundle D ∼ = B × Dn . Then (E, D) is classiﬁed by a map φ : B → BDiff (M rel Dn ). Since the universal bundle over BDiff (M rel Dn ) with ﬁber M is unipotent, there is a universal diﬀerence torsion class τ δ (M ) ∈ H 4k (BDiff (M rel Dn ); R) which pulls back to τ δ (E) = φ∗ (τ δ (M )). This is a general fact about characteristic classes which follows from the universal coeﬃcient theorem and the fact that every element of H4k (BDiff (M rel Dn )) is supported on a ﬁnite subcomplex. We will show that this universal class is trivial. This will follow from two statements which we claim are true. First, we need to go to the universal cover of BDiff (M rel Dn ) which is (M rel Dn ) = BDiff0 (M rel Dn ), BDiff where Diff0 (M rel Dn ) is the identity component of Diff (M rel Dn ). Claim 1. π0 Diff (M rel Dn ) ∼ = π1 BDiff (M rel Dn ) is ﬁnite for large n. Suppose for a moment that this is true. Then BDiff0 (M rel Dn ) will be a ﬁnite covering of BDiff (M rel Dn ) and the covering map BDiff0 (M rel Dn ) → BDiff (M rel Dn ) will induce a monomorphism in rational (and real) cohomology. Consequently, it will suﬃce to show that the pull-back of the universal class τ δ (M ) to BDiff0 (M rel Dn ) is zero. Next, we will reduce to the h-cobordism case. Let M0 be M minus an open collar neighborhood of ∂M . We may assume that M0 contains the disk Dn . So Diff (M rel M0 ) ⊆ Diff (M rel Dn ). By a theorem of Cerf [8], we know that Diff (M rel M0 ) ∼ = Diff (∂M × I rel ∂M × 0) is connected. Therefore, Diff (M rel M0 ) ⊆ Diff0 (M rel Dn ). Claim 2. For suﬃciently large n, the induced map of classifying spaces p : BDiff (M rel M0 ) → BDiff0 (M rel Dn ) is rationally 4k-connected in the sense that its homotopy ﬁber is rationally 4k − 1 connected. Suppose for a moment that this second claim is also true. Then the mapping p induced a monomorphism in degree 4k rational (and real) cohomology. So, τ δ is zero on BDiff0 (M rel Dn) if and only if it is zero on BDiff (M rel M0 ). However, the universal bundle over BDiff (M rel M0 ) with ﬁber M contains a trivial subbundle with ﬁber M0 as a deformation retract. Being a ﬁber homotopy invariant, this forces τ δ to be zero as claimed. It remains to prove the two claims. We prove them simultaneously assuming that n = dim M is suﬃciently large.

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Claim 1 is that π0 Diff (M rel Dn ) is ﬁnite. Consider the ﬁbration sequence π

Diff (M rel M0 ) → Diff (M rel Dn ) − → Embh (M0 , M rel Dn ),

(8.2)

where Emb (M0 , M rel D ) is the space of smooth embeddings ψ : M0 → intM which are the identity on Dn and so that ψ is a homotopy equivalence. By excision and Whitehead’s theorem, the complement of the image of ψ is an h-cobordism of ∂M and therefore, by the h-cobordism theorem, ψ will extend to a diﬀeomorphism ψ : M → M . Thus, the restriction map π is surjective. π is a ﬁbration by Cerf ([7], Appendix). Thus, h

n

π0 Diff (M rel Dn ) ∼ = π0 Embh (M0 , M rel Dn ). Claim 1 is that this group is ﬁnite. The ﬁbration sequence (8.2) gives another ﬁbration sequence p

Emb0 (M0 , M rel Dn ) → BDiff (M rel M0 ) − → BDiff0 (M rel Dn ) where, as before, the subscripts 0 denote the identity components of the embedding and diﬀeomorphism spaces. Claim 2 is that p is rationally 4k-connected. For this, it suﬃces to show that πi Emb0 (M0 , M rel Dn ) is ﬁnite for all i < 4k. Therefore, Claims 1 and 2 together follow from the statement that πi Embh (M0 , M rel Dn ) is ﬁnite for all i < 4k. When n is large, the homotopy dimension of M0 will be much smaller than n. Therefore, by transversality we have that the embedding space is homotopy equivalent in low degrees to the corresponding space of immersions: πi Embh (M0 , M rel Dn ) ∼ = πi Immh (M0 , M rel Dn ). By immersion theory, Immh (M0 , M rel Dn ) is homotopy equivalent in low degrees to the product of the space of all pointed homotopy equivalences M0 → M and the space of all pointed maps M0 → O(n). Since M0 is the suspension of a rationally trivial ﬁnite complex, both of these mapping spaces have ﬁnitely many components each of which is rationally trivial and thus, the low degree homotopy groups of the immersion space are ﬁnite as claimed. This completes the proof of Lemma 8.5 which implies the Main Theorem 4.4. 9. Existence of higher torsion In this section, we show that higher Miller–Morita–Mumford classes M2k and higher Franz– Reidemeister torsion τ2k are linearly independent higher torsion invariants. 9.1. MMM classes If p : E → B is any smooth bundle with compact ﬁber F , the Miller–Morita–Mumford (MMM) classes M2k are deﬁned to be the integral cohomology classes given by E M2k (E) := trB ((2k)!ch4k (T v E)) ∈ H 4k (B; Z)

for k 1, where T v E is the vertical tangent bundle of E and ch4k (ξ) := 12 ch4k (ξ ⊗ C) is the Chern character. In degree 0, this formula gives a half integer: n n M0 (E, ∂0 ) = χ(F, ∂0 ) = (χ(F ) − χ(∂0 F )), 2 2 where n = dim F . The properties of the transfer ([2]) give the following properties of the MMM classes. Proposition 9.1. (i) (stability) M2k (E × I) = M2k (E) for k > 0.

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(ii) (additivity) If E1 , E2 are smooth bundles over B with the same vertical boundary E1 ∩ E2 = ∂ v E1 = ∂ v E2 , then M2k (E1 ∪ E2 ) = M2k (E1 ) + M2k (E2 ) − M2k (∂ v E1 × I). (iii) (transfer) If q : D → E is a bundle with ﬁber X, then the MMM classes M2k (D)B ∈ H 4k (B) and M2k (D)E ∈ H 4k (E) are related by E M2k (D)B = χ(X)M2k (E) + trB (M2k (D)E ).

Proof. The ﬁrst two are easy. The transfer formula follows from the formula TBv D = TEv D ⊕ q ∗ T v E, the additivity of ch4k and the fact that trED ◦ q ∗ is multiplication by χ(X). D (2k)!ch4k TBv D M2k (D)B = trB E D E D ∗ = trB trE (2k)!ch4k TEv D + trB trE (q (2k)!ch4k (T v E)) E = trB (M2k (D)E ) + χ(X)M2k (E)

Proposition 9.2. If S 2n (ξ) → B is the S 2n -bundle associated to an SO(2n + 1)-bundle ξ over B, then M2k (S 2n (ξ)) = 2(2k)!ch4k (ξ). Proof. This follows from additivity and the observation that M2k (Dm (ξ)) = (2k)!ch4k (ξ) for any linear disk bundle Dm (ξ). Theorem 9.3. The Miller–Morita–Mumford class M2k is a nontrivial even higher torsion invariant. Proof. If we consider additivity (Proposition 9.1, part (ii)) in the case E1 = E2 , we get M2k (DEi ) = 2M2k (Ei ) − M2k (∂ v Ei × I). The additivity axiom follows from this equation and Proposition 9.1, part (ii) in the case E1 = E2 . The transfer axiom is a special case of the transfer formula of Proposition 9.1, part (iii). 9.2. Higher FR-torsion Higher Franz–Reidemeister (FR)-torsion invariants are real characteristic classes τ2k (E, ∂0 ) ∈ H 4k (B; R) for k 1 deﬁned for smooth bundle pairs (F, ∂0 ) → (E, ∂0 ) → B which are relatively unipotent in the sense that H∗ (F, ∂0 F ; Q) is unipotent as a π1 B module. (See [17].) Theorem 9.4 [17]. (i) (additivity) If (E1 , ∂0 ), (E2 , ∂0 ) are relatively unipotent smooth bundles over B with E1 ∩ E2 = ∂0 E2 ⊆ ∂1 E1 , then τ2k (E1 ∪ E2 , ∂0 E1 ) = τ2k (E1 , ∂0 ) + τ2k (E2 , ∂0 ).

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(ii) (stability) If q : D → E is a linear disk bundle and ∂0 D = q −1 (∂0 E), then the higher FR-torsion of (D, ∂0 ) as a bundle pair over B is equal to the higher torsion of (E, ∂0 ): τ2k (D, ∂0 )B = τ2k (E, ∂0 ). Higher FR-torsion and the higher MMM classes are related by the following theorem. Theorem 9.5 [19]. ﬁbers and k 1, then

If E → B is a unipotent smooth bundle with closed even dimensional τ2k (E) = (−1)k

Here n = dim F and ζ(2k + 1) =

m

1

ζ(2k + 1) M2k (E). 2(2k)!

1 m2k + 1

is the Riemann zeta function.

In other words, M2k is proportional to the even part of τ2k . The following calculation shows that the odd part of τ2k has the same size (but opposite sign). Theorem 9.6 [17]. FR-torsion

The S n -bundle associated to an SO(n + 1)-bundle ξ over B has higher τ2k (S n (ξ)) = (−1)n +k ζ(2k + 1)ch4k (ξ).

Remark 9.7. It follows from the stability of τ2k that τ2k = 0 on all linear disk bundles over B. Corollary 9.8. Higher Franz–Reidemeister torsion τ2k satisﬁes the additivity axiom (3.1) and the transfer axiom (3.2) and is therefore, a higher torsion invariant. Proof. The additivity axiom follows from the additivity of τ2k . The transfer axiom requires the following for any smooth unipotent bundle E → B with closed ﬁber F and any oriented linear sphere bundle S m → S m (ξ) → E over E. E (τ2k (S m (ξ))E ) τ2k (S m (ξ))B = χ(S m )τ2k (E) + trB

(9.1)

We prove this in each of the four cases depending on the parity of m and n = dim F . If n, m have the same parity, then the transfer formula (9.1) is equivalent to the transfer formula for M2k which we already proved. (If n, m are both odd, the RHS is zero for both τ2k and M2k . This implies that the LHS is zero in the odd–odd case for both since they are proportional by Theorem 9.5.) If n is even and m is odd, then, by the previous case, we know that E (τ2k (S m +1 (ξ))E ). τ2k (S m +1 (ξ))B = 2τ2k (E) + trB

The LHS of this is τ2k (S m +1 (ξ))B = 2τ2k (Dm +1 (ξ))B − τ2k (S m (ξ))B = 2τ2k (E) − τ2k (S m (ξ))B

(9.2)

by Theorem 9.4. In other words, E τ2k (S m (ξ))B = −trB (τ2k (S m +1 (ξ))E ).

But this is equal to the RHS of (9.1) since χ(S m ) = 0 and τ2k (S m +1 (ξ)E ) = −τ2k (S m (ξ)E ) by Theorem 9.6. In the case n odd and m even, the LHS of (9.2) is zero as we noted in the analysis of the E = 0. odd–odd case. Therefore, the RHS of (9.2) is also zero. (9.1) follows since trB

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Corollary 9.9.

185

The odd part of higher FR-torsion is given by od τ2k (E) = τ2k (E) − (−1)k

ζ(2k + 1) M2k (E). 2(2k)!

Remark 9.10. A corollary of the main theorem is that any odd torsion invariant must be proportional to the above expression. In particular, it is expected that the analytic torsion od . For the of Bismut and Lott [4] is an odd torsion invariant and therefore proportional to τ2k Bismut–Goette normalization of analytic torsion [3], we expect: BG τ2k =

k! od τ . (2π)k 2k

9.3. Tangential and exotic torsion Any higher torsion invariant in degree 4k can be written uniquely as τ = aM2k + bτ2k . T

We call τ = aM2k the tangential component of τ and τ x = bτ2k the exotic component of τ . In terms of the scalars s1 , s2 for τ , the tangential component τ T is given by E (ch4k (T v E)) τ T (E) = (s1 + s2 )trB

for any unipotent bundle E. Note that this depends only on the homotopy type of the bundle E and its vertical tangent bundle. Since τ T is an even torsion invariant which is determined by s1 + s2 , it follows from Theorem 7.1 that τ T is also characterized by the following proposition. Proposition 9.11. The tangential component τ T is the unique even torsion invariant, so that τ T (Dn (ξ)) = τ (Dn (ξ)) ∈ H 4k (B; R) for any oriented linear disk bundle Dn (ξ) → B. By the boundary case transfer formula, this gives the following characterization of the exotic component τ x = τ − τ T . Proposition 9.12. τ x (E) is independent of the vertical tangent bundle of E in the sense that τ x (E) = τ x (D) for any oriented linear disk bundle D over E. Thus, the exotic component of τ can also be given by τ x (E) = τ (D(ν)), where ν is the vertical normal bundle of E. (ν is a vector bundle over E which is complementary to T v E.) Any higher torsion invariant having this property (that τ (E) = τ (D)) will be called exotic. The main theorem can then be restated as follows. Theorem 9.13. Any exotic higher torsion invariant is a scalar multiple of higher FRtorsion. The Dwyer–Weiss–Williams (DWW) smooth torsion as deﬁned in [11] is ‘exotic’ for the following simple reason. They take their bundle E and embed it in B ×RN for some large N and

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take a regular neighborhood. This kills the vertical tangent bundle and therefore trivializes the tangential component. For this reason, we believe that the DWW smooth torsion is proportional to higher FR-torsion whenever it is deﬁned. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

J. C. Becker and R. E. Schultz, ‘Axioms for bundle transfers and traces’, Math. Z. 227 (1998), 583–605. J. C. Becker and D. H. Gottlieb, ‘The transfer and ﬁber bundles’, Topology 14 (1975) 1–12. J.-M. Bismut and S. Goette, ‘Families torsion and Morse functions’, Ast´ erisque 275 (2001). J.-M. Bismut and J. Lott, ‘Flat vector bundles, direct images and higher real analytic torsion’, J. Amer. Math. Soc. 8 (1995) 291–363. ¨ kstedt, The rational homotopy type of ΩW h D i ff (∗), Lecture Notes in Mathematics 1051, (Springer, M. Bo Berlin, 1984), 25–37. U. Bunke, ‘Higher analytic torsion of sphere bundles and continuous cohomology of Diﬀ(S 2 n −1 )’, Preprint, 1998, math.DG/9802100. J. Cerf, Sur les diﬀ´eomorphismes de la sph`ere de dimension trois (γ4 = 0) (Springer, Berlin, 1968). J. Cerf, ‘La stratiﬁcation naturelle des espaces de fonctions diﬀ´erentiables r´eelles et le th´eor`eme de la ´ pseudo-isotopie’, Pub. Math. Inst. Hautes Etudes Sci. 39 (1970) 5–173. L. J. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications, Part I, Cambridge Studies in Advanced Mathematics 18 (Cambridge University Press, Cambridge, 1990). W. Dorabiala and M. W. Johnson, ‘The product theorem for parametrized homotopy Reidemeister torsion’, J. Pure Appl. Algebra 196 (2005) 53–90. W. Dwyer, M. Weiss, and B. Williams, ‘A parametrized index theorem for the algebraic K -theory Euler class’, Acta Math. 190 (2003) 1–104. F. T. Farrell and W. C. Hsiang, ‘On the rational homotopy groups of the diﬀeomorphism groups of discs, spheres and aspherical manifolds’, Algebraic and geometric topology, Stanford, CA, 1976, Part 1, Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, RI, 1978) 325–337. S. Goette, ‘Morse theory and higher torsion invariants I’, Preprint, 2001, math.DG/0111222. S. Geotte, ‘Morse theory and higher torsion invariants II’, Preprint, 2003, math.DG/0305287. A. Haefliger, ‘Rational homotopy of the space of sections of a nilpotent bundle’, Trans. Amer. Math. Soc. 273 (1982) 609–620. K. Igusa, ‘The stability theorem for smooth pseudoisotopies’, K -Theory in (1–2) 2 (1988) vi+355. K. Igusa, Higher Franz–Reidemeister torsion, AMS/IP Studies in Advance Mathematics 31 (International Press, 2002). K. Igusa, ‘Axioms for higher torsion’, Preprint, 2003. http://people.brandeis.edu/∼igusa/Papers/ AxiomsI.pdf. K. Igusa, ‘Higher complex torsion and the framing principle’, Mem. Amer. Math. Soc. (835) 177 (2005) xiv+94. K. Igusa and J. Klein, ‘The Borel regulator map on pictures. II. An example from Morse theory’, K Theory 7 (1993) 225–267. J. Klein, ‘The cell complex construction and higher R-torsion for bundles with framed Morse function’, PhD Thesis, Brandeis University, 1989. J. Klein, ‘Higher Franz–Reidemeister torsion and the Torelli group’, Mapping Class Groups and Moduli Spaces, Contemporary Mathematics 150 (American Mathematical Society, 1993) 195–204. X. Ma, ‘Formes de torsion analytique et familles de submersions’, C. R. Acad. Sci. Paris S´er. I Math. 324 (1997) 205–210. S. Morita, Geometry of Characteristic Classes, Translations of Mathematical Monographs 199 (American Mathematical Society, 2001). J. Roitberg, ‘The signature of quasi-nilpotent ﬁber bundles’, Invent. Math. 39 (1977) 91–94. J. B. Wagoner, ‘Diﬀeomorphisms, K 2 , and analytic torsion’, Algebraic and geometric topology, Stanford, CA, 1976, Part 1, Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, RI, 1978) 23–33.

Kiyoshi Igusa Department of Mathematics Brandeis University P O Box 9110 Waltham, MA 02454-9110 USA [email protected]

c 2007 London Mathematical Society doi:10.1112/jtopol/jtm011

Axioms for higher torsion invariants of smooth bundles Kiyoshi Igusa Abstract This paper attempts to explain the relationship between various characteristic classes for smooth manifold bundles which are known as ‘higher torsion’ classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz–Reidemeister torsion and higher Miller–Morita–Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples. We also show how any higher torsion invariant, that is, any characteristic class satisfying the two axioms, can be computed for a smooth bundle with a ﬁberwise Morse function with distinct critical values. Finally, we explain the statements of the conjectured formulas relating higher analytic torsion classes, higher Franz–Reidemeister torsion and Dwyer–Weiss–Williams smooth torsion.

Contents 1. Introduction . . . . 2. Preliminaries . . . . . . . . . 3. Axioms 4. Statement . . . . 5. Extension to relative case . 6. Stability of higher torsion . 7. Computation of higher torsion 8. Proof of the main theorem 9. Existence of higher torsion References . . . . .

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159 162 163 165 167 172 173 177 182 186

1. Introduction Higher analogues of Reidemeister torsion and Ray–Singer analytic torsion were developed by J. Wagoner, J.R. Klein, the author, M. Bismut, J. Lott, W. Dwyer, M. Weiss, E.B. Williams, S. Goette and many others [3, 4, 11, 13, 14, 17, 21, 20, 26]. This paper develops higher torsion from an axiomatic viewpoint. There are three main objectives to this approach: (1) Make higher torsion easier to understand. (2) Simplify the computation of these invariants. (3) Explain the theorems relating higher Franz–Reidemeister torsion, Miller–Morita– Mumford (tautological) classes, Dwyer–Weiss–Williams higher torsion and higher analytic torsion classes. Received 22 February 2007; published online 25 October 2007. 2000 Mathematics Subject Classiﬁcation 55R40 (primary), 57R50, 19J10 (secondary). I am in great debt to John R. Klein and E. Bruce Williams for their help in completing the ﬁnal crucial steps in the proof of the main theorem. I also beneﬁtted greatly from conversations with Sebastian Goette, Xiaonan Ma, Wojciech Dorabiala and Gordana Todorov. Also, I would like to thank Bernard Badzioch for his inspired presentation of this work during the Arbeitsgemeinshaft at Oberwolfach in April, 2006. Finally, I should not forget to thank the organizers Thomas Schick, Ulrich Bunke and Sebastian Goette of the conference on higher torsion in G¨ o ttingen in September 2003 at which the ﬁrst version of these results were developed and announced. Research for this paper was supported by NSF grants DMS 02-04386 and DMS 03-09480.

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The following theorems are examples of results which will make much more sense from the axiomatic viewpoint. Theorem 1.1 (Hain–I–Penner [17, 19]). The higher Franz–Reidemeister torsion invariants for the Torelli group τ2k (Tg ) ∈ H 4k (Tg ; R) are proportional to the Miller–Morita–Mumford classes. This had been conjectured by J.R. Klein [22]. The precise proportionality constant was computed in [17]. We will see that this theorem is an example of the uniqueness theorem for even higher torsion invariants. The next theorem is Theorem 0.2 in [13]. See [3], [13], [14] for more details. Theorem 1.2 [14]. Suppose that p : M → B is a smooth bundle with closed oriented manifold ﬁber X, F is a Hermitian coeﬃcient system on M and H ∗ (X; F) admits a π1 B invariant metric. Suppose further that there exists a ﬁberwise Morse function on M . Then the Chern normalizations (from [3]) of the higher analytic torsion classes T2k (E) ∈ H 4k (B; R) and the higher Franz–Reidemeister torsion class are deﬁned and agree up to a correction term which is a multiple of the transfer to B of the Chern character of the vertical tangent bundle of M : ch E T2k (E) = τ2k (E) + ζ (−2k)rk(F)trB (ch4k (T v M )).

This theorem, together with the uniqueness theorem for odd higher torsion invariants proved below, suggests that nonequivariant higher analytic torsion classes are odd torsion invariants. Recently, Sebastian Goette has claimed that he can prove his theorem in general, that is, without the existence of a ﬁberwise Morse function. Finally, in the case where we compare diﬀerent smooth structures on the same topological manifold bundle, both Dwyer–Weiss–Williams higher torsion of [11] and the higher Franz– Reidemeister torsion of [17] are deﬁned and must be proportional. Details will be given in forthcoming joint work with Sebastian Goette. In this paper, we deﬁne a higher torsion invariant to be a characteristic class τ (E) ∈ H 4k (B; R) of ‘unipotent’ smooth bundles E → B (deﬁned in Section 2) which satisﬁes two axioms (Section 3). We show in Section 4 that each such invariant is the sum of even and odd parts τ = τ ev + τ od . The main theorem (Theorem 4.4) is shown below. Theorem 1.3. Nontrivial even and odd torsion invariants τ ev , τ od exist in degree 4k for all k > 0 and they are uniquely determined up to scalar multiples. The uniqueness statement is simply a reﬂection of the fact that, given the value of a higher torsion invariant on the universal oriented S 1 bundle over CP ∞ and the associated S 2 bundle, the higher torsion invariant can be computed or shown to be uniquely determined in all other cases. Since the cohomology of CP ∞ is one dimensional in degree 4k, these universal examples are given by two scalars s1 and s2 (Subsection 4.2). The ﬁrst step is to extend the deﬁnition to the case where the ﬁber of E → B has a boundary giving a subbundle ∂ v E which we call the vertical boundary of E. The higher torsion in this case is deﬁned in Subsection 5.1 to be half the torsion of the double of E plus half the torsion of ∂ v E. In Subsection 5.2, we extend to the relative case (E, ∂0 E) when the vertical boundary is a union of two subbundles ∂ v E = ∂0 E ∪ ∂1 E each of which is unipotent. The relative higher

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torsion is simply deﬁned to be τ (E, ∂0 E) = τ (E) − τ (∂0 E). In Section 7, we use the transfer axiom to compute the higher torsion for any oriented linear disk or sphere bundle in terms of s1 and s2 . We extend this to ﬁberwise products of linear disk bundles, such as those that arise from ﬁberwise Morse functions (E, ∂0 E) → (I, 0). If the critical points have distinct critical values, then the additivity axiom allows us to express the higher torsion of E as a sum of higher torsions of disks which we have already computed. In Subsection 7.3, we make crucial use of Hatcher’s construction of disk bundles with exotic smooth structures. Since these admit ﬁberwise Morse functions, their higher torsion can be computed. By a theorem of B¨ okstedt, these examples rationally generate all possible smooth disk bundles in the stable range. In Section 6, we show that higher torsion is stable. This implies that the higher torsion is determined (by the two scalars s1 , s2 ) for all disk bundles. By additivity, this also implies that the higher torsion is determined on all h-cobordism bundles. At this point, we are ready to prove the main theorem. We deﬁne the diﬀerence torsion to be τ δ = τ − aM2k − bτ2k , where M2k is the higher Miller–Morita–Mumford class and τ2k is the higher Franz–Reidemeister torsion and the real numbers a, b are uniquely determined by the requirement that τ δ = 0 on all oriented linear sphere bundles. In that case, the computation of Hatcher’s example shows that τ δ = 0 on all disk bundles and the h-cobordism calculation implies that τ δ (E, ∂0 E) depends only on the ﬁber homotopy type of the pair (E, ∂0 E). In Section 8, we show that this ﬁber homotopy invariant must be trivial. The main theorem follows. In Section 9, we show that there are two linearly independent higher torsion invariants of degree 4k given by the higher Miller–Morita–Mumford (MMM) classes E ((2k)!ch4k (T v E)) ∈ H 4k (B; Z) M2k (E) = trB

and the higher Franz–Reidemeister (FR) torsion invariants τ2k (E) ∈ H 4k (B; R). Using basic properties of higher FR-torsion proved in [17] and [19], in particular the framing principle and the transfer theorem, it is easy to show that τ2k satisﬁes the axioms. Basis properties of the transfer map imply that M2k also satisﬁes these axioms. It is easy to see that M2k is an even higher torsion invariant, that is, it is zero when the ﬁber is a closed odd dimensional manifold. However, τ2k has both even and odd components. The uniqueness theorem implies the known theorem [17] that τ2k and M2k are proportional whenever the ﬁber is a closed oriented even dimensional manifold, for example, an oriented surface. The proportionality constant is determined by computing one example, for example, the universal S 2 bundle over CP ∞ and we get: τ2k (E) =

(−1)k ζ(2k + 1) M2k (E). 2(2k)!

This is not a new proof of this formula since we use this formula in the proof of the uniqueness theorem. However, the point is that it is now very transparent. Finally in Subsection 9.3, we characterize the tangential and exotic components of higher torsion given by τ = τT + τx, where τ T is an even higher torsion invariant and τ x is proportional to higher FR-torsion. We will see that DWW smooth torsion is exotic by deﬁnition. This explains why we believe that DWW smooth torsion is proportional to higher FR torsion. Bismut and Lott [4] showed that nonequivariant higher analytic torsion classes are trivial for bundles with closed even dimensional ﬁbers. Thus, we believe that they are odd higher torsion

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invariants. S. Goette’s theorem above says that, assuming the existence of a ﬁberwise Morse function, T2k is proportional to the odd part of τ2k . And the proportionality constant is 1 if they are normalized in the same way. Finally, we want to point out that X. Ma [23] has shown that higher analytic torsion satisﬁes the transfer axiom and W. Dorabiala and M. Johnson [10] have shown that DWW homotopy torsion satisﬁes the product formula (Corollary 5.10) which is a variation of the transfer axiom. Becker and Schultz [1] have shown that the transfer in stable homotopy, especially for smooth bundles, can be characterized by axioms very similar to ours. The main diﬀerence is that they use a product formula instead of the formula analogous to our transfer axiom. This paper combines two earlier works: my lecture notes [18] and the preliminary manuscript ‘Axioms for higher torsion II.’ 2. Preliminaries We consider smooth ﬁber bundles p

→ B, F →E− where E and B are compact smooth manifolds and p is a smooth submersion. We usually assume that the ﬁber F is a closed oriented manifold of dimension n. But, we also consider the case when F is oriented with boundary ∂F . This gives a subbundle ∂F → ∂ v E → B of E. We call ∂ v E the vertical boundary of E. (The boundary of E is the union of ∂ v E and p−1 (∂B).) We assume that B is connected. We assume that the action of π1 B on F preserves the orientation of F . We will assume that the bundle E → B is unipotent in the sense that the rational homology of its ﬁber F is unipotent as a π1 B-module. In other words, H∗ (F ; Q) has a ﬁltration by π1 B-submodules so that the subquotients have trivial π1 B actions. In particular, π1 B does not permute the components of F . Note that unipotent π1 B-modules form a Serre category. In fact, it is the Serre category generated by the trivial modules. A unipotent module is the same as a nilpotent module. However, ‘unipotent’ is used in the case of a representation of a group over a ﬁeld. A unipotent ﬁbration is not the same as a nilpotent ﬁbration. Recall that a ﬁbration E → B is nilpotent if there is a sequence of ﬁbrations p1

p2

p3

B = E0 ←− E1 ←− E2 ←− · · · converging to E (so that E is the inverse limit of the Ei ), so that each pi is a principal ﬁbration with ﬁber an Eilenberg–MacLane space K(πi , ni ) with ni → ∞ as i → ∞. (See, for example, [15].) This implies that the action of π1 B on the homotopy groups of the ﬁber is nilpotent. Quasi-nilpotent means that the action of π1 B on the homology of the ﬁber is nilpotent. A unipotent ﬁbration could thus correctly be called a rationally quasi-nilpotent ﬁbration. The action of π1 B on H∗ (X; Q) is nilpotent which is the same as saying that it is a unipotent representation over the ﬁeld Q. (See, for example, [9].) Many theorems about nilpotent and quasi-nilpotent ﬁbrations apply to unipotent ﬁbrations. For example, in [25] it is noted that we only need a ﬁnite index subgroup of π1 B to act nilpotently on the homology of the ﬁber. Similarly, Sebastian Goette pointed out to me that any characteristic class of unipotent bundles such as the higher torsion discussed in this paper extends uniquely to bundles for which a ﬁnite index subgroup of π1 B acts unipotently on the ﬁber. Proposition 2.1. If E → B is a unipotent bundle with oriented ﬁber F having boundary ∂F , then H∗ (∂F ; Q) and H∗ (F, ∂F ; Q) are also unipotent π1 B-modules. In particular, the vertical boundary ∂ v E → B is a unipotent bundle.

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Proof. By Poincar´e duality, H∗ (F ; Q) ∼ = H ∗ (F ; ∂F ; Q) is a unipotent π1 B-module. Its dual H∗ (F, ∂F ; Q) must also be unipotent. Since unipotent modules form a Serre category, the long exact homology sequence of (F, ∂F ) implies that H∗ (∂F ; Q) is also unipotent. Let T v E denote the vertical tangent bundle of E. This is the subbundle of the tangent bundle T E of E consisting of all tangent vectors which go to zero in B, that is, T v E is the kernel of T p : T E → T B. The Euler class e(E) ∈ H n (E; Z) of the bundle E is deﬁned to be the usual Euler class of T v E. The transfer E : H ∗ (E; Z) → H ∗ (B; Z) trB

(2.1)

is given by E trB (x) = p∗ (x ∪ e(E)),

where p∗ : H ∗+n (E; Z) → H ∗ (B; Z) is the push-down operator given over R by integrating along ﬁbers. Over Z, it is given as the composition of two maps H k +n (E; Z) → H k (B; H n (F ; Z)) → H k (B; Z), where the ﬁrst map comes from the Serre spectral sequence of the bundle and the second map is induced by the coeﬃcient map H n (F ; Z) → Z given by evaluation on the orientation class of the ﬁber. For details, see [24] or [19]. If the orientation of the ﬁber F is reversed, both e(E) and p∗ change sign. Thus, the transfer is independent of the choice of orientation of F . For the basic properties of the transfer, see [2]. The main property that we need is that, for closed ﬁbers F , E E trB = (−1)n trB . E So, rationally, trB = 0 if n = dim F is odd.

3. Axioms We deﬁne a higher torsion invariant (in degree 4k) to be a real characteristic class τ (E) ∈ H 4k (B; R) for unipotent smooth bundles E → B with closed oriented ﬁbers satisfying the additivity and transfer axioms described below. When we say that τ is a ‘characteristic class’ we mean it is a natural cohomology class, that is, τ (f ∗ E) = f ∗ (τ (E)) ∈ H 4k (B ; R) if f ∗ E is the pull-back of E along f : B → B. Naturality implies that τ is zero for trivial bundles: τ (B × F ) = 0. 3.1. Additivity If E = E1 ∪ E2 where E1 , E2 are unipotent bundles over B with the same vertical boundary E1 ∩ E2 = ∂ v E1 = ∂ v E2 , then the Additivity Axiom says that τ (E) = 12 τ (DE1 ) + 12 τ (DE2 ), where DEi is the ﬁberwise double of Ei . This wording of the Additivity Axiom comes from Ulrich Bunke.

(3.1)

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3.2. Transfer Suppose that p : E → B is a unipotent bundle with closed ﬁber F and q : S n (ξ) → E is the S bundle associated to an SO(n + 1) bundle ξ over E. Then S n (ξ) is a unipotent bundle over both E and B. The Transfer Axiom says that the higher torsion invariants of these bundles τB (S n (ξ)) ∈ H 4k (B; R) and τE (S n (ξ)) ∈ H 4k (E; R) are related by the formula: n

E τB (S n (ξ)) = χ(S n )τ (E) + trB (τE (S n (ξ))).

(3.2)

Note that χ(S n ) = 2 or 0 depending on whether n is even or odd respectively. 3.3. Examples As stated in the introduction, two examples of higher torsion invariants are the higher Miller–Morita–Mumford classes M2k (E) and the higher Franz–Reidemeister torsion invariants τ2k (E). The MMM classes, for closed ﬁber F , are given by E M2k (E) = trB ((2k)!ch4k (T v E)),

where ch4k (T v E) = 12 ch4k (T v E ⊗C). Although this is an integral cohomology class (for k > 0), we consider it as a real characteristic class. This invariant is deﬁned for any smooth bundle E is zero. with closed oriented ﬁber F . If n = dim F is odd, then twice the transfer map trB Proposition 3.1. M2k (E) = 0 for closed odd dimensional ﬁbers F . Theorem 3.2. M2k is a higher torsion invariant for every k 1. The higher FR torsion invariants ([17, 19]) τ2k (E, ∂0 E) ∈ H 4k (B; R) are deﬁned for any relatively unipotent bundle pair (E, ∂0 E) → B. By this we mean that the vertical boundary ∂ v E is a union of two subbundles ∂0 E, ∂1 E with the same vertical boundary ∂0 E ∩ ∂1 E = ∂ v ∂0 E = ∂ v ∂1 E and that the rational homology of the ﬁber pair (F, ∂0 F ) of the bundle pair (E, ∂0 E) is a unipotent π1 B-module. Theorem 3.3. The higher FR torsion invariants τ2k are higher torsion invariants for unipotent bundles with closed manifold ﬁbers. These two theorems will be proved later. J.-M. Bismut and J. Lott [4] constructed even diﬀerential forms on B called analytic torsion forms. In some cases, these are closed and topological (that is, independent of the metric and horizontal distribution used to deﬁned the form). For example, if F is a closed oriented manifold and π1 B acts trivially on H∗ (F ; Q), then they obtain a (nonequivariant) analytic torsion class BL T2k (E) ∈ H 4k (B; R).

They showed, that is, the following Proposition. BL (E) = 0 for closed even dimensional ﬁbers F . Proposition 3.4. T2k

We also have the following theorems of X. Ma and U. Bunke. Theorem 3.5 [23].

T2k satisﬁes the transfer axiom.

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Theorem 3.6 [6]. Let E → B be the S 2n −1 bundle associated to an U (n) bundle ξ over B. Then (4k + 1)! B unke ζ(2k + 1)ch4k (ξ). (E) = 4k T2k 2 (2k)! Remark 3.7. Bunke uses a diﬀerent normalization of the analytic torsion. We should multiply by (2πi)−2k to get the Bismut–Lott normalization: BL (E) = (−1)k (2π)−2k T2k

(4k + 1)! ζ(2k + 1)ch4k (ξ). 24k (2k)!

As we noted in the introduction, S. Goette has extended the deﬁnition of the higher analytic torsion class to the case when π1 B acts orthogonally on H∗ (F ; Q), that is, preserving some metric. However, we need to extend it to the unipotent case. 4. Statement We give the statement of the main theorem. We begin with the following elementary observations. Lemma 4.1. For each k, the set of all higher torsion invariants τ of degree 4k is a vector space over R. Proof. The axioms are homogeneous linear equations in τ . Lemma 4.2. If τ is a higher torsion invariant, then so is (−1)n τ , where n = dim F considered as a function of the bundle E → B. Proof. We need to show that (−1)n τ satisﬁes the axioms: (−1)n τ (E) = (−1)n 12 τ (DE1 ) + (−1)n 12 τ (DE2 ) E (−1)m +n τB (S m (ξ)) = (−1)n χ(S m )τ (E) + (−1)m trB (τE (S m (ξ))).

The additivity axiom (the ﬁrst equation) is the same as before. The transfer axiom (the second equation) is the same as before if both n = dim F and m are even. If one or both are odd, then E =0 the terms on the right with the wrong sign are zero since χ(S m ) = 0 for odd m and trB for odd n. 4.1. Even and odd higher torsion These lemmas imply that higher torsion invariants can always be expressed as a sum of odd and even parts: τ = τ ev + τ od , where τ ev =

τ + (−1)n τ , 2

τ od =

τ − (−1)n τ . 2

Here τ ev is even and τ od is odd in the sense of the following deﬁnition. Definition 4.3. A higher torsion invariant τ is called even (respectively odd) if τ (E) = 0 for all unipotent bundles with closed odd (respectively even) dimensional ﬁbers. The main theorem of this paper is the following.

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Theorem 4.4 (Main Theorem). Nontrivial even and odd torsion invariants exist in degree 4k for all k > 0 and they are unique up to a scalar factor. Corollary 4.5. Every even torsion invariant is a scalar multiple of M2k and every odd torsion invariant is a scalar multiple of the odd part of the higher FR-torsion τ2k . 4.2. The scalars s1 , s2 The main theorem is that a higher torsion invariant is determined by two scalars in every degree. These scalars are given as follows. Let λ be the universal U (1) = SO(2) bundle over CP ∞ . Let S 1 (λ) → CP ∞ be the circle bundle associated to λ and S 2 (λ) → CP ∞ the associated S 2 bundle (the ﬁberwise suspension of S 1 (λ)). Since the cohomology of CP ∞ is a polynomial algebra in c1 (λ), H 4k (CP ∞ ; R) ∼ = R is generated by ch4k (λ) = c1 (λ)2k /(2k)!. Given any higher torsion invariant τ , there exist scalars s1 , s2 ∈ R, so that (1) τ (S 1 (λ) = τ od (S 1 (λ) = 2s1 ch4k (λ) (2) τ (S 2 (λ) = τ ev (S 2 (λ) = 2s2 ch4k (λ) The main theorem says that τ is uniquely determined by s1 and s2 . For example, we have the following calculations which will be explained later. Proposition 4.6. M2k (S 2 (λ)) = 2(2k)!ch4k (λ). Thus, s2 = (2k)! for the higher MMM classes M2k (and s1 = 0). Proposition 4.7. τ2k (S n (λ)) = (−1)k +n ζ(2k + 1)ch4k (λ). So, sn = 12 (−1)k +n ζ(2k + 1) for the higher FR torsion invariants τ2k . 4.3. Consequences of uniqueness The uniqueness of even torsion gives us the following. Corollary 4.8 [19]. If E → B is a unipotent bundle with closed even dimensional ﬁbers, then (−1)k ζ(2k + 1) τ2k (E) = M2k (E). 2(2k)! Theorem 1.1 in the introduction is a special case of this corollary. The uniqueness of odd torsion can now be expressed as follows. Corollary 4.9. Any odd torsion invariant is a scalar multiple of the odd part of higher Franz–Reidemeister torsion which is given by od τ2k = τ2k −

(−1)k ζ(2k + 1) M2k . 2(2k)!

Theorem 1.2 in the introduction says that analytic torsion classes are odd torsion invariants on certain bundles. We expect that the same formula should hold in general. The uniqueness theorem also tells us something about higher Franz–Reidemeister torsion. Corollary 4.10.

The higher FR torsion invariant τ2k has coeﬃcients in ζ(2k + 1)Q.

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Proof. Let π : R → ζ(2k + 1)Q be any Q linear retraction. Then, by linearity of the axioms, π∗ τ2k is a higher torsion invariant with coeﬃcients in ζ(2k+1)Q. Proposition 4.7 gives the same value of s1 , s2 for π∗ τ2k and τ2k . Therefore, the uniqueness theorem tells us that π∗ τ2k = τ2k .

The rest of this paper is devoted to the proof of the main theorem. We show that, given the values of the scalars s1 , s2 above, any higher torsion invariant can be computed in suﬃciently many cases to determine it completely. 5. Extension to relative case In order to compute the higher torsion invariant τ (E), we need to cut E into simpler pieces and compute the relative torsion of each piece. To do this, we need to extend τ ﬁrst to the case when F has a boundary and then to the case of a unipotent bundle pair (F, ∂0 F ) → (E, ∂0 E) → B. 5.1. Higher torsion in the boundary case Suppose that E → B is a unipotent smooth bundle with ﬁber F an oriented compact manifold with boundary. By Proposition 2.1, the vertical boundary ∂ v E is also unipotent. And it follows from the Mayer–Vietoris sequence that the ﬁberwise double DE is also unipotent. A higher torsion invariant τ can now be extended to the boundary case by the formula τ (E) := 12 τ (DE) + 12 τ (∂ v E). We will show that this extension of τ satisﬁes boundary analogues of the additivity and transfer axioms. These axioms refer to bundles whose ﬁbers have corners. If E is such a bundle, we deﬁne τ (E) to be equal to τ (E ), where E is E with the vertical corners rounded oﬀ in a standard way to be explained at the end of this section. We need the following lemmas. Lemma 5.1. Suppose that Ei are smooth unipotent bundles over B with the same vertical boundary. Then τ (E1 ∪ E2 ) + τ (E3 ∪ E4 ) = τ (E1 ∪ E3 ) + τ (E2 ∪ E4 ). Proof. Both sides are equal to

1 2

τ (DEi ) by the additivity axiom.

Lemma 5.2. τ (∂ v E) = τ (∂ v (E × D2 )) assuming E is unipotent. Proof. Since ∂ v (E × D2 ) = ∂ v E × D2 ∪ E × S 1 we have, by the additivity axiom, that τ (∂ v (E × D2 )) = 12 τ (∂ v E × S 2 ) + 12 τ (DE × S 1 ). But τ (∂ v E × S 2 ) = 2τ (∂ v E) and τ (DE × S 1 ) = 0 by the transfer axiom. Lemma 5.3 (additivity of transfer). If E = E1 ∪ E2 is a union of two smooth bundles along their common vertical boundary ∂ v E1 = ∂ v E2 = E1 ∩ E2 , then v

E1 E2 ∂ E trB (x) = trB (x|E1 ) + trB (x|E2 ) − trB

E1

(x|∂ v E1 ).

Proposition 5.4 (additivity for boundary case). If (E1 , ∂0 ), (E2 , ∂0 ) are unipotent bundle pairs over B with E1 ∩ E2 = ∂0 E1 = ∂0 E2 , then τ (E1 ∪ E2 ) = τ (E1 ) + τ (E2 ) − τ (E1 ∩ E2 ).

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Proof. We expand each term using the deﬁning equation: τ (Ei ) := 12 τ (DEi ) + 12 τ (∂ v Ei ) τ (E1 ∪ E2 ) := 12 τ (∂ v (E1 ∪ E2 )) + 12 τ (D(E1 ∪ E2 )) τ (E1 ∩ E2 ) := 12 τ (D(E1 ∩ E2 )) + 12 τ (∂ v (E1 ∩ E2 )). The terms in the last two equations are arranged so that the vertical sums match the following two examples of Lemma 5.1: τ (∂ v E1 ) + τ (∂ v E2 ) = τ (∂ v (E1 ∪ E2 )) + τ (D(E1 ∩ E2 )) τ (DE1 ) + τ (DE2 ) = τ (D(E1 ∪ E2 )) + τ (D(I × (E1 ∩ E2 ))), where the last term is τ (D(I × (E1 ∩ E2 ))) = τ (∂ v (D2 × (E1 ∩ E2 ))) = τ (∂ v (E1 ∩ E2 )) by Lemma 5.2. The proposition follows. Proposition 5.5 (transfer for boundary case). If X → D → E is an oriented linear disk or sphere bundle, then E (τE (D)). τB (D) = χ(X)τ (E) + trB

Proof. We consider ﬁrst the case when D = D(ξ) is an oriented linear Dn -bundle and F is E (τE (D(ξ))). This is just half of the sum of closed, that is, we will show: τB (D(ξ)) = τ (E) + trB the following two examples of the original transfer axiom. E τB (S n (ξ)) = χ(S n )τ (E) + trB (τE (S n (ξ))) E τB (S n −1 (ξ)) = χ(S n −1 )τ (E) + trB (τE (S n −1 (ξ)))

The transfer axiom takes care of the case when D is a sphere bundle and F is closed. The remaining case when ∂F is nonempty is given by the following lemma. Lemma 5.6. With the ﬁber X of q : D → E ﬁxed, the transfer formula for F closed implies the transfer formula for F with boundary. Proof. Write DE = E ∪ E as the union of two copies of E along its vertical boundary. Let D, D be two copies of D with D ∩ D = q −1 (∂ v E). Then the transfer formula τB (D) = E χ(X)τ (E)+trB (τE (D)) is half the sum of the following two transfer formulas with closed ﬁbers DF, ∂F respectively.

E ∪E τB (D ∪ D ) = χ(X)τ (E ∪ E ) + trB (τE ∪E (D ∪ D ))

E ∩E (τE ∩E (D ∩ D )) τB (D ∩ D ) = χ(X)τ (E ∩ E ) + trB

The additivity of transfer (Lemma 5.3) is used here. 5.2. Relative torsion Suppose that (F, ∂0 F ) → (E, ∂0 E) → B is a unipotent smooth bundle pair. By this, we mean that the vertical boundary ∂ v E is the union of two subbundles ∂ v E = ∂0 E ∪ ∂1 E which meet along their common vertical boundary: ∂0 E ∩ ∂1 E = ∂ v ∂0 E = ∂ v ∂1 E and that both E and ∂0 E are unipotent. This implies the weaker condition that the pair (E, ∂0 E) is relatively unipotent, that is, that H∗ (F, ∂0 F ) is a unipotent π1 B-module. We use the abbreviation (E, ∂0 ) for (E, ∂0 E).

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Suppose that τ is a higher torsion invariant which has been extended to the boundary case as above. Then for any unipotent smooth bundle pair (E, ∂0 ) → B, we deﬁne the relative torsion by τ (E, ∂0 ) := τ (E) − τ (∂0 E). Proposition 5.7 (additivity in the relative case). Suppose that E → B is a smooth bundle which can be written as a union of two subbundles E = E1 ∪ E2 which meet along a subbundle of their respective vertical boundaries: E1 ∩ E2 = ∂0 E2 ⊆ ∂ v E1 . Let ∂ v E1 = ∂0 E ∪ ∂1 E be a decomposition ∂ v E1 , so that ∂0 E2 ⊆ ∂1 E1 and (Ei , ∂0 ) → B, i = 1, 2 are unipotent smooth bundle pairs. Then (E, ∂0 E1 ) → B is unipotent and τ (E1 ∪ E2 , ∂0 E1 ) = τ (E1 , ∂0 ) + τ (E2 , ∂0 ).

∂ 0 E2

∂ 1 E1 E2 E1

← ∂ 0 E1

Proof. By Proposition 5.7, both sides of the equation are equal to τ (E1 ) + τ (E2 ) − τ (E1 ∩ E2 ) − τ (∂0 E1 ). Here is another variation of the additivity axiom which is also trivial to prove. Proposition 5.8 (horizontal additivity). Suppose that (E, ∂0 ) → B is a union of two unipotent bundle pairs (Ei , ∂0 Ei ) in the sense that E = E1 ∪ E2 and ∂0 E = ∂0 E1 ∪ ∂0 E2 with E1 ∩ E2 ⊆ ∂1 E1 ∩ ∂1 E2 . Let E0 = E1 ∩ E2 and ∂0 E0 = E0 ∩ ∂0 E and suppose (E0 , ∂0 ) is a unipotent bundle pair. Then (E, ∂0 ) is unipotent and τ (E, ∂0 ) = τ (E1 , ∂0 ) + τ (E2 , ∂0 ) − τ (E0 , ∂0 ).

E1

E0

∂ 0 E1

E2 ∂ 0 E2

∂ 0 E0 To state the transfer axiom in the relative case, we need the relative transfer: (E ,∂ 0 )

trB

: H ∗ (E; Z) → H ∗ (B; Z),

given by (E ,∂ 0 )

trB

(x) = p∗ (x ∪ e(E, ∂0 )),

where p∗ : H ∗+n (E, ∂ v E; Z) → H ∗ (B; Z)

(5.1)

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is the push-down operator (given over R by integrating along ﬁbers) and e(E, ∂0 ) ∈ H n (E, ∂ v E; Z) is the relative Euler class given by pulling back the Thom class of the vertical tangent bundle T v E along any vertical tangent vector ﬁeld which is nonzero along the vertical boundary ∂ v E and which points inward along ∂0 E and outward along ∂1 E. The relative transfer satisﬁes the following two equations for any x ∈ H ∗ (E; Z). (E ,∂ 0 )

trB

∂0 E E (x) = trB (x) − trB (x|∂0 E)

(E ,∂ 1 )

trB

(5.2)

(E ,∂ 0 )

= (−1)n trB

(5.3)

And, ﬁnally, trB 0 ◦p∗ : H ∗ (B) → H ∗ (B) is multiplication by the relative Euler characteristic of the ﬁber pair (F, ∂0 F ): (E ,∂ )

χ(F, ∂0 ) := χ(F ) − χ(∂0 F ). p

→ B and (X, ∂0 ) → Proposition 5.9 (transfer in the relative case). Let (F, ∂0 ) → (E, ∂0 ) − q (D, ∂0 ) − → E be unipotent smooth bundle pairs, so that the second is an oriented linear S n or Dn bundle with ∂0 X = S n −1 , Dn −1 or ∅. Then τB (D, ∂0 D ∪ q −1 ∂0 E) = χ(X, ∂0 )τ (E, ∂0 ) + trB

(E ,∂ 0 )

(τE (D, ∂0 )).

Proof. We already did the case when both ∂0 F and ∂0 X are empty. The case when ∂0 X is empty follows easily from the formula τB (D, q −1 ∂0 E) = τB (D) − τB (q −1 ∂0 E). The general case follows from the following two examples of the ∂0 X = ∅ case. τB (∂0 D, ∂0 D ∩ q −1 ∂0 E) = χ(∂0 X)τ (E, ∂0 ) + trB

(E ,∂ 0 )

τB (D, q

−1

∂0 E) = χ(X)τ (E, ∂0 ) +

(τE (∂0 D)) (E ,∂ 0 ) trB (τE (D))

Take the second formula minus the ﬁrst to prove the proposition. A useful special case is the case when D → E is the pull-back of a linear bundle over B. In this case, D is the ﬁber product of two bundles. Corollary 5.10 (product formula). Suppose that (F, ∂0 ) → (E, ∂0 ) → B and (X, ∂0 ) → (E , ∂0 ) → B are unipotent smooth bundle pairs, so that the second is an oriented linear S n or Dn bundle with ∂0 X = S n −1 , Dn −1 or ∅. Let D = E ×B E be the ﬁber product of these bundles and let ∂0 D = ∂0 E ×B E ∪ E ×B ∂0 E . Then τ (D, ∂0 ) = χ(X, ∂0 )τ (E, ∂0 ) + χ(F, ∂0 )τ (E , ∂0 ).

Proposition 5.11. For any unipotent bundle pair (E, ∂0 ) → B, we have τ (E, ∂0 ) + (−1)n τ (E, ∂1 ) = 2τ ev (E, ∂0 ), where n = dim F is the ﬁber dimension of E.

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Proof. Using the deﬁnition of relative torsion and additivity resulting from the decomposition ∂ v E = ∂0 E ∪ ∂1 E, we can write each term as a linear combination of higher torsions of the bundles DE, D∂0 E, D∂1 E, ∂ v ∂0 E = ∂ v ∂1 E. For example, 1 1 1 1 τ (E, ∂0 ) = τ (DE) + τ (D∂1 E) − τ (D∂0 E) − τ (∂ v ∂0 E). 2 4 4 2 The proposition follows immediately. 5.3. Further extension Given any higher torsion invariant τ initially deﬁned only for unipotent bundles with closed ﬁbers, we have extended the deﬁnition to all unipotent bundle pairs (E, ∂0 E). If we assume the main theorem, we can extend it further. Theorem 5.12. Any higher torsion invariant can be extended uniquely to all relatively unipotent smooth bundle pairs (E, ∂0 E), so that the relative versions of the axioms (additivity, horizontal additivity and transfer) hold. Proof. Suppose that (E, ∂0 ) → B is a relatively unipotent smooth bundle pair. Then the torsion τ (E, ∂0 ) can be deﬁned as follows. (i) Let D(ν) → E be the normal disk bundle and let ∂0 D = D(ν)|∂0 E. (ii) Embed ∂0 E ﬁberwise into the northern hemisphere of B × S N for N large. (iii) Thicken this embedding to a codimension zero embedding ∂0 D → B × S N . Then the union of B × DN +1 with D along ∂0 D is unipotent. (iv) Deﬁne the higher torsion of (D, ∂0 ) by τ (D, ∂0 ) := τ (B × DN +1 ∪ D). (v) Deﬁne the higher torsion of (E, ∂0 ) by E τ (E, ∂0 ) := τ (D, ∂0 ) − (s1 + s2 )trB (ch4k (ν)),

where s1 , s2 are as given in Subsection 4.2. These formulas show that the extension of τ to the relatively unipotent case is unique if it exists. However, this extension does exist in the cases τ = M2k and τ = τ2k which span all possibilities by the main theorem. Therefore, the extension is unique for all τ . The additivity, horizontal additivity and relative transfer formulas hold for M2k and τ2k when extended to the relatively unipotent case. Therefore, these axioms hold for all τ . 5.4. Rounding oﬀ corners Here is a brief description of how corners can be rounded oﬀ in a canonical way. First, we need a deﬁnition. Let Ck be one of the subsets of Rn given by the conditions x1 0, x2 0, · · · , xk 0 connected in that order by a sequence of ‘and’s and ‘or’s. For k = 0, there is only one such subset: C0 = Rn . But C4 = {x ∈ Rn | (x1 0 or (x2 0 and x3 0)) and x4 0} is one of the many possibilities when k = 4. Suppose that M is a compact topological n-manifold with boundary which has a smooth structure given by some embedding into a smooth open manifold of the same dimension. Then M is a smooth manifold with corners if every point x ∈ M has a neighborhood diﬀeomorphic to a neighborhood of 0 in some Ck . Consequences: (1) The product of manifolds with corners is a manifold with corners. (2) If M n is embedded in the interior of N n , then N × [0, 1] ∪ M × [0, 2] is also a manifold with corners. This is locally equivalent to unions such as E1 ∪ E2 in Proposition 5.7.

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In our model Ck , we deﬁned a positive vector at 0 to be any vector whose ﬁrst k coordinates are all positive. Note that a positive vector always points outward, that is, Ck ⊆ Ck + v for any positive vector v. A positive vector ﬁeld on the boundary of M is any smooth vector ﬁeld deﬁned in some neighborhood of ∂M which maps to a positive vector in the model at every point of ∂M . If we take the smooth ﬂow generated by this vector ﬁeld, we get a homeomorphism F : ∂M × (− , 0] ∼ =U sending (x, 0) to x for all x ∈ ∂M , where U is some neighborhood of ∂M in M . For each point x ∈ ∂M , we can ﬁnd a small open neighborhood Vx of x in ∂M and a homeomorphism gx : Wx → Vx , where Wx is an open set in Rn −1 and a continuous function hx : Wx → (− , 0), so that φx = F ◦ (gx , hx ) : Wx → Vx × (− , 0) → U is a smooth embedding. Since F is smooth in the second coordinate, φx will remain smooth if we alter hx by any operation which is smooth in x. Since ∂M is compact, it is covered by a ﬁnite number of the open sets Vi and we get a ﬁnite number of smooth embeddings φi : Wi → U which meet all of the ﬂow lines given by the vector ﬁeld. Let W be the abstract closed smooth n − 1 manifold given by pasting together the Wi by identifying points which map to the same point in ∂M . Take a smooth partition of unity ψi : Wi → [0, 1] on W . Then the continuous function h : W → (− , 0) given by h(x) = ψi (x)hi (x) gives a smooth embedding φ : W → U by φ(x) = F (g(x), h(x)). Furthermore, the set C = {F (g(x) × [h(x), 0]) | x ∈ W } is homeomorphic to ∂M × I and the closure of the complement of C in M is a smooth manifold with smooth boundary φ(W ). We call this the manifold obtained from M by smoothing the corners. This construction can be carried out for bundles E. We just need to choose the positive vector ﬁeld to be vertical. We can also extend any choice of data on E × 0 ∪ E × 1 to all of E × [0, 1] without any trouble: Extend the vector ﬁeld, then choose more embeddings φi , then patch them together using a partition of unity using the already given partition of unity near E × 0 and E × 1. So, the rounding of corners is canonical. 6. Stability of higher torsion Smooth bundles are stabilized by taking products with disks. The following special case of the product formula says that higher torsion is a stable invariant. Corollary 6.1 (stability of torsion). If (E, ∂0 ) → B is a unipotent smooth bundle pair, then so is (E × Dn , ∂0 E × Dn ) and the relative torsion is the same: τ (E × Dn , ∂0 E × Dn ) = τ (E, ∂0 ). If M is a compact smooth manifold, then we recall that a concordance of M is a diﬀeomorphism of M ×I which is the identity on M ×0∪∂M ×I. Let C(M ) be the space of concordances of M with the C ∞ topology: C(M ) = Diff (M × I rel M × 0 ∪ ∂M × I).

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The classifying space BC(M ) is the space of h-cobordisms of M rel ∂M . Recall that an hcobordism of M rel ∂M is a compact smooth manifold W with boundary ∂W = M × 0 ∪ ∂M × I ∪ M , where M is another compact smooth manifold with the same boundary as M , so that the inclusion M × 0 → W is a homotopy equivalence. A mapping B → BC(M ) is a smooth bundle over B whose ﬁbers are all h-cobordisms of M rel ∂M , so that the subbundle with ﬁber M × 0 ∪ ∂M × I is trivial. We will call such a bundle an h-cobordism bundle over B. There is a suspension map σ : C(M ) → C(M × I) which is highly connected when dim M is large by the concordance stability theorem ([16] or the last chapter of [19]). The limit is the stable concordance space P(M ) = lim C(M × I n ) →

which is well known to be an inﬁnite loop space. Therefore, the set [B, BP(M )] of homotopy classes of maps from B to the classifying space BP(M ) is an additive group. By the concordance stability theorem, this group is isomorphic to [B, BC(M × I n )] for suﬃciently large n. Proposition 6.2. For any h-cobordism bundle E → B, let ∂0 E be the trivial subbundle with ﬁber M × 0 ∪ ∂M × I. Then the higher torsion invariant E → τ (E, ∂0 ) gives an additive map τ : [B, BC(M × I n )] → R. Proof. This is an immediate consequence of the horizontal additivity of τ (Proposition 5.8) since the H-space structure on the h-cobordism space BC(M × I n ) is given by lateral union that is, the sum of two mappings B → C(M × I n ) is given by lateral union of the corresponding h-cobordism bundles followed by rescaling. (The lateral union is an h-cobordism of M × I n −1 × [0, 2]. The last coordinate needs to be rescaled down to [0, 1].) Corollary 6.3. Suppose that Eα is a collection of h-cobordism bundles which spans the Q vector space [B, BC(M × I n )] ⊗ Q. Suppose also that τ (Eα , ∂0 ) = 0 for all α. Then τ (E, ∂0 ) = 0 for any h-cobordism bundle E classiﬁed by a map B → BC(M × I n ). 7. Computation of higher torsion We will now show how the higher torsion invariants of unipotent bundles can be computed in many cases given the values of the parameters s1 , s2 . We recall that these parameters are given by τ (S n (λ)) = 2sn ch4k (λ), where S n (λ) is the S n bundle associated to a complex line bundle λ over B. 7.1. Torsion of disk and sphere bundles Theorem 7.1. The higher torsion of the Dn -bundle Dn (ξ) associated to an SO(n)-bundle ξ over B is given by τ (Dn (ξ)) = (s1 + s2 )ch4k (ξ). Proof. For n = 2, this is by deﬁnition of τ in the boundary case: τ (D2 (ξ)) = 12 τ (S 2 (ξ)) + 12 τ (S 1 (ξ)) = (s2 + s1 )ch4k (ξ).

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The general case follows from the product formula (Corollary 5.10) and the splitting principle. If n = 2m + 1, then we may assume by the splitting principle that ξ is a direct sum of m complex line bundles λi and a trivial real line bundle. Then τ (Dn (ξ)) = τ D2 (λ1 ) ×B · · · ×B D2 (λm ) ×B (B × I) = (s1 + s2 )

ch4k (λi ) = (s1 + s2 )ch4k (ξ)

by the product formula. The even case is similar. From the formula τ (Dn (ξ)) = 12 τ (S n ) + 12 τ (S n −1 (ξ)) = (s1 + s2 )ch4k (ξ), we get the following by induction on n. Corollary 7.2. For n > 0, the higher torsion of the S n -bundle S n (ξ) associated an SO(n + 1)-bundle ξ over B is given by τ (S n (ξ)) = 2sn ch4k (ξ), where sn depend only on the parity of n. Comparing this with Proposition 9.2 and Theorem 9.6 below, we get the following. Lemma 7.3. Let k 1 and let a, b ∈ R be given by a :=

s1 + s2 , (2k)!

b :=

(−1)k +1 2s1 . ζ(2k + 1)

Then τ (E) = aM2k (E) + bτ2k (E) for E any oriented linear sphere and disk bundle over B. Remark 7.4. If the value of τ on a bundle E is determined by the known values of τ on disk and sphere bundles, then this lemma implies that τ (E) = aM2k (E) + bτ2k (E) for that bundle. 7.2. Morse bundles Given a Morse function f : (M, ∂0 M ) → (I, 0), a compact n-manifold M will be decomposed as a union of handles Di × Dn −i attached along S i−1 × Dn −i to the union of lower handles and the base ∂0 M × [0, ]. Each such handle has a critical point at its center with index i. The core of the handle is Di × 0. This is also the union of trajectories of the gradient of f (with respect to some metric on M ) which converge to the critical point. The tangent plane is the negative eigenspace of the second derivative D2 f at the critical point. Suppose that (E, ∂0 ) → B is a smooth bundle pair and f : E → I = [0, 1] is a ﬁberwise Morse function with f −1 (0) = ∂0 E and with distinct critical values. In other words, for each b ∈ B, the restriction fb : (Fb , ∂0 ) → (I, 0) is a Morse function with critical points x1 (b), . . . , xm (b) having critical values fb (x1 ) < fb (x2 ) < · · · < fb (xm ). For 1 j m, let ξj be the negative eigenspace bundle of the ﬁberwise second derivative of f along xj . Let ηj be the complementary positive eigenspace bundle. Then E has a ﬁltration E = Em ⊃ Em −1 ⊃ · · · ⊃ E0 , where E0 ∼ = ∂0 E × I and Ej = Ej −1 ∪ D(ξj ) ×B D(ηj ), where Ej −1 ∩ D(ξj ) ×B D(ηj ) = S(ξj ) ×B D(ηj ). By additivity we get the following.

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Lemma 7.5. If ∂0 E → B is unipotent and the bundles ξj , ηj are oriented, then (E, ∂0 ) → B is a unipotent bundle pair with torsion invariant τ ((D(ξj ), S(ξj )) ×B D(ηj )). τ (E, ∂0 ) = Each summand in the above lemma can be determined using the product formula (Corollary 5.10) as follows. Lemma 7.6. The value of τ on the ﬁber product of an oriented linear disk bundle D(η) and the oriented relative i-disk bundle (D(ξ), S(ξ)) is given by τ ((D(ξ), S(ξ)) ×B D(η)) = (−1)i τ (D(η)) + τ (D(ξ), S i−1 (ξ)) = (−1)i (s1 + s2 )ch4k (η) + (si − si−1 )ch4k (ξ). Remark 7.7. Since si −si−1 = (−1)i (s2 −s1 ), the even and odd parts of the above formula are: τ ev ((D(ξ), S(ξ)) ×B D(η)) = (−1)i s2 (ch4k (η) + ch4k (ξ)) τ od ((D(ξ), S(ξ)) ×B D(η)) = (−1)i s1 (ch4k (η) − ch4k (ξ)). Putting these together, we get the following theorem which is a mild improvement over the obvious. Namely, the negative eigenspace bundles need not be oriented. Of course, the sum ξj ⊕ ηj of negative and positive eigenspace bundles must be oriented since it is the vertical tangent bundle along the jth component of the Morse critical set. Theorem 7.8. Suppose that (E, ∂0 ) → B is a unipotent smooth bundle pair and f : (E, ∂0 E) → (I, 0) is a ﬁberwise Morse function with distinct critical values. Let ξj , ηj be the negative and positive eigenspace bundles associated to the jth critical point whose index we denote by i(j). Then (−1)i(j ) (s1 + s2 )ch4k (ηj ) + (−1)i(j ) (s2 − s1 )ch4k (ξj ). τ (E, ∂0 ) = Proof. Suppose ﬁrst that the bundles ξj , ηj are oriented. Then the two lemmas above apply to prove the theorem. If these bundles are not oriented, then there is a ﬁnite covering of B so that, on the pull-back E, the function E → E → I is a Morse function with B oriented eigenspace bundles ξj , ηj which are pull-backs of ξj , ηj . Thus, the theorem applies to However, the induced map in real cohomology H ∗ (B; R) → H ∗ (B; R) is a monomorphism E. and the two expressions in our theorem are elements of H 4k (B; R) which go to the same element R). So, they must be equal. of H 4k (B; 7.3. Hatcher’s example One crucial example of a Morse bundle to which the above theorem holds is Hatcher’s construction. This constructs an exotic disk bundle over B = S n out of an element of the kernel of the J-homomorphism J : πn −1 O → πns −1 (S 0 ). B¨okstedt [5] interpreted this as a mapping from G/O to the stable concordance space of a point H(∗) = BP(∗). This means that a mapping B → G/O gives an exotic disk bundle over B. Since G/O is the homotopy ﬁber of the map BO → BG, a map B → G/O is unstably the same as an n-plane bundle ξ : B → SO(n) together with a homotopy trivialization of the

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associated sphere bundle, that is, we have a ﬁber homotopy equivalence g : S n −1 (ξ) B × S n −1 . We write this as a family of homotopy equivalences gt : Stn −1 (ξ) → S n −1 , t ∈ B and extend to the disk Dtn (ξ) (the ﬁber over t ∈ B of the disk bundle Dn (ξ)) g t : Dtn (ξ), Stn −1 (ξ) → (Dn , S n −1 ). Assuming that n and m are large enough, we can lift g t up to a family of embeddings gt : Dtn (ξ), Stn −1 (ξ) → (Dn , S n −1 ) × Dm . If we let η be the complementary bundle to ξ, we can extend this to a family of codimension zero embeddings Gt : Dtn (ξ), Stn −1 (ξ) × Dtm (η) → (Dn , S n −1 ) × Dm . n Embed Dn into a larger n-disk D+ by adding an external collar along ∂Dn = S n −1 , that is, n ∼ n n −1 × I). For each t ∈ B, the image of Gt together with the thickened collar D+ = D ∪ (S (S n −1 × I) × Dm forms an n + m disk (if the corners are rounded). ∆t (ξ) := Gt Dtn (ξ) × Dtm (η) ∪ (S n −1 × I) × Dm .

This is the ﬁber over t ∈ B of a smooth disk bundle ∆(ξ) → B which we call a Hatcher disk S n −1 × I Dm

G t (D tn (ξ) × D tm (η)) n D+

Dn

bundle. By Theorem 7.8, the torsion of Hatcher’s bundle is: τ (∆(ξ)) = (−1)n (s1 + s2 )ch4k (η) + (−1)n (s2 − s1 )ch4k (ξ). Since ch4k (η) + ch4k (ξ) = 0 (ξ ⊕ η being trivial), this simpliﬁes to the following. Lemma 7.9. The higher torsion of Hatcher’s example is given by τ (∆(ξ)) = (−1)n +1 2s1 ch4k (ξ) where s1 is deﬁned in Subsection 4.2. As pointed in [17], this calculation gives another proof of B¨ okstedt’s theorem. Theorem 7.10 [5]. Hatcher’s construction gives a rational homotopy equivalence

→ H(∗) = BP(∗). ∆ : G/O − Proof. By Farrell and Hsiang [12], we know that H(∗) is rationally homotopy equivalent to BO. Therefore, it suﬃces to show that ∆ induces a rational isomorphism on homotopy groups. But the rational homotopy groups of these spaces are zero except in degree 4k where they have

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rank 1. Thus, it suﬃces to show that ∆ is rationally nontrivial on π4k . This follows from the od and Lemma 7.9 when applied to a nontrivial existence of nontrivial odd torsion, such as τ2k element of the kernel of J : π3k −1 O → π3k −1 G. Lemma 7.9 also implies the following. Lemma 7.11. For any smooth oriented disk bundle D → B, we have τ (D) = aM2k (D) + bτ2k (D), where a, b are given in Lemma 7.3 Proof. By Corollary 6.1, we can stabilize D without changing the value of the three higher torsion invariants τ (D), τ2k (D), M2k (D). After stabilizing, D becomes an n + 1 disk bundle over B, where n > dim B and we can ﬁnd a smooth section B → ∂ v D. We can thicken this up to get a linear n disk bundle D(ξ) ⊆ ∂ v D. Suppose ﬁrst that ξ is a trivial bundle. Then (D, D(ξ)) is an h-cobordism bundle classiﬁed by an element of [B, BC(Dn )] which is rationally generated by Hatcher’s examples by B¨okstedt’s theorem. So, by Corollary 6.3, the ‘diﬀerence torsion’ τ − aM2k − bτ2k is equal to zero on D by Lemma 7.9 and the choice of a, b. If the linear bundle ξ is nontrivial, we simply take the ﬁber product of D with the disk bundle of the complementary bundle η. The argument above applies to the higher torsion of D ×B D(η) which is related to the torsion of D by: τ (D ×B D(η)) = τ (D) + τ (D(η)). Since the lemma holds for D ×B D(η) and D(η), it holds for D. 8. Proof of the main theorem We have enough calculations to prove the main theorem, namely that τ = aM2k + bτ2k or, equivalently, that the diﬀerence is zero. 8.1. The diﬀerence torsion We deﬁne the diﬀerence torsion by τ δ := τ − aM2k − bτ2k . This is a linear combination of higher torsion invariants and therefore, a higher torsion invariant. Furthermore, this new invariant has the property that s1 = s2 = 0. So, by Lemma 7.11 and Corollary 7.2, we have the following. Lemma 8.1. The diﬀerence torsion is zero on all disk bundles and linear sphere bundles. Lemma 8.2. If q : D → E is an oriented linear disk bundle, then the diﬀerence torsion of any unipotent bundle pair (E, ∂0 ) is equal to that of (D, ∂0 ) as a bundle over B, where ∂0 D = q −1 (∂0 E)), that is, τBδ (D, ∂0 ) = τ δ (E, ∂0 ). Proof. By the relative transfer formula, we have

δ E τE (D) . τBδ (D, ∂0 ) = τ δ (E, ∂0 ) + trB

But τEδ (D) = 0, since D is a linear disk bundle over E. Lemma 8.3. The diﬀerence torsion is zero on any unipotent bundle pair (E, ∂0 ) → B whose ﬁbers (F, ∂0 ) are h-cobordisms.

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Proof. Choose a smooth ﬁbered embedding of ∂0 E into B × DN for some large N . Let ∂0 D be the normal disk bundle of ∂0 E in B × DN . Since E is ﬁber homotopy equivalent to ∂0 E, the linear disk bundle ∂0 D extends to a linear disk bundle D over E. By the above lemma, τ δ (E, ∂0 ) = τ δ (D, ∂0 ). So, it suﬃces to show that τ δ (D, ∂0 ) = 0. However, this follows by additivity since the union of B × DN with D along the inclusion ∂0 D → B × DN is a disk bundle over B: τ δ (D, ∂0 ) = τ δ ((B × DN ) ∪ D) = 0.

Lemma 8.4 (First Main Lemma). of unipotent smooth bundle pairs.

The diﬀerence torsion τ δ is a ﬁber homotopy invariant

Proof. Suppose that (E1 , ∂0 ) and (E2 , ∂0 ) are unipotent smooth bundle pairs over B that are ﬁber homotopy equivalent. Then we want to show that τ δ (E1 , ∂0 ) = τ δ (E2 , ∂0 ). If we replace (E2 , ∂0 ) by a large dimensional disk bundle (D2 , ∂0 ), we can approximate the ﬁber homotopy equivalence by a ﬁberwise smooth embedding g : (E1 , ∂0 ) → (D2 , ∂0 ). Let D1 be the normal disk bundle of E1 in D2 . Then the complement of D1 in D2 is a unipotent ﬁberwise h-cobordism which has trivial τ δ by the previous lemma. Thus, τ δ (D1 , ∂0 ) = τ δ (D2 , ∂0 ) by additivity. By Lemma 8.2, this implies that τ δ (E1 , ∂0 ) = τ δ (E2 , ∂0 ). 8.2. Vanishing of the ﬁber homotopy invariant The ﬁnal step in the proof of Theorem 4.4 is to show the following. Lemma 8.5 (Second Main Lemma). Any higher torsion invariant τ δ which is also a ﬁber homotopy invariant of unipotent smooth bundle pairs must be zero. This is the crucial step in the proof of the main theorem which I completed with the help of John Klein and Bruce Williams. The proof is in three parts. First, we show that τ δ satisﬁes excision. Then we use obstruction theory to reduce to the case when the ﬁbers are rationally trivial. Finally, we use immersion theory to reduce to the h-cobordism case (Lemma 8.3), where it holds by the assumption that τ δ is a ﬁber homotopy invariant. 8.2.1. Excision. Since τ δ is a ﬁber homotopy invariant, it is well deﬁned on any ﬁbration pair (Z, C) → B with ﬁber (X, A) which is unipotent in the sense that H∗ (X; Q) and H∗ (A; Q) are unipotent as π1 B-modules and smoothable in the sense that it is ﬁber homotopy equivalent to a smooth bundle pair (E, ∂0 ) with compact manifold ﬁber (F, ∂0 ). By deﬁnition, the relative torsion is related to the absolute torsion by τ δ (Z, C) = τ δ (Z) − τ δ (C).

(8.1)

Lemma 8.6. Given unipotent smoothable ﬁbrations (Z, C) and Y over B and any continuous mapping f : C → Y over B, the union Y ∪C Z is also unipotent and smoothable with τ δ (Y ∪C Z) = τ δ (Y ) + τ δ (Z, C).

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Proof. Unipotence follows from the Mayer–Vietoris sequence for the ﬁber homology of Y ∪C Z. The formula is additivity of torsion. Smoothing of Y ∪C Z, very similar to the construction in Subsection 5.3, goes as follows. First, choose smoothings (E, ∂0 ) and E for (Z, C) and Y which are both codimension 0 submanifolds of B × Rn . The continuous map f : C → Y gives a map ∂0 E → E over B which is ﬁberwise homotopic to a smooth embedding φ : ∂0 E → E × Dm . Since the normal bundle of φ(∂0 E) in E × Dm is trivial, we have a codimension 0 embedding φ : ∂0 E × Dm +1 → E × Dm . The smoothing of Y ∪C Z is E × Dm +1 ∪φ E × Dm × I with corners rounded. Lemma 8.7. For any unipotent smoothable ﬁbration pair (Z, C) the ﬁberwise smash Z/B C = B ∪C Z and ﬁberwise suspension ΣB (Z/B C) → B, with ﬁbers X/A and Σ(X/A) respectively, are unipotent and smoothable with τ δ (Z/B C) = τ δ (Z, C) = −τ δ (ΣB (Z/B C)). Remark 8.8. By taking an iterated ﬁberwise suspension Σm B (Z/B C) → B, this lemma reduces the proof of Lemma 8.5 to the case where the ﬁber is highly connected and pointed. However, it would be incorrect to take the ﬁberwise limit and assume that ﬁbers are contractible. Proof. The coﬁbration sequence X/A ∨ X/A → X/A → Σ(X/A) and additivity give: τ δ (ΣB (Z/B C)) = τ δ (Z/B C) − 2τ δ (Z/B C) = −τ δ (Z/B C). This is equal to −τ δ (Z, C) by the previous lemma. 8.2.2. Making ﬁbers rationally trivial. Following suggestions of Bruce Williams, we will reduce the proof of Lemma 8.5 to the case of smoothable ﬁbrations with rationally trivial ﬁbers. Such ﬁbrations will necessarily be unipotent. The proof was reorganized following suggestions of the referee. Lemma 8.9. Suppose that X → E → B is a unipotent ﬁbration with section, where B and X have the homotopy type of ﬁnite complexes. Then there is a ﬁnite ﬁltration of unipotent ﬁbrations over B Σm B E = E 0 → E 1 → · · · → Ek starting with an iterated ﬁberwise suspension Σm X → Σm B E → B of E satisfying the following conditions. (i) The map Ek → B is a rational weak homotopy equivalence. (ii) There is a coﬁbration sequence of ﬁbrations over B: αj

B × S n j −→ Ej → Ej +1 . (iii) The total rank of the rational homology of the ﬁber of Ej +1 is one less than that of Ej . Furthermore, if the original ﬁbration E → B is smoothable, then so is each Ej → B. Remark 8.10. By Lemma 8.7, if E → B is smoothable, then so is E0 = Σm B E. Each Ej would also be smoothable by induction on j since Ej +1 = Ej ∪B ×S n j B × Dn j +1 . Since B × (Dn j +1 , S n j ) is a trivial bundle pair, its torsion is zero. So, τ δ (Ek ) = τ δ (Ek −1 ) = · · · = τ δ (E0 ) = (−1)m τ δ (E).

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Therefore, the statement that τ δ (E) = 0 for all smoothable unipotent E is reduced to the case of the bundle Ek → B which is a rational weak equivalence. Proof. By induction on the total rank of the rational homology of the ﬁber X, it suﬃces to construct a mapping α = α0 : B × S n 0 → E0 = Σm B E over B, so that the induced homomorphism (substituting n0 = n + m) α∗ : H n +m (S n +m ; Q) → H n +m (Σm X; Q) ∼ = H n (X; Q) is nonzero. We will construct the mapping α to be pointed on ﬁbers (that is, α preserves the section). Since B is a ﬁnite complex, the existence of α for some m 0 is equivalent to the existence of a section of the bundle (Ωn Q)B E → B, which is associated to E → B with ﬁber Ωn QX = colimΩn +m Σm X. We will use obstruction theory to construct this section. Suppose that sk is a section of the bundle (Ωn Q)B E over the k-skeleton B k of B. Then we have an obstruction class θ(sk ) ∈ Hk +1 B; πns +k X which is the obstruction to extending sk −1 = sk |B k −1 to B k +1 . The key point is that πns +k X is rationally isomorphic to H n +k (X). Therefore, if we take n to be maximal so that H n (X; Q) = 0 (such an n exists since X is equivalent to a ﬁnite complex), these obstruction groups will be rationally trivial for k 1. But we still need to construct s1 and we need to show that rational triviality of the obstruction group is suﬃcient. Since πns (X) ⊗ Q ∼ = H n (X; Q) is unipotent, it contains a nonzero element which if ﬁxed under the action of π1 B. A nonzero multiple of this element lifts to a stable homotopy class β ∈ πns (X). Since X is homotopy ﬁnite, the torsion subgroup T of πnS (X) is ﬁnite. The group π1 B permutes the elements in the coset β + T . Therefore, the sum, say γ ∈ πns (X), of all elements in this coset is ﬁxed under the action of π1 B and is still rationally nontrivial. Note that any nonzero multiple of γ has the same properties. These properties imply that γ deﬁnes a component (Ωn Qγ )B E of (Ωn Q)B E with connected ﬁber Ωn Qγ (X). We let s1 be any section of this subbundle over the 1-skeleton B 1 . Given sk for k 1, we need to kill the obstruction class θ(sk ) by a geometric construction. Let be the order of the ﬁnite group πns +k (X) and let f be the operation on the ﬁbration (Ωn Q)B E given by right multiplication by a stable self-map of S n of degree . This operation sends (Ωn Qγ )B E to (Ωn Qγ )B E. We claim that θ(f ◦ sk ) = 0 and therefore, we get a section sk +1 of (Ωn Qγ )B E over B k +1 . To see that the obstruction vanishes, we look at the operation f on ﬁbers. Composing with the homotopy equivalence Ωn Q0 X Ωn Qγ X given by addition with γ, we get a map

f

Ωn Q0 X − → Ωn Qγ X −→ Ωn Qγ X which sends x to f (x ∨ γ) f (x) ∨ γ. But the map f is multiplication by on πk Ωn Q0 X = πns +k X and therefore is zero on πk both on Ωn Q0 X and on Ωn Qγ X. This implies that θ(f ◦ sk ) = f θ(sk ) = 0 as claimed. 8.2.3. Proof for rationally trivial ﬁbers. The statement and proof of the following lemma which is the last step in the proof of Lemma 8.5 follows several discussions I had with John Klein. Lemma 8.11. τ δ (Z) = 0 for smoothable ﬁbrations with rationally trivial ﬁbers.

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Proof. We use the fact that a ﬁbration Z → B with rationally trivial ﬁber X is necessarily unipotent. This implies the existence of a universal torsion class which, by immersion theory, can be lifted to a space of h-cobordisms where it is trivial by assumption. The ﬁrst step is to construct a universal class. By ﬁberwise suspension using Lemma 8.7 and Remark 8.8, we may assume that the map Z → B has a section and the ﬁber X is simply connected. We choose a smooth bundle E → B ﬁber homotopy equivalent to Z. By embedding E in B × Rn for large n and taking a tubular neighborhood, we may assume that the vertical tangent bundle of E is trivial and the ﬁber is a compact n-manifold M n embedded in Rn so that both M X and ∂M are simply connected. Also, the image of the section B → E will have a neighborhood which is a trivial disk bundle D ∼ = B × Dn . Then (E, D) is classiﬁed by a map φ : B → BDiff (M rel Dn ). Since the universal bundle over BDiff (M rel Dn ) with ﬁber M is unipotent, there is a universal diﬀerence torsion class τ δ (M ) ∈ H 4k (BDiff (M rel Dn ); R) which pulls back to τ δ (E) = φ∗ (τ δ (M )). This is a general fact about characteristic classes which follows from the universal coeﬃcient theorem and the fact that every element of H4k (BDiff (M rel Dn )) is supported on a ﬁnite subcomplex. We will show that this universal class is trivial. This will follow from two statements which we claim are true. First, we need to go to the universal cover of BDiff (M rel Dn ) which is (M rel Dn ) = BDiff0 (M rel Dn ), BDiff where Diff0 (M rel Dn ) is the identity component of Diff (M rel Dn ). Claim 1. π0 Diff (M rel Dn ) ∼ = π1 BDiff (M rel Dn ) is ﬁnite for large n. Suppose for a moment that this is true. Then BDiff0 (M rel Dn ) will be a ﬁnite covering of BDiff (M rel Dn ) and the covering map BDiff0 (M rel Dn ) → BDiff (M rel Dn ) will induce a monomorphism in rational (and real) cohomology. Consequently, it will suﬃce to show that the pull-back of the universal class τ δ (M ) to BDiff0 (M rel Dn ) is zero. Next, we will reduce to the h-cobordism case. Let M0 be M minus an open collar neighborhood of ∂M . We may assume that M0 contains the disk Dn . So Diff (M rel M0 ) ⊆ Diff (M rel Dn ). By a theorem of Cerf [8], we know that Diff (M rel M0 ) ∼ = Diff (∂M × I rel ∂M × 0) is connected. Therefore, Diff (M rel M0 ) ⊆ Diff0 (M rel Dn ). Claim 2. For suﬃciently large n, the induced map of classifying spaces p : BDiff (M rel M0 ) → BDiff0 (M rel Dn ) is rationally 4k-connected in the sense that its homotopy ﬁber is rationally 4k − 1 connected. Suppose for a moment that this second claim is also true. Then the mapping p induced a monomorphism in degree 4k rational (and real) cohomology. So, τ δ is zero on BDiff0 (M rel Dn) if and only if it is zero on BDiff (M rel M0 ). However, the universal bundle over BDiff (M rel M0 ) with ﬁber M contains a trivial subbundle with ﬁber M0 as a deformation retract. Being a ﬁber homotopy invariant, this forces τ δ to be zero as claimed. It remains to prove the two claims. We prove them simultaneously assuming that n = dim M is suﬃciently large.

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Claim 1 is that π0 Diff (M rel Dn ) is ﬁnite. Consider the ﬁbration sequence π

Diff (M rel M0 ) → Diff (M rel Dn ) − → Embh (M0 , M rel Dn ),

(8.2)

where Emb (M0 , M rel D ) is the space of smooth embeddings ψ : M0 → intM which are the identity on Dn and so that ψ is a homotopy equivalence. By excision and Whitehead’s theorem, the complement of the image of ψ is an h-cobordism of ∂M and therefore, by the h-cobordism theorem, ψ will extend to a diﬀeomorphism ψ : M → M . Thus, the restriction map π is surjective. π is a ﬁbration by Cerf ([7], Appendix). Thus, h

n

π0 Diff (M rel Dn ) ∼ = π0 Embh (M0 , M rel Dn ). Claim 1 is that this group is ﬁnite. The ﬁbration sequence (8.2) gives another ﬁbration sequence p

Emb0 (M0 , M rel Dn ) → BDiff (M rel M0 ) − → BDiff0 (M rel Dn ) where, as before, the subscripts 0 denote the identity components of the embedding and diﬀeomorphism spaces. Claim 2 is that p is rationally 4k-connected. For this, it suﬃces to show that πi Emb0 (M0 , M rel Dn ) is ﬁnite for all i < 4k. Therefore, Claims 1 and 2 together follow from the statement that πi Embh (M0 , M rel Dn ) is ﬁnite for all i < 4k. When n is large, the homotopy dimension of M0 will be much smaller than n. Therefore, by transversality we have that the embedding space is homotopy equivalent in low degrees to the corresponding space of immersions: πi Embh (M0 , M rel Dn ) ∼ = πi Immh (M0 , M rel Dn ). By immersion theory, Immh (M0 , M rel Dn ) is homotopy equivalent in low degrees to the product of the space of all pointed homotopy equivalences M0 → M and the space of all pointed maps M0 → O(n). Since M0 is the suspension of a rationally trivial ﬁnite complex, both of these mapping spaces have ﬁnitely many components each of which is rationally trivial and thus, the low degree homotopy groups of the immersion space are ﬁnite as claimed. This completes the proof of Lemma 8.5 which implies the Main Theorem 4.4. 9. Existence of higher torsion In this section, we show that higher Miller–Morita–Mumford classes M2k and higher Franz– Reidemeister torsion τ2k are linearly independent higher torsion invariants. 9.1. MMM classes If p : E → B is any smooth bundle with compact ﬁber F , the Miller–Morita–Mumford (MMM) classes M2k are deﬁned to be the integral cohomology classes given by E M2k (E) := trB ((2k)!ch4k (T v E)) ∈ H 4k (B; Z)

for k 1, where T v E is the vertical tangent bundle of E and ch4k (ξ) := 12 ch4k (ξ ⊗ C) is the Chern character. In degree 0, this formula gives a half integer: n n M0 (E, ∂0 ) = χ(F, ∂0 ) = (χ(F ) − χ(∂0 F )), 2 2 where n = dim F . The properties of the transfer ([2]) give the following properties of the MMM classes. Proposition 9.1. (i) (stability) M2k (E × I) = M2k (E) for k > 0.

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(ii) (additivity) If E1 , E2 are smooth bundles over B with the same vertical boundary E1 ∩ E2 = ∂ v E1 = ∂ v E2 , then M2k (E1 ∪ E2 ) = M2k (E1 ) + M2k (E2 ) − M2k (∂ v E1 × I). (iii) (transfer) If q : D → E is a bundle with ﬁber X, then the MMM classes M2k (D)B ∈ H 4k (B) and M2k (D)E ∈ H 4k (E) are related by E M2k (D)B = χ(X)M2k (E) + trB (M2k (D)E ).

Proof. The ﬁrst two are easy. The transfer formula follows from the formula TBv D = TEv D ⊕ q ∗ T v E, the additivity of ch4k and the fact that trED ◦ q ∗ is multiplication by χ(X). D (2k)!ch4k TBv D M2k (D)B = trB E D E D ∗ = trB trE (2k)!ch4k TEv D + trB trE (q (2k)!ch4k (T v E)) E = trB (M2k (D)E ) + χ(X)M2k (E)

Proposition 9.2. If S 2n (ξ) → B is the S 2n -bundle associated to an SO(2n + 1)-bundle ξ over B, then M2k (S 2n (ξ)) = 2(2k)!ch4k (ξ). Proof. This follows from additivity and the observation that M2k (Dm (ξ)) = (2k)!ch4k (ξ) for any linear disk bundle Dm (ξ). Theorem 9.3. The Miller–Morita–Mumford class M2k is a nontrivial even higher torsion invariant. Proof. If we consider additivity (Proposition 9.1, part (ii)) in the case E1 = E2 , we get M2k (DEi ) = 2M2k (Ei ) − M2k (∂ v Ei × I). The additivity axiom follows from this equation and Proposition 9.1, part (ii) in the case E1 = E2 . The transfer axiom is a special case of the transfer formula of Proposition 9.1, part (iii). 9.2. Higher FR-torsion Higher Franz–Reidemeister (FR)-torsion invariants are real characteristic classes τ2k (E, ∂0 ) ∈ H 4k (B; R) for k 1 deﬁned for smooth bundle pairs (F, ∂0 ) → (E, ∂0 ) → B which are relatively unipotent in the sense that H∗ (F, ∂0 F ; Q) is unipotent as a π1 B module. (See [17].) Theorem 9.4 [17]. (i) (additivity) If (E1 , ∂0 ), (E2 , ∂0 ) are relatively unipotent smooth bundles over B with E1 ∩ E2 = ∂0 E2 ⊆ ∂1 E1 , then τ2k (E1 ∪ E2 , ∂0 E1 ) = τ2k (E1 , ∂0 ) + τ2k (E2 , ∂0 ).

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(ii) (stability) If q : D → E is a linear disk bundle and ∂0 D = q −1 (∂0 E), then the higher FR-torsion of (D, ∂0 ) as a bundle pair over B is equal to the higher torsion of (E, ∂0 ): τ2k (D, ∂0 )B = τ2k (E, ∂0 ). Higher FR-torsion and the higher MMM classes are related by the following theorem. Theorem 9.5 [19]. ﬁbers and k 1, then

If E → B is a unipotent smooth bundle with closed even dimensional τ2k (E) = (−1)k

Here n = dim F and ζ(2k + 1) =

m

1

ζ(2k + 1) M2k (E). 2(2k)!

1 m2k + 1

is the Riemann zeta function.

In other words, M2k is proportional to the even part of τ2k . The following calculation shows that the odd part of τ2k has the same size (but opposite sign). Theorem 9.6 [17]. FR-torsion

The S n -bundle associated to an SO(n + 1)-bundle ξ over B has higher τ2k (S n (ξ)) = (−1)n +k ζ(2k + 1)ch4k (ξ).

Remark 9.7. It follows from the stability of τ2k that τ2k = 0 on all linear disk bundles over B. Corollary 9.8. Higher Franz–Reidemeister torsion τ2k satisﬁes the additivity axiom (3.1) and the transfer axiom (3.2) and is therefore, a higher torsion invariant. Proof. The additivity axiom follows from the additivity of τ2k . The transfer axiom requires the following for any smooth unipotent bundle E → B with closed ﬁber F and any oriented linear sphere bundle S m → S m (ξ) → E over E. E (τ2k (S m (ξ))E ) τ2k (S m (ξ))B = χ(S m )τ2k (E) + trB

(9.1)

We prove this in each of the four cases depending on the parity of m and n = dim F . If n, m have the same parity, then the transfer formula (9.1) is equivalent to the transfer formula for M2k which we already proved. (If n, m are both odd, the RHS is zero for both τ2k and M2k . This implies that the LHS is zero in the odd–odd case for both since they are proportional by Theorem 9.5.) If n is even and m is odd, then, by the previous case, we know that E (τ2k (S m +1 (ξ))E ). τ2k (S m +1 (ξ))B = 2τ2k (E) + trB

The LHS of this is τ2k (S m +1 (ξ))B = 2τ2k (Dm +1 (ξ))B − τ2k (S m (ξ))B = 2τ2k (E) − τ2k (S m (ξ))B

(9.2)

by Theorem 9.4. In other words, E τ2k (S m (ξ))B = −trB (τ2k (S m +1 (ξ))E ).

But this is equal to the RHS of (9.1) since χ(S m ) = 0 and τ2k (S m +1 (ξ)E ) = −τ2k (S m (ξ)E ) by Theorem 9.6. In the case n odd and m even, the LHS of (9.2) is zero as we noted in the analysis of the E = 0. odd–odd case. Therefore, the RHS of (9.2) is also zero. (9.1) follows since trB

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Corollary 9.9.

185

The odd part of higher FR-torsion is given by od τ2k (E) = τ2k (E) − (−1)k

ζ(2k + 1) M2k (E). 2(2k)!

Remark 9.10. A corollary of the main theorem is that any odd torsion invariant must be proportional to the above expression. In particular, it is expected that the analytic torsion od . For the of Bismut and Lott [4] is an odd torsion invariant and therefore proportional to τ2k Bismut–Goette normalization of analytic torsion [3], we expect: BG τ2k =

k! od τ . (2π)k 2k

9.3. Tangential and exotic torsion Any higher torsion invariant in degree 4k can be written uniquely as τ = aM2k + bτ2k . T

We call τ = aM2k the tangential component of τ and τ x = bτ2k the exotic component of τ . In terms of the scalars s1 , s2 for τ , the tangential component τ T is given by E (ch4k (T v E)) τ T (E) = (s1 + s2 )trB

for any unipotent bundle E. Note that this depends only on the homotopy type of the bundle E and its vertical tangent bundle. Since τ T is an even torsion invariant which is determined by s1 + s2 , it follows from Theorem 7.1 that τ T is also characterized by the following proposition. Proposition 9.11. The tangential component τ T is the unique even torsion invariant, so that τ T (Dn (ξ)) = τ (Dn (ξ)) ∈ H 4k (B; R) for any oriented linear disk bundle Dn (ξ) → B. By the boundary case transfer formula, this gives the following characterization of the exotic component τ x = τ − τ T . Proposition 9.12. τ x (E) is independent of the vertical tangent bundle of E in the sense that τ x (E) = τ x (D) for any oriented linear disk bundle D over E. Thus, the exotic component of τ can also be given by τ x (E) = τ (D(ν)), where ν is the vertical normal bundle of E. (ν is a vector bundle over E which is complementary to T v E.) Any higher torsion invariant having this property (that τ (E) = τ (D)) will be called exotic. The main theorem can then be restated as follows. Theorem 9.13. Any exotic higher torsion invariant is a scalar multiple of higher FRtorsion. The Dwyer–Weiss–Williams (DWW) smooth torsion as deﬁned in [11] is ‘exotic’ for the following simple reason. They take their bundle E and embed it in B ×RN for some large N and

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take a regular neighborhood. This kills the vertical tangent bundle and therefore trivializes the tangential component. For this reason, we believe that the DWW smooth torsion is proportional to higher FR-torsion whenever it is deﬁned. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

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Kiyoshi Igusa Department of Mathematics Brandeis University P O Box 9110 Waltham, MA 02454-9110 USA [email protected]

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