Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and Bo Eckmann, Z0rich
161 James Stasheff The Institute for Advanced Study Princeton / NJ / USA
H-Spaces from a Homotopy Point of View
$ Springer-Verlag Berlin-Heidelberg • New York 1970
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar meac~s,and storage in data banks. Under §.54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to he determined by agreement with the publisher. © by Springe~r-VerlagBerlin. Heidelberg 1970. Library of Congress Catalog Card Number 71-154651 Printed in Germany. Title No. 3318
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of G e o r g e
Yuri Rainich
to m e t h e d e e p s i g n i f i c a n c e of a s s o c i a t i v i t y
Preface
T h e s e n o t e s h a v e t h e i r o r i g i n in a c o u r s e g i v e n a t P r i n c e t o n U n i v e r s i t y in t h e f a l l of 1968; I a m m o s t g r a t e f u l t o P r i n c e t o n f o r p r o v i d i n g t h e o p p o r t u n i t y to g i v e s u c h a c o u r s e a n d to t h o s e i n s t i t u t i o n s w h i c h p r o v i d e d s u p p o r t d u r i n g the p r e p a r a t i o n of t h e s e n o t e s :
P r i n c e t o n , t h e U n i v e r s i t y of N o t r e D a m e ,
The
I n s t i t u t e for A d v a n c e d Study, the A l f r e d P. Sloan F o u n d a t i o n and NSF G r a n t GP-9590.
T h e c o u r s e i t s e l f w a s p r e s a g e d b y a l e c t u r e at t h e M i c h i g a n
C o n f e r e n c e on the T o p o l o g y of M a n i f o l d s [ S t a s h e f f ] .
State
The p r e s e n t a r r a n g e m e n t
of t o p i c s o w e s m u c h t o a s h o r t c o u r s e g i v e n a t B o s t o n C o l l e g e in t h e f a l l of 1969.
F i n a l l y , m y d e e p g r a t i t u d e to M r s .
A n n G o s l i n g of P r i n c e t o n U n i v e r s i t y
a n d M i s s E v e l y n L a u r ~ n t of T h e I n s t i t u t e of A d v a n c e d S t u d y f o r t h e i r f i n e p r e p a r a t i o n of p r e l i m i n a r y
and final t y p e s c r i p t s
of t h e s e n o t e s .
An a t t e m p t h a s b e e n m a d e to b r i n g t h e s e n o t e s up to d a t e , b u t c u r r e n t a c t i v i t y is s u c h t h a t t h e d a t e in q u e s t i o n is at b e s t t h e e a r l y spring of 1970. It is h o p e d t h a t m o r e r e c e n t r e s u l t s w i l l b e c o v e r e d a t t h e C o n f e r e n c e on H - s p a c e s to b e h e l d a t the U n i v e r s i t y of N e u c h a t e l in A u g u s t , 1970, f o r w h i c h t h e s e n o t e s should provide adequate background. Bibliographic references
a r e g i v e n in t h e f o r m [Hopf].
Where a given
a u t h o r h a s m o r e t h a n one e n t r y in the b i b l i o g r a p h y , the v a r i o u s e n t r i e s a r e d i s t i n g u i s h e d by u n d e r l i n i n g s ,
Princeton,
e.g.,
[Hopf] v s .
[_Hopf].
N e w J e r s e y and
Lansdale, Pennsylvania
S p r i n g , 1970
TABLE
OF
CONTENTS
Chapter PREFACE
...............................................
INTRODUCTION
..........................................
THE
HOPF
THE
PROJECTIVE
MAPS
CONSTRUCTION
INTO
iNVERSES,
AN
PLANE H-SPACE:
OTHER
4
ASSOCIATIVITY:
5
H-SPACES
6
THE
7
HOMOTOPY
8
MAPS
9
SPACES
3
.................................
7
ALGEBRAIC
SPACES
ARE
FINITE
CONSTRUCTION
H-SPACES INDUCED
I0
DLFFERENTIAI~
Ii
A
-SPACES
AND
STRUCTURE, ETC ................ TOPOLOGICAL
COMPLEXES.
SPECTRAL
ASSOCIATIVITY
OF
I
................................
MULTIPLICATIONS, LOOP
WHICH
BAR
V
GROUPS
. . 14
.................
SEQUENCE
.............
...............................
.....................................
BY
H-MAPS
IN T H E
BAR
10
................
..........................................
23 27 31
............................. CONSTRUCTION
20
38 44
• • • • 48
n
IZ
MASSEY
13
HOMOTOPY
14
STRUCTURE
15
INFINITE
16
OPERATIONS
IN I T E R A T E D
REFERENCES
............................................
PRODUCTS
AND
GENERALIZED
COMMUTATIVITY ON LOOP
BAR
CONSTRUCTION..
59
..............................
65
B x ...................................... SPACES
................................... LOOP
SPACES
...................
71 75 81 89
H-SPACES
FROM
A HOMOTOPY
POINT OF VIEW
by J a m e s Stasheff
The concept of H-space
e v o l v e d f r o m t h a t of t o p o l o g i c a l g r o u p .
w a s Hopf [Hopf] w h o f i r s t c a l l e d a t t e n t i o n to m a n i f o l d s cations,
and many basic
It
with continuous multipli-
i d e a s i n t h e f i e l d a r e d u e to h i m .
The
H
in H-space
m a y b e t a k e n in h i s h o n o r . The concept of H-space as a significant generalization results
is valuable both because
of its p a r e n t
and because
in t o p o l o g i c a l g r o u p s w h i c h a r e n o t a c c i d e n t s
in the c a s e of L i e g r o u p s , This course
it occurs
in n a t u r e
it h e l p s to e l u c i d a t e
of the e x t r a a l g e b r a
or,
the extra analyticity. is a survey
homotopy point of view.
of the c u r r e n t
s t a t u s of H - s p a c e s
Homology or cohomology,
ordinary
from a
or extraordinary,
w i l l b e u s e d a s a t o o l , b u t w e a r e n o t i n t e n t o n t h e t h e o r y of, f o r e x a m p l e , algebras
per se.
ing s p a c e s
We w i l l b e p a r t i c u l a r l y
and their
as
concerned
with loop spaces,
Hopf
classify-
iterates.
Let us begin.
D e f i n i t i o n 1.
An H-space
that for some point for
e
consists
we have
of a s p a c e
e x = x = x e.
X
and a map
m : X × X-~ X
[Where reasonable,
we write
such x7
m(x,y). ] Several
of "topological and existence
comments
group"
a r e in o r d e r .
That " H-space"
is a generalization
is obvious; we have dropped the conditions on associativity
of i n v e r s e s .
S i n c e w e a r e a d o p t i n g t h e p o i n t o f v i e w of h o m o t o p y t h e o r y ,
it w o u l d
be natural and
to require
m IX × e
only a homotopy unit,
be homotopic
i.e. , require
to the identity rel
has the homotopy extension property, can be deformed
then m'
X
to one with precise
has the homotopy
with exact unit
e'
t y p e of
= I
e.g. , the category
will say means
"X
is
CW"
X'
extending
W e i n t e n d to o p e r a t e nice,
of s p a c e s
For
to indicate
a similar
the s e t of p a t h c o m p o n e n t s
with homotopy unit e
i s c l o s e d in
m
has
e
where
spaces
are at least this
on s p a c e s
of greater
locally is
out elsewhere
in t h e C W - c a s e ,
this
A comparable
generality
we will often assume
[We
Essentially
and r u l e out l o c a l p a t h o l o g y .
can be studied,
X,
as homotopy unit.
X belongs to this category.]
reason,
X)
as a branch X
of
is connected;
as a discrete
set
operation.
From is o f t e n i r r e l e v a n t ,
t h i s h o m o t o p y p o i n t of v i e w , as we shall see,
will play a significant
any such additional structure, general.
(X × X , X v
which can be given a multiplication
to h o m o t o p y t h e o r y b u t is c a r r i e d
point set topology.
homotopy,
if
if
if
m Ie × X
of the h o m o t o p y t y p e of C W - c o m p l e x e s .
s t u d y of c o n t i n u o u s m u l t i p l i c a t i o n s
with a binary
More directly,
-- XeUoI m,
However,
a multiplication
in a c a t e g o r y
we f o c u s on g l o b a l p r o p e r t i e s
less appropriate
unit.
(e,e}.
that the maps
the existence
but associativity,
of h o m o t o p y i n v e r s e s
b o t h s t r i c t a n d up to
role in our development.
Before
considering
we look at what can be said about H-spaces
in
THE HOPF
Geometrically, the Hopf construction. famous
fibratfons=
D e f i n i t i o n 1.1. H
in
the outstanding
S 3 - + S 2, S 7 - ~ S 4 a n d
of t h e m u l t i p l i c a t i o n
l i k e t h i s to p r o d u c e
is
his
S 15-~ S 8.
m : X X Y -* Z, t h e H o p f c o n s t r u c t i o n
is the map given by
join : X × I × Y / R
consequence
H o p f [Hopf] u s e d s o m e t h i n g
Given a map
: XsY-~ SZ
CONSTRUCTION
where
R
(x,t,y)-~
(t,xy).
is t h e r e l a t i o n
[Here
X~Y
is t h e
{(x, 0, y) ~ (x, 0 , y ' ) ,
(x, t , y )
(x, ,1,y)} Theorem
1. 2. [ S u g a w a r a ] .
If (X, m )
i s a CW H - s p a c e ,
then
H
is a q u a s i m
fibration,
i.e.
'
H
: ~ri{X~X, H
m ~
- l ( b ) ) -~ wi(SX, b)
m
is a n i s o m o r p h i s m
for any
be SX. Rather
than prove this theorem,
we study an alternate
f o r m of t h e
Hopf construction. Definition 1.3.
Given
H(ml : X × CY~JZ-~ m Theorem
1.4.
equivalence
SY
Proof.
y, t h e n Since
5.
If p :
If X = Z
H(m)
m(
and
,y}
Y.
is a w e a k h o m o t o p y
Z-~ # are qnasifihrations
follows from fundamental
P I : P - l ( U ) -~ U, p I : P - I ( V ) -~ V sois
and
of q u a s i f i b r a t i o n s
E-~ B
and
Of X ~ Y o n t o
is a q u a s i f i b r a t i o n .
X × CY -~ CY
the theorem
T h o m on t h e c o n s t r u c t i o n Theoreml.
is i n d u c e d b y p r o j e = t i o n
[D_old a n d L a s h o f ] .
for each
fact, bundles),
m : X X Y -~ Z , t h e H o p f c o n s t r u c t i o n
B = U~.JV and
theorems
(in
of D o l d a n d
[Dold and Thorn]: where
p I : P-l(U~V)
U, V
are openin
-~ U ( - ~ V
are
B
andif
q. f. s
p.
Theorem
1.6.
If p : E - - B D A
exist deformations
Dt : E-~ E
and and
p] : p ' l ( A ) dt : B-~ B
: D -~ A such that
is a
q. f a n d t h e r e
then
D 1 : id, D 0(E) C D d I = id, d 0 ( B ) C A
and
d0P = p D 0
DOI . : p ' l ( b ) - . p ' l ( d o ( b ) ) then
p
is a
is a weak homotopy equivalence
q.f. T h e c o n d i t i o n on
m(
e n c e of a r i g h t u n i t b e l o n g i n g to
, y) Y.
follows for connected It a l s o h o l d s if X = Y
Y from the existis CWas
we now
see.
Theorem map:
1.7.
~ugawara].
(x, y) -~ (xy, y) Proof.
If X
is a c o n n e c t e d
H-space,
then the shearing
is a w e a k h o m o t o p y e q u i v a l e n c e . I c l a i m t h e i n d u c e d m a p of h o m o t o p y g r o u p s :
~r.1(X)~ Tri(X) -~ wi(X ) O ~ri(X) is given by 4, ~ -~ = + 6, ~, which is clearly an isomorphism.
To
verify the claim, w e use a lernrna. Lemma
1.8 [Hilton].
If X
is an H-space,
the usual operation in ~r.(X) is 1
induced by the multiplication Proof.
T h i s c a n b e p r o v e d a s in t h e u s u a l p r o o f f o r t h e c o m m u t a -
t i v i t y of lr1 of a n H - s p a c e . l o o p s p a c e of X
in X.
In f a c t , b y r e g a r d i n g
Iri+l(X)
and using the induced multiplication
as
of l o o p s ,
w1 of t h e i - t h the usual proof
applies directly. The restriction group under
"X connected"
shearing Remark.
CW H - s p a c e ,
a n d h e n c e s o is t r a n s l a t i o n
m a p : x, y - ~ x, x y By altering
m(
the shearing
, y).
is a
is c o n n e c t e d
CW.
map is a homotopy
The left unit is relevant
to t h e
in t h e a n a l o g o u s w a y .
t h e t o t a l s p a c e of H ( m )
but without changing the homotopy
type, we can obtain a map with the weak covering X
to " l r 0 ( X )
m.". Thus for a connected
equivalence
can be weakened
This follows from results
homotopy property
of D o l d if
of D o l d on t h e c o n s t r u c t i o n
of
such fibrations
[Dotd].
being c~ntent ourselves Theorem unit
1.9.
and both
X X CYUZ m
Proof.
Both spaces
sequence,
1.10.[S.ugawara].
Proof.
are simply connected.
x-~ (e,t,x). If
X
X X CYUZ
via the Meyer-
t h e f i b r e in
H(m)
is c o n t r a c t i b l e
T h i s i s in f a c t a c h a r a c t e r i z a t i o n .
is connected
CW, then
X
is an H-space
of s p a c e s
if
which are CW
of X
in
of p E
to
b y t h e s u m of ~ p
is g i v e n b y e ¢ X. and
H(m).
Map
For the converse,
X-~ ~B
X-~ ~B.
by
kx(t)=p
If t h e d i a g r a m
kt(x).
of h o m o t o p y
sequences: ,
,r. ( x )
.... Ir.(E) i
z
~ Ir.(B) i
~ ~ri - I(X)
~ i . l ( ~ E ) ~ - " ~ Tri_I{QB)
?
iri.,l(~E × X)--
•i s c o m m u t a t i v e ,
it follows that
~B
p
is a
q.f.,
@[f] = [h I S n] then
h
let
h:CS n-~ E such that
f o r a n y c h o i c e of h.
can be taken to be
In p a r t i c u l a r ,
h(t,x) =kt(x),
sothat
is t h e i d e n t i t y .
Tri_I(~B) -~ ~ri.l(X)
g : S ( S n) -~ B b e t h e a d j o i n t ,
there exists
~i_I(X)
~ri(X) -~ w i ( ~ B ) -~ ~i{X)
The most direct way to see this is to recall how f:S n-~ ~B,
] •
h a s t h e w e a k h o m o t o p y t y p e of ~ E X X.
T h e p o i n t t o c h e c k is t h a t
Given
-~
in h o m o l o g y .
is a n H - s p a c e ,
The existence
~E × X-* ~B
The map
in t h e t o t a l s p a c e .
kt be a contraction
Map
and there is a left
t y p e of X * Y .
i s t h e f i b r e of p : E - ~ B', a q u a s i f i b r a t i o n
with fibre contractible
let
Y are connected
induces an isomorphism
in t h e t o t a l s p a c e ; e. g. , v i a
and only if X
and for the time
m a p is a m a p of t r i a d s w h i c h ,
N o t i c e t h a t if X
Theorem
X and
has the weakhomotopy
induced by the shearing
Vietoris
result
with quasifibrations.
If Y = Z
e • Y, t h e n
X~Y
We will not need this stronger
is defined.
i . e . , g ( t , x ) = f(x) (t).
ph~ g if f
rel
t = 1.
comes
hlSn ;f '.
from [If p
Since
We d e f i n e f':S n-* X isa
Hurewicz
fihring,
the homotopy
then
a
h t ( k ) -- k ( t ) , Now all spaces
of ~ E
X X.
Theorem
Theorem
1.11.
If X
is realized
by a map
starting
f 0 ( ~ ) -- e e X C E . ]
at
being CW,
~ B -~ X
the loop space
1.10 c a n n o w b e c o m p l e t e d is a retract
of a n H - s p a c e
~B
given by a covering
has the homotopy
Use
type
by the following remark. Y, t h e n
X
admits
a multiplica-
tlon with unit. Proof.
of
X X X-~ Y × Y-* Y-~ X
and check on units.
THE
PROJECTIVE
The full significance most
clearly
Definition
in the mapping
2.1.
p i n g cone of H(m), i . e . ,
E.G.
to
Z. 2.
H*(XP(2))
plane
which is isomorphic
XP(2)
of a n H - s p a c e
XP(2) = RP z
X -- S 1
XP(2)
X = S3
XP(2) = QpZ
X = S7
XP(2)
= Cp 2 = SZUe
i s usually e x p l o i t e d
through
are classes
non-trivially, secondary unless
giving a proof,
[Adams].
Proof.
If u
therefore,
or extraordinary,
n = 0,1,3,7.
its cohomology. S u , S v c H v (SX) p u l l b a c k
from
S(uSv) E H*(S(X*X))
H$ ( X P (2), S X ) .
S (u'v)
2.3.
16
such that comes
Su
Theorem
is the map-
8
= Kp 2 = S8Ue
-~ H* (S (X*X) ) --,- H * ( X P ( 2 ) ) - ~
Before
(X,m)
4
S4Ue
( S u ) ~ J (Sv) ~ H # ( X P ( 2 ) ) to
shows up
C ( X x C X U X ) . W=~m )SXm
If u , v e H $ ( X ) then
of t h e H o p f f i b r a t i o n s
H(m).
X = SO
This space Theorem
of t h e e x i s t e n c e
c o n e of
The projective
PLANE
Sn
-*
Su
S u -~ Sv
let us look at the consequences.
is an H-space
generates SuUSu
-~
H * (SX)-~
Hn(sn), # 0.
one shows,
if a n d o n l y if n = 0 , 1 , 3 , 7 . Su
pulls back and
Now using operations, following Adem
S(u*u)
maps
primary,
and Adams,
SuUSu
= 0
Theorem
2.4.
sPt..)eq~.Je p+q
a r e o n e of t h e f a m i l y
(1, 3),
Most cases tions [Adams], and
in conjunction
XP(2)
(3, 5),
with
q>
can be eliminated
with Theorem
Theorem
approximation
can be defined,
2.2.
2. 2.
by using cohomology The remaining
[Hubbuck],
Following
Y -~ Y X Y
up to homotopy,
being given in barycentric
where
as
,x : y - . y × y approximation
coordinates,
can be deformed to the diagonal
on the chain level,
Applying this to
keeping
o2
opera(7, 11)
[Douglas and Sigrist].
We
induces
track
n
maps
to = 0
(t0, t z , x l x 2 )
if
t1 = 0
(t0, t 1 , x l )
if
t2 = 0
if x 1 = e
(t0, t l + t 2 , x l)
if x 2 = e
o
n
For to
on
example,
~
approximation
of t h e i d e n t i f i c a t i o n s ,
we read
note that
points
of
o2
map by taking a nice cellular consider
(first p-face) on
for
the usual
one
@ (last (n-p)-face).
X 2 ' we have induced a
a 2 × X 2 which is compatible
a diagonal
First
a
identifications:
~- ( t 0 + t l , t 2 , x 2 )
g .
we construct
Y = XP(2).
if
to a cellular
on
[Milgram],
we have the
and using the true diagonal
approximation
and hence
(p,q)
cases,
SX~.) o 2 X X 2 w h e r e ,
(t o , t l , t 2 , x I , x 2) ~ (t 1 , t 2 , x 2)
diagonal
if a n d o n l y i f
to this later.
diagonal
which,
p+l
(3, 7).
of u s i n g K - t h e o r y
Now to prove specific
(1, 7),
not listed
(7, 15), a r e d i s p o s e d
will return
is anH-space
XP(2).
with the identifications On the chain level,
off
A~(sZ @ u ® v) = (a2 ® u ® v ) @ * + (el®v)@ (al@ u) + * ® (s2 @ u®v).
w h i c h in t u r n i m p l i e s original representation
Su~.~Sv
is c a r r i e d b y
0.2 0 u @ v, c o r r e s p o n d i n g in t h e
to S(u*v).
T h e a b o v e p r o c e d u r e is t y p i c a l of m a n y r e s u l t s in the t h e o r y of H-spaces.
T h e c o m b i n a t i o n of t o p o l o g y and a l g e b r a in q u e s t i o n is u s e d to
c o n s t r u c t a n o t h e r s p a c e in w h i c h the combination.
h o m o t o p y alone r e f l e c t s the o r i g i n a l
T h i s a u x i l l i a r y s p a c e is t h e n s t u d i e d b y t h e a v a i l a b l e f u n c t o r s of
a l g e b r a i c t o p o l o g y , c o n v e r t i n g the p r o b l e m thus into a p u r e l y a l g e b r a i c one.
MAPS INTO AN H-SPACE: ALGEBRAIC
STRUCTURE,
The fact that X h o m o t o p y classes of m a p s Theorems
INVERSES,
OTHER
MULTIPLICATIONS,
ETC.
is an H - s p a c e is immediately reflected in the set of [K,X]
3.1. [Copeland].
of a space
The functor
K
into X.
[ ,X,x0]
takes values in the category
of sets with binary operation and Z-sided unit w h e n defined on a category including
~X,x0) and
~X X X, x0 z) if and only if X
unit if and only if (ilviz)~:[A × B;X] -~ [AVB;X]
is an H - s p a c e with x 0 as
is onto for all A , B
in the
category. The crucial idea is to realize that the multiplication allows us to add m a p s of A and B tion i t s e l f a p p e a r s If Theorem
into X to get a m a p of A × B and that the m u l t i p l i c a -
a s t h e s u m of t h e two p r o j e c t i o n s
K
is CW,
3. Z. [ J a m e s ] .
[K,X]
of maps
of
binary
operation
K
the set If
K
[K;X]
X
h' and
as a trivial
: K-~ X X X hw~
f hence If
structure
on
Definition
3.3.
homotopic
to
Theorem
3.4.
CW-complex,
X
hu~
X.
an algebraic
loop,
classes
i.e. , has a
identity and left and right inverses.
quasifibration.
such that
on to
t h e s e t of h o m o t o p y
forms
T h e i d e a of t h e p r o o f is to r e g a r d X X X
X X X
structure.
is a CW-complex,
into an H-space
with Z-sided
has more
of
s h' ~
Given (f,u).
the shearing
map
f, u : K -- X , t h e r e T h u s if
h'
= (h,w),
s : X X X -~
is thus a map we have
w~
u
f.
itself is CW, the inverses
come
from
the corresponding
X. A map
I : X -~ X
is a left homotopy
inverse
if
m(l
X 1)
is
e. [Sugawara,
Sibson].
left and right homotopy
If
X
has the homotopy
inverses
always
exist.
t y p e of a c o n n e c t e d If
m
is homotopy
11
associative,
left and right homotopy Proof:
(Theorem q(e,
We know the shearing
1.1); l e t
q : X × X-~ X × X
)_~ I × 1 i s a l e f t h o m o t o p y If m
homotopy
inverses
is homotopy
inverses
map
up to homotopy.
is a homotopy
be an inverse.
inverse
since
associative,
is completely
agree
A map
~
satisfying
m ( ~ × 1) i s h o m o t o p i c
the agreement
analogous
equivalence
to
e.
of l e f t a n d r i g h t
to the agreement
of s t r i c t
inverses
in a monoid. Similarly, Theorem [K,X]
3.5.
If X
is a group,
is a homotopy
natural
Regarding maps
we have the following result.
are fibrewise
associative
with respect
projections
maps.
of
to maps
Hence the shearing
i f a n d o n l y if i t i s a f i b r e h o m o t o p y
translation
(left or right) is a homotopy
X
X
(e.g.,
admits X
a numerable
K
is CW,
then
as fibrations,
map
the shearing
(left or right) is a homotopy
equivalence
equivalence.
[DoLd] w h i c h i m p l i e s
The converse
holds pro-
by sets which are nullhomotopic
in
is CW). In light of these
Theorem
covering
and
K - ~ K ~,
X × X -~ X
equivalence
vided
H-space
remarks
1.4 and in Chapters
definition of H-space
and the importance
of translation
4 a n d U, i t m a y b e a p p r o p r i a t e
to include left and/or
to amend
right translation
in the
being a homotopy
equivalence. Multiplications. Given a space others
X , if i t a d m i t s
which are not homotopic
Theorem
3.6.
multiplications
[Copeland]. is in
Proof: [KVK;X]
1-1
to
If X
is exact for any CW
K.
m,
it may admit
m. i s C W , t h e s e t of h o m o t o p y
correspondence
The sequence
a multiplication
with the loop
-~ [ S K v S K ; X ] - ~ If X
of
[XAX;X].
[K~K;X]--
is an H-space,
classes
(ilV iz)*
[K × K ; X ]
the last three
sets are
L
12
l o o p s , a n d the m a p s a r e m o r p h i s m s t h e r e o f . (ilWi2)* ' l ( x )
It f o l l o w s t h a t the i n v e r s e i m a g e s
a r e in 1-1 c o r r e s p o n d e n c e for a l l x ~ [KVK;X]. If X = K is CW, the m u l t i p l i c a t i o n i n d u c e s a s p l i t t i n g f r o m w h i c h
the t h e o r e m f o l l o w s . T h e e a s i e s t c a s e s to c o m p u t e a r e Theorem 3.7.
K ( ~ r , n ) ' s or s p h e r e s .
Up to h o m o t o p y , K(Ir, n) a d m i t s o n l y one m u l t i p l i c a t i o n .
Up to h o m o t o p y , S 3 a d m i t s
12 d i s t i n c t c l a s s e s of m u l t i p l i c a t i o n .
Up to h o m o t o p y , S 7 a d m i t s
120 d i s t i n c t c l a s s e s of m u l t i p l i c a t i o n s .
[ J a m e s , L e m m e n s ] h a v e s h o w n a l l the c l a s s e s on S 3 and S 7 c a n b e r e p r e s e n t e d in t e r m s of the s t a n d a r d m u l t i p l i c a t i o n s b y u s i n g c o m m u t a t o r s . F o r p r o d u c t s of s p h e r e s ,
[Loibel] gives a f o r m u l a which for
S 3 × S 3 is c o m p u t e d b y [Norman] to be
220 × 316 .
[ N a y l o r ] a n d [Kees] h a v e s h o w n t h a t SO(3) = RP(3) c l a s s e s of m u l t i p l i c a t i o n s .
7 and
220.
768 d i s t i n c t
[Rees] h a s f o u n d 3 0 , 7 2 0 m u l t i p l i c a t i o n s on K P ( 7 ) ,
a n d [ M i m u r a ] h a s f o u n d the n u m b e r s f o r 215- B9 " 5 "
has
B • 55.
SU(3)
a n d Sp(2)
to b e r e s p e c t i v e l y
7.
M o r e g e n e r a l l y A r k o w i t z a n d C u r j e l [A-C] h a v e i n v e s t i g a t e d the f i n i t e n e s s of this n u m b e r for f i n i t e c o m p l e x e s .
T h e y f i n d that a m o n g the
c l a s s i c a l a n d e x c e p t i o n a l L i e g r o u p the n u m b e r is f i n i t e o n l y f o r SO(n)
with
n < 16,
SU(n)
with
n < 5
n # I0, 14
m
Sp(n)
with
G 2, F 4 a n d F i n a l l y , for a space and
n < 7 E 7.
X w i t h j u s t two n o n - t r i v a l h o m o t o p y g r o u p s
~rp, p > n, the s e t of c l a s s e s of m u l t i p l i c a t i o n s c a n b e i d e n t i f i e d w i t h
HP(x~JC;~rp) w h i c h is HP((~rn, n ) A ( ~ n , n ) ;
Tr ). P
A m o r e c l a s s i c a l a p p r o a c h to c l a s s i f y i n g m u l t i p l i c a t i o n s w o u l d
n
~3
r e g a r d a s m ~ a i v a l e n t t h o s e w h i c h c o r r e s p o n d up to h o m o t o p y u n d e r a h o m o t o p y equivalence.
J a m e s s h o w s t h a t t h e r e a r e o n l y s i x s u c h c l a s s e s on S 3, b u t in
g e n e r a l the c o m p u t a t i o n s s e e m m o r e difficult. some reasonable results are available
[Cheng].
For two-stage Postnikov systems,
ASSOCIATIVITY:
Although an important
strict
SPACES
associativity
AND
groups,
be seen to be equivalent homotopy
theory.
Definition
4.1.
s e t of p a i r s
for,
from
to spaces
[Moore].
It characterizes
o u r p o i n t of v i e w ,
of l o o p s ,
the latter
the operation
~X
is an associative
GROUPS
concept,
it plays
loop spaces
topological being more
t'~X : {X:[0, r ] - ~ X I X(0) = k ( r ) = * )
(k, r ) c X R X R .
under
TOPOLOGICAL
is not a homotopy
r o l e i n t h e s t u d y of H - s p a c e s .
hence topological
monoid)
LOOP
H-space
and
groups basic
will
in
topologized
as the
(= t o p o l o g i c a l
m = + given by k , ~ -~ k + / ~
defined by
k + ~ : [0,r+s]-~
x+. In the CW category loops are essentially
the same
That loop spaces groups
was proved
for semi-simplicial Theorem
4.2.
cW complex,
complexes
[Milnor].
Proof: homotopy
If X
then there
type as
X'.
and from
~X
a homotopy
as topological
groups
p o i n t of v i e w ,
spaces
or associative
H-spaces.
of t h e h o m o t o p y
though the result
of
t y p e of t o p o l o g i c a l
was presaged
by a similar
one
d u e to [ K a n ] . has the homotopy
is a topological
First
I [r,r,+s]:t~.(t-r).
are usually
by Milnor,
X by
group
t y p e of a c o n n e c t e d
GO()
has the homotopy
We might as well assume
of the homotopy
t y p e of ~ X .
t y p e of ~ X '
if
then that
is a countable
X
X
countable
has the same
s implic ial complex. Let The equivalence {x1 . . . . .
G(X)
be defined as a quotient
relation
of a s u b s e t
of ~_)X n n
as follows:
is
x n) ~- (x 1 . . . . .
A x i .....
x n)
if x i = x i + 1 o r
x i . 1 = x i + 1.
15
GO B x .
are the exotic multiplications
q : S 3 X S 3 --~ S 3 , t h e q u a t e r n i o n
multiplication,
S3 v $3 by an element of ~6 (s3) = ZIZ.
we can alter
This is not
on
S 3.
it a w a y f r o m
In this way, w e get the IZ different
h o m o t o p y classes of multiplications including quaternions and its opposite
q : x,y~
q(y,x)
29
which differ by a class
plication m we see
is h o m o t o p i c
H(m)
to q + no0 f o r s o m e
is h o m o t o p i c
p1 : w7(S 4)
to
Since
pl
detects
are associative, pI(H(q))
Thus
pI(H(q))
w e have
ZBC~r6(S3),
and
From
Any multi-
the c o n s t r u c t i o n ,
Now a homomorphism in $4~.) e 8 w e h a v e
we have
pl(~co) # 0.
Since
q and
m
pl(zco). In general, for m
-- q + nco,
can be h o m o t o p y associative only if
In fact, J a m e s shows in these cases the multiplication [_s_s
homotopy associative, i.e. , XP(3) XP(n)
[James].
pI(H(q)) = pl(H(q)) + pl(~co) are non-zero.
pl(H(m)) - I + n(3). Thus
these cases
n e ZlZ.
pl(a) = k where
m u s t have the s a m e sign as
n - 0 or I rood 3.
of w6(S 3)
H(q) + n ~ co e ~r7($4).
Z 3 can be defined by
p l ( u 4 ) = ku 8. q
co w h i c h is a g e n e r a t o r
can be formed.
[Slifker] goes on to s h o w in
can be f o r m e d for all n, although by the classification
theorem for Lie groups, only q and cations on the 3-manifold
S 3.
q are homotopic to associative multipli-
Thus there are topological groups of the h o m o -
topy type of S 3, but not multiplicatively equivalent to the unit quaternions in either order of multiplication. The first example of a homotopy associative H-space which is not equivalent to a loop space was given by [Adams]. Example
7.6.
Let
Y
be a M o o r e space
3 but no other primes is possible. Zn-I and for i ~ n, ITi (X) Consider
Y
Z and
has cohomology only in dimension
~r.(sZn'l)p. 3
yC~Z
structions to deforming
Thus
Y(G, Zn-I) w h e r e divisibility by
~Zy
Y X Y -~ 2
1
which is an associative H-space. ~ Z y X ~ Z ~ Z y -, ~ 2 ~ 2 y
into Y
The oblie in
Hi(y A Y; ir.(~Z ~Zy, y)). The relative groups are trivial to at least 5(Zn)-3, i
SO the obstructions vanish.
Similarly, there are no obstructions to deforming
an associating homotopy within ~ Z ~ 2 y
to one in Y.
That
space follows f r o m the decomposability of P n UZn = u2nP Further examples,
even finite complexes,
Y
is not a loop
for p > 3 and
n J-p-h
have been constructed by
30
Zabrodsky.
We w i l l p r e s e n t t h e s e e x a m p l e s w h e n we h a v e d e v e l o p e d the a p p r o p -
r i a t e m a c h i n e r y to e x p l a i n t h e m , n a m e l y , the h o m o t o p y a n a l o g u e s of h o m o morphisms.
MAPS OF H-SPACES
homomorphism
Bya
f ( x y ) = f(x}f{y). morphism.
For
of H - s p a c e s
example,
In a limited
if
sense,
we mean a map
g : W-* Z
then
~
f : X-* Y
: ~W-*
g
f~Z
such that
is ahomo-
we shall see that for associative
H-spaces
this
is the only example. Clearly
a homomorphism
S f : S X -~ S Y , b u t s u c h m a p s Definition 8.1. H-map
if
fm
If
(X,m)
is h o m o t o p i c
H-maps associative H-map
n ( f X f).
generally
are H-spaces,
maps
a map
for H-spaces
X P ( 2 } - ~ YP(2}
an H-map.
If
f: X - * Y
in g e n e r a l ,
but notnecessarily
but,
extending
is an
for An
BX-* By.
have been investigated
(shm) map"
and
n
are
strictly
f(x)f(yz)
exist for all
involving
associative,
or via
if w e h a v e t h e a p p r o p r i a t e
we are led to conditions
If s u c h m a p s
m
in t w o w a y s - v i a
are homotopic
by
map
f(xy)f(z).
f(xyz)and These
I g x X 3 -~ Y.
More
I n ' l x x n -> Y, s u i t a b l y r e l a t e d
on the
n, we have a " strongly homotopy multiplica-
[Sugawara] or an "A
-map"
[Stasheff] or an "H-homomor-
[Fuchs].
Definition 8. g.
an A -map
Let
X
and
Y
b e associative H - s p a c e s .
if t h e r e e x i s t s
a f a m i l y of m a p s
h 1 =f
and
t i _ l , X 1. . . . .
h i { t 1. . . . .
A map
h. : i i-1 ~< X i -~ Y
n
that
YP{n)
t h e y a r e n o t t h e h o m o t o p y a n a l o g of h o m o m o r p h i s m s .
are h0motopic
two homotopies
phism"
to
into
people.
f(x}f(y)f(z}
tive
(Y,n)
XP{n}
generally.
homotopy conditions for the latter
Consider
faces.
and
X-* Y will induce
The necessary several
occur more
are the relevant
H-spaces,
f:
induces a map
for
f : X -~ Y
is
1< i< n
such
1
x i)
t%
= h i - 1 {. . . .
t j. . •. , x. j x. j +. 1,.
= h .j ( t l , . " . , tj_ 1, x 1 . . . .
} if
t.j = 0
x j ) h i _ j (tj+ 1 . . . .
,ti_l, Xj+ 1 .....
x i)
if t.j = 1.
32
Such animals occur naturally.
For any space
m a k e the function space topology nice), let H(K) equivalences of K K r L s K
into itself. If K
and
L
K
(locally compact to
be the space of all homotopy
are two such spaces and
h o m o t o p y inverses, then there is a strongly h o m o t o p y multiplicative
homotopy equivalence
H(K)-~ H(L)
given by ~-~ r ~ s.
homotopy with R(0) = I and R(1) = sr, then the family
If R(t) : K-~ K
is a
{hi}, as constructed by
[Fuchs], looks like: hi(tI..... ti_l,¢l ..... ¢i ) = r #iR(tl ) CZ'''R(ti-1) ¢i s" The composition of A n - m a p s
is again an A n - m a p ,
families m u s t be fitted together, rather than composed.
families are
{h 1 : i i - I × X 1" - Y} and
to be Jlhl and
(jh) z to be
but the
For example,
if the
{Ji : ii-I X yi --* Z} then we define
(jh) 1
Jlhz + jz(hl X hl), m e a n i n g the h o m o t o p y :
Jlhl(xY) _~ Jl(hl(X)hl(Y)) _~ Jlhl(X)Jlhl(Y). For
(jh) 3, Jlh3 , jz(hl × h~), jz(h2 X hl) and J3(hl X h I X hl) a r e f i t t e d t o g e t h e r
as i n d i c a t e d below.
Full details are given (k(jh)).
y [Fuc s]
He points out
although they are homotopic.
a
((kj))1
is not the s a m e as
The a p p r o p r i a t e c a t e g o r y w h o s e o b j e c t s a r e
1
a s s o c i a t i v e H - s p a c e s has as m a p s h o m o t o p y c l a s s e s of s u c h f a m i l i e s
{hi}.
A l t e r n a t i v e l y [ D r a c h m a n ] g e t s a r o u n d the d i f f i c u l t y b y u s i n g h o m o t o p i e s of v a r i o u s lengths.
33
Of c o u r s e a h o m o m o r p h i s m
c o m p o s e d w i t h a n A - m a p is m u c h m o r e n
r e a d i l y s e e n to b e an A - m a p , the o b v i o u s f a m i l y b e i n g h o m o t o p i c to the one n d e f i n e d b y the g e n e r a l p r o c e d u r e u s i n g the " t r i v i a l " h, i . e . ,
h i ( t 1. . . . .
xi) = h(x 1 . . . x i )
family for a homomorphism
= h(Xl)...h(xi).
On the o t h e r h a n d , a m a p h o m o t o p i c to an A - m a p is i t s e l f a n A n
map, though again some fiddling with parameters Theorem 8.3.
[Fuchs].
n
is r e q u i r e d to s h o w t h i s .
A homotopy equivalence
f : X -~ Y is s t r o n g l y h o m o t o p y
m u l t i p l i c a t i v e if a n d o n l y if a n y i n v e r s e i s . Proof:
If f is a n H - m a p a n d g is a n i n v e r s e ,
we h a v e
gn ~ gn(fg X fg) ~ g f m ( g X g) ~ m ( g X g), which shows
g is a n H - m a p .
compatible with gf~
fg ~ id.
1X b y a h o m o t o p y
written as
To p r o c e e d f u r t h e r ,
we n e e d t h i s h o m o t o p y to b e
S p e c i f i c a l l y l e t fg ~ 1y b y a h o m o t o p y k.
If f m ~
n(f × f) b y
- g n ( ! X 1) - g f 2 ( g X g) + k m ( g × g).
I
and
f2' the a b o v e h o m o t o p y c a n b e F o r p u r p o s e s of i n d u c t i o n , w e
w i s h to f i l l in the d i a g r a m
fgfz(g ~
f g n ( l X 1)
[
x
~.
g)
] fz(g X g)
i)
The lower quadralateral
c a n b e f i l l e d in w i t h
l n ( l × 1) a n d t h e n the
u p p e r one b y u s i n g
l f 2 ( g X g) is we h a v e f o l l o w e d F u c h s in m a k i n g t h e c l e v e r
observation that
can be chosen so that
between
fgf a n d
l f.
fk is h o m o t o p i c to
If
as homotopies
34
The induction now proceeds,
constructing
at each state
so that
f n
{(fg)i}
is homotopic
to
(id) i.
Thus A -maps
are a reasonable
class
of m o r p h i s m s .
As expected,
n
A -maps
are nice with respect
to projective
spaces.
n
Theorem
8.4.
t e n d s to
XP(n)-@ YP(n).
Corollary t y p e of
Amap
8.5.
f : X-* Y
The homotopy
t y p e of
n
-map
BH(E)
if and only if Sf : S X - ~ S Y
is an invariant
ex-
of t h e h o m o t o p y
E. The induced map
terms
is a n A
of t h e f a m i l y
that "f respect H-map,
then
SX~)C(X
f P ( n ) : X P ( n ) -~ Y P ( n )
h i by formulas
the identifications fP(2)
X CX~)
m
[S.ugawara] which give
up to homotopy".
can be defined in terms
meaning
example,
of t h e r e p r e s e n t a t i o n
in
to the idea if
f
is an
XP(2) =
0 p + q.
The corollary admits
are all zero.
i.e. , for
forward
to f b e i n g a n H - m a p
lr A lr, t h e y m a p t o c l a s s e s
so its obstructions
an isomorphism,
is a s t r a i g h t
is in
of t h e h o m o t o p y result 1-1
h 2 g i v e r i s e to d i f f e r e n t
of T h e o r e m
3. 6, t h e s e t of h o m o t o p y
correspondence
as isomorphic
with
Fixing a homotopy
K{G,q)
and
L
m
is t h e f u n d a m e n t a l
o h 2 and its cochain
E f , El]
b c c q ' l ( K { ~ r , p ) ~ K { I r , p);G)
5b = m S f ~ ( ~ ) - (f X f)~t m ~ : ( L ) w h e r e or
[EfA
H q - 1 (K (Tr, p) A K (~r, p); G).
h 2 gives us a specific cochain
K(lr, p)
with
denotes the c o c h a i n of
b °, the correspondence
40
between multiplications
and the cohomology
E v e n if f is t r i v i a l , example,
if f = ~ t 2 w h e r e
Ef
L
g r o u p is g i v e n b y
can have more
is t h e f u n d a m e n t a l
than one multiplication. class
h a s t h e h o m o t o p y t y p e of S 1 × K ( Z , 2) b u t r e g a r d e d has a Pontrjagin
ring not ring isomorphic
The corollary
of a P o s t n i k o v
Definition 9.3.
A Postnikov
of f i b r a t i o n s
system
system
pnJn = J n - l '
[Suzuki,
fibre homotopy equivalent
to k i l l
systems
is to f o r m
~-ilX) f o r
i > n.
)
result
for a space
X
about
with maps
can be constructed
consists
of
classes Jn : X -* X n for
i 2n+Z.
together
for
9.1.
to f o r m a
n
multiplication
on
X
the multiplication
is s o l v e d b y c o n s i d e r i n g
"mixing and
however,
X ~1 Y'I"* X0
such that
I H i ( Y . ; Z ). p
the cohomology Yi+l
qi
Let
of Z a b r o d s k y '
X(P 1)~ X 0 by a succession
is a n i s o m o r p h i s m
There
i.
A
s method
of
for
becomes
r < i-1
K(Z
P
,i)
in
PI"
f o r b e i n g so d e t a i l e d .
as
and
and all a
p ~ ]t~1.
in t h e k e r n e l
in t h i s d i m e n s i o n
in t h i s d i m e n s i o n .
enough times
Repeat for all primes
is r e a s o n
in t h e k e r n e l
monomorphic
f : X ~ X0
a far.torization
Y i + l - * Y'I b e i n d u c e d b y a c l a s s
by taking a product with
i s m in d i m e n s i o n
interest.
of f i b r a t i o n s ,
homotopy equivalence
Continue killing classes
morphism
c a n b e a p p l i e d to a
to b e of o n l y t h e o r e t i c a l
is a r a t i o n a l
H r (X, Zp)
p be such a prime. 1
approach
by induction that we have constructed
qi * : Hr (Y.,Z)-~ 1 p
qi
X and constructing
Given a rational homotopy equivalence
we can construct
Assume
system
gives useful applications
homotopy types".
P1CP,
follows:
the Postnikov
H - * E - * K, b u t t h i s a p p e a r s
modification,
of
by induction.
More generally, fibration
the skeleta
to m a k e
qi+l
Let
of until
Now form an isomorph-
42
Theorem map,
9. 5. [Zabrodsky].
then
X{PI)
If f : X - * X 0 is a rational equivalence and an H -
admits a multiplication so that fl' f2 are H - m a p s .
Proof:
If f : X - ~ X'
is an H - m a p
with
f~ : H r ( X '; Z P
being an i s o m o r p h i s m
for
r < n, then ~ ~ K e r f',~ I Hn(X' , Z
)-~ Mr(x; Z ) P
) is represented by P
an H - m a p ,
for f'~ ~ = 0 is represented b y an H - m a p
being represented by an H - m a p
lie in H n ( x ' ~
and the obstructions to
X' ; Z
) w h i c h is m a p p e d
iso-
P morphically resented trivial
to
HncK ~ X; Zp)
by an H-map one) showing
will admit
Corollary H-maps,
The homotopy
showing
so as to map into a homotopy
f$¢~ i s a n H - m a p .
Thus the space
such that
Y
If t h e i n g r e d i e n t s
the space 9. 7.
multiplication
constructed
X'
by
T : X -~ Y.
s construction
are H-spaces
and
will be an H-space. is classified
by
m0, t h e n
M 10 a d m i t s
a
n
if n ~ 2 ( 4 ) . of
can be constructed
S--p(Z) a n d
maps
(e. g. , t h e
f can be lifted to an H-map
of Z a b r o d s k y '
If S 3 - ~ M 1 0 - ~ S 7
The values
or
is rep-
induced over
n
spaces
a
follows by induction.
9.6.
Application
( f : ~ f)~.
can be chosen
a multiplication
The theorem
by
n
not covered
by having
X z = S 3 X S 7 or
Sp(2) -~ K(Z,3)
in Hi(Sp(Z)~'Sp(2))
and for
previously
2 E P1
and
i = 3 or 7.
n -+3,
3 ¢ P2
X 1 = S 3 X S 7 and
Sp(2) -~ K(Z,7)
are
The H-spaces
and taking
X2. = Sp(Z)
are H - m a p s
+4(12).
These
X 1 ; Sp(g)
or Sp(Z).
The
since the obstructions lie
M I0 +3
are definitely not h o m -
m
otopy associative as ~pl : H 3 -~ H 7 is trivial w h i c h contradicts the existence of cup cubes in the 3-projective space.
43
The relation properties
of
f carries
we have studied between over to many
can also be proved
more
Theorem
has the homotop7
9.8.
Postnikov
X
system
k-invariants
of
X
Corollary
9. 9.
X
t y p e of a l o o p s p a c e
has each stage being a loop space
forms
as well as Aoki, H o n m a
structures.
of
Ef
and
The following corollary
directly.
are loop classes Various
other
properties
and the
in' Pn
of t h e t h e o r e m
if a n d o n l y if s o m e such that the
are loop maps.
have been proved:
[Suzuki],
[Iwata],
and Kaneko.
has the homotopy
t y p e of a l o o p s p a c e
if 17.(X) = 0 f o r 1
i2p-2. This follows by induction on the stages of the Postnikov s y s t e m since
the only non-trivial k-invariants are in the range w h e r e an isomorphism.
the following result.
Corollary
H-space
rr.(X) = 0 for sented by an
An i1
K(~r, 0) b e i n g t h e a b e l i a n
A class
the class
Wl~% q + ~Z~t q
"B
10.1.
consider
K(~,q)
by regarding
of
is homotopic
Theorem
for example,
That
#
K(vr, q)
with
ways,
u E Hq(x;~).
AnT representative
tion on
K(zr, n )
class
H-space.
in which case
L
and induces
we can regard
Y = K(lr,q)
and
K{w,q),
belong to
with
are the projections
tion on
homotopy
will denote an associative
occur
can be seen in several
up to homotopy, as where
X
of A - m a p s n
map can be interpreted a multiplication
IN THE BAR CONSTRUCTION
H-space
and
u • Hq(X;~v), t h e n t h e
u
45
l)
d [u] = 0 f o r
r < n
r
2)
Su
3)
u
pulls back to
XP(n)
is represented
by anA
-map. n
The equivalence differential
in a spectral
of
1) a n d
sequence
e x a c t c o u p l e of t h e f i l t r a t i o n . d [u] = 0 r class
is easy.
in
r < n, since A -maps
is a standard
u
proof that
as
f*(L
f being an An-map respect
interpretation
which can be seen most
A direct
If w e r e g a r d
K(lr, q ) , t h e n
2)
u
implies
the differentials
from
being an A -map n
) where
q
easily
of t h e
L
q
the implies
is the fundamental
dr[U ] = (fx...xf)*
dr[L q]
which are defined in terms
for
of
n
XP(s),
s < n. --
resents
L
Now
X-* ~XP(n)
Corollary
if
Su
If X
v ~ Hq+I(Bx;w)
r
since
extends
if a n d o n l y if
reasons,
L
q-r+l
Special
XP(n)-*
survives
q
cases
K(lr, q + l )
in which all maps and
dr[U ] = 0 for
of course,
i.e. , dr[U ] ~
which is zero for
to
is (p-1)-connected
The point is,
Corollary
] for all
-* ~ B K ( y r , q) -* K
10.3.
nectivity
q
to
E
where
it rep-
q+l" Finally
as
dr[t_
then
can be factored
are at least A -maps. n
u ~ Hq(x;~r), then
r + 1 < ¢L+2 -- p+l
that higher
El+r, q- r+l r
u
for
"
differentials
and
u = ~v
vanish
for con-
l+r, q=H ~ E1 r+l q-r+l
r+l
~X)
< (r+l)p. are well known.
10.4.
~ : H q + l (Y;~r) -* H q ( ~ Y ; ~ r )
is onto for
10.5.
~ : H q + l ( y ; l r ) -* H q ( ~ Y ; l r )
maps
q < 2p
where
Y
is p-
connected. Corollary q < 3p
where
Y
is p-connected.
We write "loop class"
greater these
~
rather
(Corollary than
a
content than the over-worked corollaries
yield Corollaries
9.10 is now established.
and refer
rather than as a " s u s p e n s i o n "
onto the primitive
to a class
subspace )
in the image
as a
as w e prefer terminology of
"suspension".
Applied
9. 9 - 1 0 of t h e l a s t c h a p t e r .
to k - i n v a r i a n t s ,
for
46
For class
Y = K(G,n),
it so happens that primitive
implies being a loop
if w = Z , b u t t h i s i s n o t t r u e in g e n e r a l . P
Example
10. 6. [ S t a s h e f f ] . --
a loop class.
In f a c t
Let
~ ~ 0 { HZn(z
d2(~P) ~ 0, s o
, 2n-1;Z).
p
~P
~P is n o t r e p r e s e n t e d
is p r i m i t i v e
but not
by ahomotopy
associ-
ative map. To obtain examples ferent
of m a p s w i t h
d
r
~ 0 for
r > 2, w e h a v e t w o d i f -
sources.
Example
10.7.
An-map
A non-zero
class
a
in
H2n(~cP(n))
is r e p r e s e n t e d
by an
b u t n o t b y a n A n + l - m a P. The class
cannot be a loop class
since
H2n+l(cP(n))
= 0.
Recalling
that ~CP(n) ~ S 1 X ~ S zn+l, the only possible'non-zero differential is d lot] = k[ul.., lu] where n
u ¢ HI(~cP(n)).
It follows that S I-~ COP(n)
is not an
An+l-mapE x a m p l e I0.8. [Zabrodsky]. _
Let X ~ K(Z --
, 2n-l) )< K(Z p
ative multiplication obtained as the loops on the space so that B x ~ Y. Ap_l-map,
The class
, Znp-g) have the associP
u = ~ 2np-Z c HZnP'Z(X;Zp)
Y
with k-invariant
(~ Zn )P,
is represented by an
not an A p - m a p . B y comparing the spectral sequence with that of the product structure
on K(Z
, gn-l) X K(Z P
, 2-np-Z) w e see the only w a y the class
(L Zn )P =
P
it Zn_ll... [~ gn_l ] can be killed is by E x a m p l e I0. 9. [Zabrodsky]. and k-invariant ~ l WZp_z(Y)
restricts to ~ p - 2 in K ~
and
k-invariant ~ l
= 0, there is a class
2p-3
to ~ P - 2 L 4p-6"
be the space with
WZp- 3
Now
u
4p-5"
~Y
where
2p-2 + k(L 2p-2 )2 for any
u c H2P(P'I)-3(X)
(We have -~p-2~pl
k ~ Z p"
which restricts
is not a loop class since there is no class in Y
which
+ ks 2) = -2s P + k z ~ P P - J ~ P J - Z ~
, 2p-2). ) B y the s a m e token, for k = 0, L P goes to P
W4p- 6 ~ Zp
Zp-3' which can be regarded as the loop space
w4 p_5(Y) ~ Zp
Since ]PP-Z~IL
Let X
dp_l[~ 2np_2].
zero
in Y
and
0
47
thus u
[~ I ' ' "
I t ] must
is the first
class
assasin.
Thus
Example
10.10.
u
be killed in the Eilenberg-Moore
in
X
which exists
is represented
[Gheng].
Let
for unstable
spectral
reasons,
by an Ap_l-rna pbut
Y be the space with
~
With respect
back to the fundamental
not an Azi-rna p.
class
is represented
Again
be the
k-invari-
" 2 u ~ H z l - (f~Y) w h i c h p u l l s
a class
K(Zz, zi-z)
must
w2 = w 2 i _ l = Z 2 a n d
to loop multiplication, of
u
Since
not by an Ap-map.
2 i+l ant
sequence.
by an A . -map but 21_i
u kills [~ ii... IL i] in the spectral sequence.
For
i=g,
v ~ H 7 {Y) w h i c h restricts to S g S IL is represented by an A ) - m a p q q 7
the class
but not an A4-rna p for any A4-structure on
Y.
It is possible to give chain formula for using a spectral sequence is to avoid such work.
d , although a m a j o r point of r T h e case
d g is quite m a n a g e -
able and illuminating in t e r m s of our next topic. If
dl[U ] = 0, t h e n
any representative Gq'l{x~:
X;w).
u ~ -For
u
u
we have
any choice
c = (1 × m ) ~F b - {m × 1)~
is primitive.
b
- w1 ~ -u - w/ u
m~
of b , t h e c o m p o n e n t
represents
by a coboundary
On the chain level,
dg[u].
u, w e a l t e r
b
a n d if w e a l t e r
we alter
c
by
dg[u ] is
[(1 X m ) * - ( m X 1)*] H q ' l ( x ~ x ; w )
[(1 X m ) ~: - ( m X 1 ~ ]
= 5b{u)
in
this means
where
our choice
of a c o c y c l e .
b •
G q - l { x ~b X ~ - X ; w )
Notice by altering of b
of
our choice
for a given
of u,
Thus the indeterminacy
which is
dl(Hq-l(x
for
~=X;w))
in
just as
it should be. Example
10. 6 i s w o r k e d
In C h a p t e r sented
by H-maps.
associative
out this way in [Stasheff].
8 we saw that the k-invariants
Similarly
of an H-space
one can show the k-invariants
H - s p a c e are r e p r e s e n t e d
by A3-rnaps,
are repre-
of a h o m o t o p y
so e x a m p l e s I0. 6 and 10. 8
and I0. 9 for p = 3 provide e x a m p l e s of H - s p a c e s w h i c h are not h o m o t o p y associative.
In order
associativity
to generalize
more
fully from
these
results
our homotopy
to A -maps n
we need to study
p o i n t of v i e w .
A
-SPACES n
We h a v e s e e n t h a t t h e e x i s t e n c e H-space
is e q u i v a l e n t to h o m o t o p y a s s o c i a t i v i t y ;
significance
of p r o j e c t i v e
induced by A3-maps natural
of a p r o j e c t i v e
n-space.
In b o t h c a s e s w e a r e l e d to c o n s i d e r
invariant
for an
to i n q u i r e a s to t h e
On the other hand, we have seen that fibrings
to a s k a b o u t t h e s i g n i f i c a n c e
homotopy
it is n a t u r a l
admit homotop7 associative
equation but as a conjery
three-space
again,
it i s
of a f i b r a t i o n b e i n g i n d u c e d b y a n A - m a p . n the associative
of n - v a r i a b l e
characterization
multiplications;
equations.
of s p a c e s
law not as a three variable T h i s in t u r n l e a d s to a
of t h e h o m o t o p y t y p e of a s s o c i a t i v e
H-spaces. Consider determine
five maps
a single application topy as a map
the various
w a y s of a s s o c i a t i n g
of X 4
X, e a c h of w h i c h is h o m o t o p i c
into
of h o m o t o p y a s s o c i a t i v i t y .
h : I-~ X X3, we can construct
Regarding a map
S1 as a pentagon with the five maps as vertices
If t h i s m a p c a n b e e x t e n d e d to a t w o c e l l of p r o j e c t i v e case for
~Ix,
the associating
spaces
four factors.
These
to t w o o t h e r s b y
the associating
homo-
S 1-* X X 4 b y r e p r e s e n t i n g
and the five homotopies
as edges.
,iwx e 2 with boundary
can be extended one stage further.
t h e s p a c e of l o o p s p a r a m e t e r i z e d homotopy can be represented
S 1, t h e c o n s t r u c t i o n
T h i s i s of c o u r s e
by the unit interval.
schematically
by
the
In ~ I x ,
49
s o t h e m a p of
S1 we are looking at is represented
which can be extended to
e 2 by deforming
by
all paths to
(wx) (yz)
in the obvious
way.
To proceed volving maps morphic
to
m.
1
K 2 = ~.
(K r × K s ) k
K . X X i -~ X
:
1
where
w e n e e d a f a m i l y of c o n d i t i o n s
K.
is a s p e c i a l
1
in-
cell complex borneo-
Ii-2.
D e f i n i t i o n U . 1. Let
with this approach
K. d e n o t e s a c o m p l e x 1
Let of
K. = C L . , 1
1
(Kr × K s )
symbols,
e.g.,1
Z ...
responds
to i n s e r t i n g
constructed
t h e c o n e on
L.
(k k + l . . . two pairs
w h i c h i s t h e u n i o n of v a r i o u s
1
corresponding
inductively as follows:
to inserting
k+s-1) ...
i.
of p a r e n t h e s e s
a p a i r of p a r e n t h e s e s
The intersection with no overlap
copies in
of c o p i e s c o r or with one as a
subset of the other: I ...
(k...k+s-l)
...
I...
(k... (j...j+t-l)
(j...j+t-l)
...
r
× K
s
-~ K.
1
(An a l t e r n a t i v e
sense,
is t h e i n c l u s i o n of the c o p y i n d e x e d by indexing by trees
i
k+s+t-Z) ...
Thus the foUowing definition makes K
...
or i.
where
a k ( r , s) :
I . . . (k... k+s-1) . ..
i s g i v e n a t t h e e n d of t h i s c h a p t e r . )
i.
i
50
Definition maps
11.2.
An An-space
(X;{Mi})
M . : K. X X . - ~ X , i < n
x i) = M r (p, x [ . . . . .
M.
exist and satisfy
and a family
of
M s (or, x k . . . .
Xk+ s _ l ) . . . . .
xi)
p ~ K , ~ c K . r
If t h e
X
with unit and
M i ( ~ k ( r , s) (p, a ) , x I . . . . . for
of a s p a c e
such that
1) M 2 i s a m u l t i p l i c a t i o n 2)
consists
these
S
conditions
for all
i > 2, w e s p e a k
of
{X, { M i } )
1
as anAl-space.
Where
Conditions
necessary,
we refer
approximating
The complexes
K.
to the
these were
{Mi}
as an An-fOrm.
first presented
a r e a l s o of i m p o r t a n c e
in [Sugawara].
in category
theory
in
1
relation
to coherence
morphic
to
exhibited
Ii-2
the
of f u n c t o r s
is not obvious.
K.
1
as specific
[MacLane]. Several
That the complexes
are homeo-
ways to see it are available.
convex subsets
of
Ii-2
which are
clearly
I have homeo-
K3 morphic
to the whole cube,
e.g. ,
K2 = *
~0
1~2
/"
1
/
K4
\ Adams
has computed
the homology
and fundamental
group
of
L.
and
1
thus shown
L. f o r 1
i>
shown the cell complex Z~i - 2 .
[Boardman]
5 has the homotopy
t y p e of a s p h e r e .
L . i s t h e d u a l of a c e r t a i n z
has given a cubical
decomposition
subdivision of
K. 1
idea first
suggested
by Adams.
Stallings
has
of the boundary
indexed by trees,
of an
51
Associative fined to have constant
H-spaces value
are
of c o u r s e
Before
description
proving
this,
-spaces
o0
since
M.
can be de-
1
x 1. . . x i.
T h e m a i n p o i n t of t h e d e f i n i t i o n invariant
A
of A
CO
of a s p a c e
of t h e h o m o t o p y
we present
the main theorem
-space
is that it is a homotopy
t y p e of a n a s s o c i a t i v e about A -spaces,
H-space. which is
n
w,h a t o n e s h o u l d e x p e c t . Theorem
11.3.
A connected
CW
admits
X
the structure
of a n A - s p a c e
if a n d
n
only if there
exists
a sequence
of quasLfibrations
E 0 =X-*
1-*
,., with
E.1 c o n t r a c t i b l e
in
En_ 1
. . .
B I-~ ...
Bn_ 1
E i + 1.
The construction
is not iterative,
En
although
En_l~
be
~
inductive.
We let
Kn+ z X X n+l
n
Pn Bn
Bn_l~
Kn+ z X X n n
The attaching ~n{0k(r' s)(P'c)'Xl with the
M
term
s
.....
map for
if
the first
x
factor,
. M. s (. ~ ' X. k . . . .X k. +.s -.1 ). .
k + s - 1 = n + 2.
x
coordinate
By induction we prove first
is given by
n
Xn+l) = ~ r ( P ' X l ' .
deleted
obtained by dropping
E
Pn'
is a quasifibration.
The attaching
map for
B
n
is
consistently.
induced by projection
This
Xn+l)
time
we break
onto all but the
B
into two
over-
n
lapping pieces
by considering
1
crucial
condition
weak homotopy one
onto
in proving
equivalence
and breaking
K. = CL.
Pn
is a quasifibration
occurs
as the fibre
~ -~ ~
occurs
over
over
(~,x_ .....
x ) where n
L..
This map
can be identified with mapping
1
up a cone as before.
The
1
in showing that a
(T,x z .....
is a deformation x
Xn )
is mapped
of a n e i g h b o r h o o d into
x
by
of
to L. 1
52
x-~ M r(p,x,x 2 ..... M
r
(p,x,e .....
e)
As for the limit, Theorem o n l y if
l l . 4. X
Xn) since
for fixed X
p
and
is c o n n e c t e d ,
the arguments A connected
X
admits
the structure
admitting
an associative
before,
an A
00
for some
invariant
multiplication
is not.
associating
if a n d
a s o p p o s e d to
We h a v e m a d e t h i s l a t t e r
follow.
quasifibrations
Thus
multiplicative
as in T h e o r e m
S3 with these particular
the unit. A d a m s
has given m e
-form.
remark
on
S3
chosen
Actually he works by
exotic multiplications
cannot be deformed
T h e p r o o f of T h e o r e m
o0
11. 3 f r o m w h i c h t h e A
h o m o t o p y t y p e of an a s s o c i a t i v e
these multiplications
of m u l t i p l i c a t i o n s
[Slifker] shows that a properly
h o m o t o p y c a n b e e x t e n d e d to a n A
constructing
H-space
=o
-forms
has the homotopy
t h o u g h on t h e s t a n d a r d
to be a s s o c i a t i v e .
11. 4 w h i c h i s i m p l i e d b y o u r e x p o s i t i o n u s e s
an alternative proof of a stronger result w h i c h
no use of units.
Theorem {Mi} ~
-space
b u t l e t us e x p a n d on it now.
only eight are homotopy associative.
makes
00
is a homotopy invariant while
R e c a l l t h a t of t h e t w e l v e h o m o t o p y c l a s s e s
S3
of a n A
case.
Y.
statement
-structure
to
to t h e i d e n t i t y .
t o t h o s e in t h e a s s o c i a t i v e
CW
since admitting
and hence homotopic
similar
N o t i c e t h i s is a h o m o t o p y 4.3,
T h i s in t u r n i s h o m o t o p i c
are
h a s t h e h o m o t o p y t y p e of ~ Y
Theorem
x..1
II. 5. {Adams). satisfying
If X
admits a m a p
2) of II. 2, then X
with an associative multiplication n
M z : X X X-~ X
and a family
is a deformation retract of a space
such that n IX X X
is h o m o t o p i c in Y
Y to
m. T h e proof has b e e n simplified by [Boardman]. while for A
n
-spaces with n
Definition U. 6. Bn_iVKn+
defer the proof
finite, w e again look at projective spaces.
If (X, { M i } ) is an A
Z X Xn
We
n
-space, XP(n)
constructed in proving T h e o r e m
will denote the space
II. 4.
53
Theorem
Ii. 7.
If Y
is a M o o r e
space of type
(G, Zp+l) w h e r e
abelian group in w h i c h division is possible for all p r i m e s prime
p, t h e n
Y admits
The maps
the structure
M.
for
i< p
of a n A p . l - s p a c e
are constructed
G
is an
q less than the b u t n o t of a n A p - s p a c e .
a s in t h e c a s e
p = 5
1
(Example M
P
7.6) by deforming
t h e t r i v i a l o n e s in ~ 2 ~ Z y .
follows from the decomposability
p-fold cup products
in
YP(p)
of ~ p + l
The nonexistence
contrasted
of
with the non-trivial
if i t w e r e to e x i s t .
Given two A n - s p a c e s ,
w e can again consider m a p s
which respect the
structure. Definition II. 8.
If (X;(Mi})
a homomorphism
if
and
(Y, {Ni})
are A n - S p a c e s ,
a map
f : X -~ Y
f M i (7, x 1 ..... xi) = N i (~, fx I..... fx.1)" It is also possible to consider m a p s
of A
-spaces which respect the n
structure
up to homotopy,
pletely here.
but the details are too complicated
F o r example,
to m e n t i o n
respecting a n associating h o m o t o p y
corn-
involves a 2-
cell subdivided as a hexagon, while respecting an A4-structure involves a c o m plex w h i c h looks like
7ill
z_&L
k\ \
is
54
However,
maps
of an A
n
-space into an associative H - s p a c e
or vice v e r s a are
manage able. Definition A map
U . 9.
Let
f : X -~ Y
(X, { M i } ) b e a n A n - s p a c e
is an A -map
if t h e r e
exists
and
Y
an associative
a family
H-space.
of m a p s
n
h i:
Ki+ 1 × X i-~ Y
such that
h 1 = f and h i (Ok ( r , s ) ( p , a ) ,
x 1. . . . .
= h r ( 9 , x 1. . . . .
x .1) =
Ms(a,x k .....
= h r _ l ( p , x 1. . . . .
Xk+s.1) .....
X r . 1 ) h s _ 1 (¢r, X r . . . . .
It is easy to see that an A -map
xi)
x,)l
of a s s o c i a t i v e
for
for
r + s = i+2 k < r
k = r.
H-spaces
is an A -map
n
sense
with respect
Theorem
11.10.
to the trivial If
(X;{Mi})
higher
homotopies
is an An-space,
The proof is a generalization h i : Ki+ 1 X X i-~ ~XP(n)
then
X -~ ~ X P ( n )
conveniently
by defining some
cO
-form.
is an An-map.
8.6.
The maps
in terms
of
reasonable
homeomorphisms
K i + 2. M a n y of o u r r e m a r k s
associative H - s p a c e s
Theorem
used as the A
of t h a t of T h e o r e m
can be described
K i + 1 × X i × I "*- K i + 2 × X i ' * X P ( i ) Ki+ 1 X I~
in this
n
11.11.
X
about H-maps
c a r r y over to A
admits
n
-maps
of H - s p a c e
in this m o r e
and A -maps n
of
general sense.
of a n A - s p a c e if a n d o n l y if e a c h s t a g e
the structure
n
of a n y P o s t n i k o v
system
for
homomorphisms
and the k- invariants
Corollary ll. lZ. (cf. 9. I0). space provided
X
does in such a way that the projections are represented
An An_l-space
~.(X) = 0 for
i< p
and
X
Pn
are
by An-maps.
has the h o m o t o p y
type of a loop
i > np+n-4.
I
Example
11.12.
class
of ExampIe
u
Let
W be the space 10.8
or
constructed
10. 9, t h e n
W
by using as k-invariant
admits
an Ap_i-form
the
but not an
A -form. P These
examples
used the bar
construction
spectral
sequence.
More
55
generally,
for an A -space
X
we have the spectral
sequence
derived
from
the
n
finite filtration Theorem
of
11.14.
XP(n) Let
by
XP(i),
i < n.
(X, { M i ) ) b e a n A - s p a c e
and
u e Hq(x;Tr).
Then
n
If
3)
u
is represented
holds for
1)
d
2)
Su
r
[u] = 0 f o r
r < i
pulls back to
by an A.-map
then
1
i f a n d o n l y if
XP(i)
1) a n d
2)
follow.
The converse
i < n.
The converse know to prove
~XP(i)
is stated
-* X
in this limited
is an A.-map
way because
is to use
the only way I
XP(i+I).
1
Our analysis
of A - m a p s
in terms
of c o h o m o l o g y
classes
also applies
n
to the maps
inducing the succession
Zabrodsky'
s technique.
Zabrodsky's
method
of f i b r a t i o n s
used to construct
T h u s w e f i n d if t h e i n g r e d i e n t s
are A -spaces
and maps,
first
the result
is anA
used the technique
to construct
a homotopy
b u t n o t of t h e h o m o t o p y
sky].
was not an A5-space.
Example
the example
11.15.
Let
P1 = {2,3),
That the resulting
X
~ 1 : H3 ( X z ; Z 5 ) -~ s l l ( X z ; Z 5 ) essentially A3-space marks,
the same if
G
the example
not A -spaces P
is trivial.
can he adapted
several
i d e a of i n d e x i n g b y p l a n a r in the plane,
H-space
t y p e of a l o o p s p a c e
[Zabrod-
follows from Xz
is an A3-space
Adams
divisible
by
the fact that
to show
Z and
3.
to give finite complexes
follows from
Y(G;Zn+I)
is an
As Zabrodsky
re-
which are Ap_ 1 but
p.
Boardman' are
associative
X 2 = (S 3 × S 5 X S 7 × S 9 × s l l ) ( P 2 ) .
That
used by Frank
of r a t i o n a l s
for any prime
There
X 1 = SU(5),
is not an A5-space
argument
consists
-space. n
which was a finite complex In fact,
in
to be mixed by
n
Zabrodsky
X(]t~ 1)
s Proof clever
trees,
so as to keep track
of T h e o r e m
ideas
11. 5.
in the proof.
i.e. , directed
connected
of w a y s of i n s e r t i n g
First, acyclic
parentheses.
there
is Adam's
finite graphs For
example,
56
w{(xy)z)
corresponds Second,
an associative
to t h e t r e e there
operation
m ( m × 1) = m(1 X m )
D e f i n i t i o n 11.16.
~
o
is an idea from categorical not by a multiplication
algebra
of c h a r a c t e r i z i n g
m : X × X -* Y
and a relation
but as follows:
An A-structure
on a s p a c e
X
i s a f a m i l y of c o n t i n u o u s m a p s
n
{k i : X i - ~ X , i > 2} Usually, The other trees
k.
1
s u c h t h a t if k
i
~ m. = m then k o(vk ) = k . 1 n rn. m 1 1 is t o b e t h o u g h t of a s t h e m u l t i v a r i a b l e m a p
can be indexed by the trees
can be obtained by composites The complex
where
T
K.
1
is a tree with
without disconnecting
n
the tree).
For
-//
m
.
and all the
of t h e s e .
will now be represented
branches
edge is subdivided needlessly.
~kk
x 1. . . x
a s a u n i o n of c u b e s
C(T)
(= i n p u t s = e d g e s w h i c h c a n b e r e m o v e d our present
The cube
C(T)
purposes
we will assume
will have parameters
no
indexed by
t h e e d g e s of t h e t r e e w h i c h a r e n o t b r a n c h e s .
Definition II. 17.
WA
with n-branches.
(n, I) is the union with identifications of C(T) over all trees
T h e identifications are that a face
t. = 0
of
C(T)
is to be
1
identified with edge indexing
C(T' ) w h e r e
T'
is obtained f r o m
t.. 1
F o r example, WA(Z,I) =
=
C(V)
W A ( 3 , 1) = ~ ( ~ / ) G ( ~ )
G(h) ~)
T
b y shrinking to 0 the
57
It c a n b e s h o w n t h a t Definition
U. 18.
A
WA(n,1)
WA-structure
on
is a cubical
X
is an A
subdivision
-structure
of
K . n
without units,
i.e. ,
~C
a family
of maps
l~l : W A ( n , 1 ) × X n - ~ X n
Mn(~, x I .....
x n) = lVir (p, x 1 . . . . .
if ~ = (tI ..... tn_Z)
•
C(T)
with
such that
Xk_ 1, M s (c;, x k . . . . .
t. = 0 w h e r e ,
Xk+ s) . . . . .
x n)
if the e d g e indexing
t. is
1
deleted, same
T
as
decomposes
T' [J T "
values to the c o r r e s p o n d i n g
1
while
p e C(T' ), a e C ( T " )
e d g e s as does
7.
T h e f i n a l i d e a of t h e p r o o f i s to u s e t h e nective
tissue
to build something
like a tensor
associate the
WA(n,1)
algebra
complexes
of w h i c h
X
as con-
will be a
retract. First
we let
that now we permit
For
example,
description as
WA(n,1)
trees
WA(Z,I)
be the complex
with an extended
= C(y) = ;
root,
: and
constructed
except
i.e. , "'~'"
WA(n,I)
gives a useful parameterization.
as before
= WA(n,l)
We also let
× I but the tree
WA(1,1) ~ •
regarded
C(1).
Definition MX
If. 19.
Given a
is defined by taking
(T,x I..... • ,p,a
x n)
with
(~)
WA-structure WA(n,1) × Xn (p,x I .....
on
X, the associated
for each
n
Xk_l, Ms(~,x k .....
associative
space
and identifying Xk+s_l) .....
x n)
where
are as above. If
t. = 0
on the edge corresponding
to the extended
root,
then
(•)
1
means
Mn(t I .....
~i .....
The operation
t k - l ' Xl . . . . . on
MX
= ((p,a),x I ..... regarded
as being in
tive involves combinatories
essentially
only the
aptly called tree
is given by
Xr+ s)
C ( T 1 v TZ).
Xn)"
where
(p,x I .....
now for
That the operation WA-parameters,
surgery.
x r ) • (a, X r + I . . . . .
p c C(TI),
ae
X r + s)
C(Tz), (p,a)
is well defined and associaan exercise
in parameterized
is
MASSEY PRODUCTS The differentials were
usefu~ in analyzing
the homology
spectral
AND GENERALIZED in the cohomology
k - invariants
sequence,
BAR (31)NSTRUCTION
Eilenberg-Moore
in terms
of
spectral
A -maps. n
we have in particular
sequence
If we turn to
differentials
of the form
dr[all .. l Ur+ l] represented These
by homology
are closely
ucts originally introduced,
related
of t h e a s s o c i a t i v e
to the Pontrjagin
defined in the cohomology
these homology
the duality; however, algebra,
classes
products
Massey'
ring analogues of a n a r b i t r a r y
were
s procedure
so we will use the term
"Massey
should now expect,
strict
homotopy
we save such generality
analogue;
associativity
As for the differentials
H-space
called
in question.
of t h e [ M a s s e ~ p r o d space.
Yessam
When first
products
product"
generically.
as one
by an appropriate
f o r t h e e n d of t h i s c h a p t e r .
more
generally,
it turns
products
out [ May]
by appropriate
Massey
allow matrices
classes
than single homology
rather
differential
In fact,
that they are all determined of h o m o l o g y
to emphasize
is valid in any associative
can be replaced
dr
X
if w e a r e w i l l i n g t o classes
as
arguments. Until further
notice,
with differential
d.
Definition
Let. u,v,w
product
12.1. ~
Remark.
If defined,
has the larger remarks higher
where
.
u
E H(A)
.
indeterminacy
Massey
be an associative s original
such that
of H ( A )
represents
like this continue order
A
We start with Massey'
is the coset
B
ux(.1)deg u
let
by
uH(A) + H(A)w
.
.
[Uehara
.
triple
by
dy = uv.
Notice that
d2[ulvlw]
this situation,
The differentials are usually
and Massey]-
The Massey
As we generalize
but the latter
algebra
determined
dx = vw,
d2[ulvlw]
represents
to be applicable.
products,
product
uv = 0 = vw.
u, e t c . , a n d
H(A). H(A).
differential
more
are determined delicate,
less
by
60
often defined and with s m a l l e r i n d e t e r m i n a c y .
choice, for of course
T h u s in g e n e r a l i z i n g we h a v e s o m e
d r [ . 1 1 . . , l ur+ 1] can itself be regarded as a generalization.
We w i s h to d e f i n e h i g h e r o r d e r M a s s e y p r o d u c t s < u l. . . . . , u i + s >
a r e defined and z e r o for
l w h e r e
[May] h a s
d e v e l o p e d the f o l l o w i n g n o t a t i o n .
D e f i n i t i o n 12.2.
aij ~ A
The M a s s e y p r o d u c t < a 1. . . . , a t >
for l is d e f i n e d , t h e n a n y of its r e p r e s e n t a t i v e s
dr'l[all...
]ar].
The p r o o f is s t r a i g h t f o r w a r d , though t e d i o u s .
The d e f i n i n g s y s t e m
aij is used directly to show d s [all " ' " l a r ] = 0 for s < r - 1 a n d to o b t a i n a representative
X i n B'{A) of
[ a l l . . . J a r ] , w h i c h is a c y c l e u n d e r the t o t a l
d i f f e r e n t i a l in B(A).
E x a m p l e lZ. 4.
In H . ( ~ C
P ( n ) ) , if u g e n e r a t e s
Hi, t h e n < u , . ~ n+l
generates
HZn.
S i n c e n - f o l d M a s s e y o p e r a t i o n s a r e e a s i l y s e e n to be n a t u r a l
w i t h r e s p e c t to A - m a p s , t h i s a g a i n shows n
S1-~ ~ C
P(u)
i s - n o t a n A n + l - m a p.
61
T h e c o m p u t a t i o n m a y be done by o b s e r v i n g
H2n+I(CP{n)) = 0 w h i c h c a n b e
a c h i e v e d o n l y if the g e n e r a t o r of H Z n ( ~ C P(n)) The m a t r i c M a s s e y product,
is k i l l e d by
dn.
i n t r o d u c e d b y M a y , is a f a i r l y
s t r a i g h t f o r w a r d g e n e r a l i z a t i o n in w h i c h
a.
V..
is r e p l a c e d by a m a t r i x
l
obtain a reasonable definition, certain conventions about matrices
To
1
will be
observed. If V is a m a t r i x
(v..), t h e n ~
w i l l b e the m a t r i x
((-I) l + d e g v i i v . . ) .
U
An o r d e r e d p a i r of m a t r i c e s n × q and for each
i,j,
matric Massay product Vn
is
q × 1 and
U
(X,Y)
is m u l t i p l i c a b l e if X is
deg Xik + deg Ykj w i l l be c o n s i d e r e d only if V 1 is n
V 1 . . . V . , j V j+ 1 . . .
Vk
is m u l t i p l i a b l e f o r e a c h
G i v e n the a b o v e c o n v e n t i o n s , D e f i n i t i o n 12. g c a r r i e s d e f i n e the m a t r i c M a s s e y p r o d u c t
"
Y is
1 × P,
j , k _ < n. o v e r v e r b a t i m to
T h e i n d e t e r m i n a c y is the s e t
of a l l p o s s i b l e d i f f e r e n c e s c o r r e s p o n d i n g to d i f f e r e n t c h o i c e s of t h e s y s t e m
A... U
M a y (in p a r t f o l l o w i n g [ K r a i n e s ] ) g i v e s b o u n d s on the i n d e t e r m i n a c y , linearity formulas,
associativity formulas,
Massey products, permutation rules. As b e f o r e ,
" s l i d e " r u l e s and for o r d i n a r y
He a l s o d i s c u s s e s n a t u r a l i t y .
t h e s e M a s s e y p r o d u c t s c a n b e r e l a t e d to the d i f f e r e n t i a l s
in the s p e c t r a l s e q u e n c e .
Matric products are particularly relevant when
dr
is
d e f i n e d on a c o m b i n a t i o n of t e r m s w i t h o u t b e i n g d e f i n e d on the i n d i v i d u a l t e r m s . F i n a l l y M a y s h o w s t h a t m a t r i c M a s s e y p r o d u c t s d e t e r m i n e the s p e c t r a l s e q u e n c e in t h e f o l l o w i n g s e n s e : such that for each element
x
if x
d x P
s u r v i v e s to
<W 0 . . . . .
E , then P
Wp_I,V>.
all
q there are matrices
of E p ' q t h e r e is a c o l u m n m a t r i x is r e p r e s e n t e d
The differentials
of p a r t i a l d e f i n i n g s y s t e m s f o r
For each
drX f o r
<W0 . . . .
x ; on the o t h e r h a n d , t h e y a r e h u g e .
'
W
p-l'
V
such that
by a suitable r e p r e s e n t a t i v e r .
W.
of
c a n b e e v a l u a t e d in t e r m s
N o t e t h a t the
W.
i
work for
62
Of course
A -spaces
can be mimiced
on t h e c h a i n l e v e l ,
just as
n
associative
differential
algebras
Definition 12.5 [Stasheff]:
mimic
topological monoids.
( A , m . , 1 < i < n)
is a n A - a l g e b r a
1
--
if m . : A x -* A
n
i
such that 0 = T
(-I)
m r {l@. . . @ m s @ ' ' ' @ I )
k,s
k-1 where for
al@
[email protected] the
(s+l)k + s (i + ~
~ is
d i m a.) 2
/
I
m 1 p l a y s t h e r o l e of d i f f e r e n t i a l . )
(Note:
For an A
-algebra,
there
is a g e n e r a l i z a t i o n
of t h e b a r c o n s t r u c t i o n .
00
Given an A
Definition 12.6 [Stasheff]:
-algebra
(A,m.),
o0
B'(A)
is
@ i=0
A i with the differential
~[all-.-lanl : where
the tilde construction
1
(-1)x [all... [ms(ak®...®%+s_l)l'-" I%1.
k-1 )~ = (s+l)(k+l) + s(i + >
dim aj).
1 A l l of o u r m a c h i n e r y and Massey
products,
is d e f i n e d b y Finally,
connected
C QC P q
uy+xw+_m3(u,v,w)
differential (1~)
Let
including the spectral
for
coalgebra,
JC = C/A
~
Let
x e C
P
For example,
Let
~
P,
and define
E@...®C
n>O ~ --
be a simply
graded,
denote the component
> 0 that
•
(C, ~)
i . e . , if p o s i t i v e l y
%, q
7-Cp
~(C) = T(~) =
sequence
[Stasheff].
the dual situation.
= (&®l)~.
and we shall assume
£~0, p(X) = x@l.
over,
albeit with additional complication.
l e t us c o n s i d e r
associative
C O = A, C 1 = 0 and
carries
n
o(X) = l ~ x
and
of
~
in
63
with
@c = - d c =~'---- (-l)P A
Z__
p,n-q
c for c e C
and e x t e n d m u l t i p l i c a t i v e l y to
n
l Zl
1
3.
Q u a t e r n i o n i c m u l t i p l i c a t i o n on S 3
4.
S 7 with the Z - c o m p o n e n t s of ~.(S 7) k i l l e d f o r
i > 14
1
5.
S 7 with c e r t a i n p - c o m p o n e n t s of ~. (S7) k i l l e d for
i > Zl and Z-components
1
k i l l e d for
i > 14
6.
s7
7.
AMoore
space
Y(Z[I/Z],7)
8.
A Moore
space
Y{Z[I/Z,I/3],5)
9.
Y(Z[1/ Z], 5)
10.
[o, lz]
[0,8]
S1 If X is a f i n i t e c o m p l e x , the s i t u a t i o n c h a n g e s r a d i c a l l y .
a r e p o s s i b l e only for
1, Z, 3, a n d 10, of w h i c h a l l b u t
Examples
Z are given above.
For
a n e x a m p l e is p r o v i d e d b y [ Z a b r o d s k y ] : the H - s p a c e w h i c h m o d 2 a n d 3 is SU(6) b u t m o d
a l l o t h e r p r i m e s is
S 3 X S 5 X S 7 X S 9 X S 11.
A t t e m p t s h a v e b e e n m a d e to c h a r a c t e r i z e h o m o t o p y c o m m u t a t i v i t y in t e r m s of a u n i v e r s a l e x a m p l e .
C o n j e c t u r e 13.14.
X a d m i t s a h o m o t o p y c o m m u t a t i v e m u l t i p l i c a t i o n if a n d o n l y
if X is a r e t r a c t of ~ZSZX. [ W i l l i a m s ] h a s g i v e n c o n d i t i o n s w h i c h a r e e q u i v a l e n t to X b e i n g a n A ° ° - r e t r a c t of ~ Z s Z x .
STRUCTURE Since an associative on
BX
implies
homotopy
homotopy
commutative
H-space
X
commutativity
and homotopy
ON
Bx
is essentially
of
X.
~B x,
a multiplication
O n t h e o t h e r h a n d if m
associative,
then
m
is at least
is
an H-map,
since we have
(wx) (yz) _~ w(x(yz)) _~ w((xy)z) _~ w((yx)z) _~ w(y(xz)) _~ (wy)(xz).
Theorem
1 4 . 1 [Su_._gawara].
multiplication
(X,m)
if a n d o n l y i f m
Notice that for x y ..~ y x
If
by taking
m
is strongly
Similarly, regarded
xyz
yzx
Proof B X × By
and
of T h e o r e m . B X X Y"
homotopy
to be an H-map
w = z = e.
fill in the following triangles
is a n a s s o c i a t i v e
m
H-space,
Bx
implies
being an A3-ma p implies of
we can
S 1 X X 3 -* X :
xyz
zxy
The key to the proof is the equivalence
A specific
a
multiplicative.
(i. e . , w x y z _~ w y x z )
as maps
admits
equivalence
called the shuffle map
of is
induced by : A p × (X X e)p × Ziq )< (e × Y)q-~ A p + q )< Of × Y)P+q w h i c h triangulates (e X Y)q
~P X ~q
and shuffles
(X)< e) p
and
together according to w h i c h s i m p l e x of the triangulation is involved. Specific f o r m u l a s are e a s y to write d o w n [Sugawara, Iv[ilgram,
Steenrod] if w e r e p a r a m e t e r i z e • .. < s
< I.
We
An
by n-tuples
(sI ..... s ) s/t
set up the c o r r e s p o n d e n c e so that the face
0 < sI < s 2
k and l a s t d e g r e e
p > 2, we c o n s i d e r s e q u e n c e s £
1
lr), we
q
--
with e. = 0 or
I = (i 1. . . . .
We s a y QI is a d m i s s i b l e if
r
excess of I is > q and the last degree i raises to the
For
e(I) = il - Zi2 + i Z - Zi3 + ... + i r = 2iI- ~i"3
e(I) < i . Notice that QI --
H . (~nsn+k).
so
We let QI denote
S
¢'_
~ 1Q 16 ~ . . . ~
£
I = (e 1, s 1. . . . . S
kQ k, w h e r e
< n+k.
e k ' Sk)
84
: Hq (X) -~ Hq_l(X) 0-~ Z p - ~ Z p 2 - ~ Z the r e l a t i o n
is the B o c k s t e i n b o u n d a r y f o r the s e q u e n c e -~ 0.
[For
~ = 2, ~ n e e d not be u s e d e x p l i c i t l y s i n c e we have
~Qgi+l = oZi. ] [ D y e r and L a s h o f ] d e s c r i b e g e n e r a t o r s f o r
H , {"Ansn+k ~ ; Z p ) in t e r m s of a l l o w a b l e s e q u e n c e s .
May has translated" allowable"
into the following.
k Definition 16.8.
QI is m - a d m i s s i b l e
if PSi+ 1 " ¢ i+l >-- s i > [ m + >
Zsj(p-l)-¢j]l 2.
i+l The following two t h e o r e m s a r e then t r a n s l a t i o n s of t h o s e g i v e n by D y e r and L a s h o f .
T h e o r e m 16. 9.
F o r any
connected
X we have that H.(l~m 6 ~ n s n x ; z ) is f r e e P commutative on {x, Olx I x , basis of H,(X;Zp), Ol is d i m x-admissible}. N o w let O X
denote the base point component of lirn onsnx.
T h e o r e m 16.10.
H , ( Q S 0 ; Z ) is f r e e c o m m u t a t i v e on g e n e r a t o r s y(I) of d i m e n s i o n P e q u a l to deg I w h e r e I r u n s o v e r all 0 - a d m i s s i b l e s e q u e n c e s . [ F o r p = Z, k I = (s 1. . . . . Sk) is 0 - a d m i s s i b l e if 2si+ 1 > s i > ~ ' - - sj. ] The D y e r - L a s h o f o p e r a t i o n s b e h a v e as f o l l o w s : J = (¢l, Sl . . . . .
If
i+l
~k_l, Sk_l), then Q J ~ k y ( s k )
Y(S k) e HSk(QS0;Z z) and if J = (s 1. . . . . The g e n e r a t o r QI
=y(£1, Sl . . . . .
Sk_l)
then
Ck, Sk).
[For
Q J y ( s k) = y(s 1. . . . .
y(I) c a n not be i n t e r p r e t e d as
p = Z, Sk).]
QI(x) f o r a n y x s i n c e
is t r i v i a l on H0(QS 0) a l t h o u g h an i n t e r p r e t a t i o n in t e r m s of QI is p o s s i b l e
if we c o n s i d e r all the c o m p o n e n t s of
lira 0nsn.
Kudo and Araki prove their result by mimicing H*(Z, n;Zz). Theorem
for
The crucial machine is a dual t o ~ o r e ~ s transgression theorem.
16. U.
If H,(X;Z2)
Serre
Let X
be a simply connected, homotopy associative
has a simple system of transgressive generators
x. then I-I,(~X;Z 2) 1
is a polynomial ring on generators
Yi such that Yi £ Tx..1
H-space.
85
Dyer and Lashof proceed somewhat differently. H.(~ksix;Zp)
in the range < 3i- Zk which determines
They compute
H~(lim..f~nsn+kx;Zp).
They do the computation by analyzing the homology as a tensor product of monogenic Hopf algebras, mapping the corresponding tensor product of model spectral sequences into the Serre spectral sequence for ~ k + I s i x - J ~ k s i x - ~ ~ksix
and applying the comparison theorem.
Finally, they identify the various
classes at hand in terms of the operations QJ. ~nsn
The final stage from
~n-Isn
to
requires s o m e special effort. N o w let us return to a space of m o r e geometric interest.
The space
Q S 0 has the s a m e homotopy type as S F = lira SH(Sn), S denoting m a p s of degree I, but the equivalence is not one of infinite loop spaces. multiplications are not equivalent.
In fact even the
A major accomplishment of the last few years
has been the determination of the Hopf algebra H $ ~ ; Z z) and the algebra # H ( B F ; Z ) for p >_.Z. The operations Qi have played an important role in this P development. That H * 0BSF) was not the s a m e as H *(BQS 0) has tong been k n o w n for H ( B S F ; Z 2 ) D Z 2 [w i I i [ 2] w h e r e H~'(BQS0;Zz)
{wi}
is a n e x t e r i o r a l g e b r a .
a r e the S t i e f e l - W h i t n e y c l a s s e s .
While
The c o m p l e t e r e s u l t s c a n n o w b e s t a t e d ,
a l t h o u g h the p r o o f s a r e s o i n v o l v e d a l g e b r a i c a l l y a s to b e i n a p p r o p r i a t e f o r p r e sentation here. Let
o denote composition,
l o o p a d d i t i o n in QS °.
the m u l t i p l i c a t i o n in S F
We h a v e c o r r e s p o n d i n g
operations
t h e f i l t r a t i o n of H~ (SF) b y p o w e r s of t h e a r g u m e n t a t i o n
and l e t
Q~ and Q I. i d e a l and l e t
# denote Consider
E"
denote
the associated graded.
T h e o r e m 16.12. [ M i l g r a m ] . of H~(QS °)
For
p = 2, l e t
o r the i s o m o r p h i c c l a s s e s
I Q,~y(k) b e the D y e r - L a s h o f
in H ~ ( S F ) .
generators
86
y(k)
o y(k) ~ 0 ,
y~)o
y(k) o y e ) .
y(k) : 0
Q y(k) ° Q,y(k) = 0 in E ° (y(k) * y(k)) ° (y(k) * y(k)) : 0
Corollar~" 16.13.
H
(BSF;Z2) ~. Zz[Wi] Q C
E°
in
where
C
is isomorphic to
E(ezi+t Ii > I) ® r(g([)i I is l-admissible of length > 1). Here
eZi+l is dual to sly(i) • y(i)] and g([) is dual to ~y(1). The spectral sequence f r o m
ExtH~(SF)
to H
(BSF) has no choice but
to collapse since, being a spectral sequence of Hopf algebras, only primitive relations can be added.
This m e a n s only 2i-th powers of primitive classes could
be killed, but the only nontrivial ones present in E 2 are in ZZ[wi] which w e k n o w survives untouched. For
p > Z, there is a striking difference which is really a subtle
s i m i l a r ity.
T h e o r e m 16.14. [May].
For
T h e o r e m 16.15. [May].
In the E i ! e n b e r g - M o o r e
E2 ~ EXtH ( S F ; Z ) p If I = (1, j , J )
we h a v e
is 0 - a d m i s s i b l e ,
dp_l[Y(J) I . . . I
p > 2, H , ( S F ; Z
E 2 ~ E p . 1 with
P
)~--H(QS0;Z
P
) as Hopf a l g e b r a s .
spectral sequence for
BSF
with
dp_ t g i v e n as follows:
J odd of d e g r e e Zj-1 and l e n g t h > 1, then
Y(J)] =y(I)-
J.
C o r o l l a r y , 16.16. 1) 2)
H (BSF;Zp) ~ Zp[qi ] O E(~q i) O E O F w h e r e
{qi } a r e the
Wu c l a s s e s
E is a n e x t e r i o r a l g e b r a on p r i m i t i v e g e n e r a t o r s
e. dual to 1
~y(1, Z(p-1) 1, i) in H Z P i ( p - 1 ) - I ( B s F ) a n d
g(I) dual to ~y(I) w h e r e
I r u n s o v e r all 1 - a d m i s s i b l e s e q u e n c e s of e v e n d e g r e e and length
>
1.
87
3)
1~ is a divided polynomial algebra on primitive generators
~e. i
and g(J) where
J runs over all 1-admissible sequences of odd
degree and length> I. [The W u
class qi is dual to ~(l,i) and
~qi to a(0, i). ] Milgram' s and May' s proofs are rather unusual exercises in manipulating Hopf algebras over the Steenrod algebra or its dual. the C a f t a n f o r m u l a e and A d e m r e l a t i o n s for the t h e [ N i s h i d a ] r e l a t i o n s b e t w e e n the operations. structures
Theorem
Qi
Qi
It is important to have
[ D y e r - L a s h o f ] and e s p e c i a l l y
a n d t h e h o m o l o g y d u a l s of t h e S t e e n r o d
An old f a c t in h o m o t o p y t h e o r y c r u c i a l in r e l a t i n g the v a r i o u s a l g e b r a i c _ S n-l, •
involved. 16.17.
(~nsnx, e) is a module over
Corollary 16.18.
He{~nsnx)
Theorem
(~nsnx, e) is a module over
(sn-l,e
is a Hope algebra over
,°).
H e(s n'l,eS
n-I • , ).
S n-l, • 16.17.
(Sn-l,e
, .).
S n-l, Corollary 16.18.
I-Ie {~nsnx)
is a Hopf algebra over
T h e t h e o r e m is a r e s t a t e m e n t
(leg) . ~.a = f • r.c~eg . r.~. ~nsnx
is commutative, corresponding
e).
of the r i g h t d i s t r i b u t i v i t y of c o m p o s i t i o n
if f , g : S n -~ s n x
over track addition, i.e.,
He (Sn'l, •
and
a : S n ' l -~ S n-1
D i a g r a m a t i c a U y we h a v e w i t h
X ~ n s n X × F(n-1) -~ ~ n s n x
then
F ( n ) = Sn , e S n ' e , t h a t
X ~ n ~ n x X F(n)
~ n s n x ~'~F(n)
~ n S n "X - -X ~ n S n~X X F(n) X F(n)
~-~ n x
~'~nsnx X F (n) X ~ n s n x X F (n)
~
at least up to homotopy.
The diagram helps in describing the
c o n d i t i o n in h o m o l o g y .
T h e h i g h e r o r d e r p h e n o m e n a i n v o l v e d in h a n d l i n g t h e o p e r a t i o n s n e c e s sitate studying this distributive t e r n a t e d e s c r i p t i o n s of H n
or
l a w up to h i g h e r h o m o t o p i e s . Hn-structures P
on BO
and
BF
There are also al[Boardman,
T s u c h i y a , M i l g r a m ] w h i c h s h o u l d g i v e the s a m e h o m o l o g y o p e r a t i o n s b u t at the
88
m o m e n t a r e n o t k n o w n to do s o . characterizeable problems,
FinaUT, infinite loop spaces should be
in t e r m s of t h e m a p s J ( n )
*...*
~ ( n ) X X n - ~ X.
In a l l t h e s e
we a r e f a c e d w i t h a n a l T z i n g a f a m i l i a r a l g e b r a i c s t r u c t u r e f r o m a
h o m o t o p 7 p o i n t of v i e w , b u t p e r h a p s t h e s p i r i t of t h a t p o i n t of v i e w is by n o w sufficientl 7 clear.
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