i.
I.i.
Complexes
We c o n s i d e r sets endowed w i t h an order relation denoted b y
C
and read "is a face of" o...
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i.
I.i.
Complexes
We c o n s i d e r sets endowed w i t h an order relation denoted b y
C
and read "is a face of" or "is contained in" . Such a set is called a complex if the o r d e r e d subset of all faces of any given element is isomorphic w i t h the ordered set of all subsets o f a set, and if any two elements denoted b y
A,B
have a greatest lower bound,
A N B . A complex has a smallest element w h i c h we shall always denote b y
0 . The n u m b e r of m i n i m a l non and denoted b y
0
faces of an element
A
is called the rank of
rk A . The elements of rank i are called vertices.
A ,
Since an element of
a complex is c o m p l e t e l y c h a r a c t e r i z e d b y the set of its vertices, we m a y also define a complex as a set (x}6 A
for all
~
of subsets of a set
x 6 V , and that
of
&
by
rk A , is b y d e f i n i t i o n
V
B C A e A
is its c a r d i n a l i t y (as a subset of
(the set of vertices), implies
such that
B e A ; the rank of an element
V ) . The rank of a complex
A , denoted
sup {rk A I A r AS 9 A complex is called a simplex if
it is isomorphic to the set of all subsets of a given set, o r d e r e d b y inclusion. Hereafter,
~
always denotes a complex.
We define a m o r p h i s m
G
: & ~ ~'
of
~
into another complex
a m a p p i n g of the u n d e r l y i n g sets such that, for e v e r y to the simplex of all faces of simplex of all faces of
G(A)
A
is a subset of
in
A
b e l o n g to
If
A 6 ~ , the restriction of
. (About later use of the w o r d ' m o r p h i s m ' i n these notes,
~
9
is b y definition a complex
A'
w h o s e u n d e r l y i n g set
Z~ , and such that the inclusion is a m o r p h i s m (this means that the
o r d e r relation of A
as
is an i s o m o r p h i s m of ordered sets onto the
see also the general c o n v e n t i o n at the end of n ~ 1.3)
A s u b c o m p l e x of
~
A ~ is induced b y that o f
~ , and that if
A e &' , all faces of
A~ ) .
A r f~ , the set o f all elements of
with the o r d e r r e l a t i o n induced b y that of
A
w h i c h contain
Z~ , is a complex,
A
, together
called the star of
A
(in
A ) and denoted by
St A .(Notice
commonly used in topology,
where
B 9 St A , the rank of
in
denoted by
B
the slight deviation from the terminology
St A
is often called the link of
St A is called the codimension of
A
A
in
in
& .) If
B , and
codim B A .
If two elements
A, B 9 A
have an upper bound, they are called incident
(another deviation from the standard terminology!); upper bound, denoted by
A sequence if, for every
in that case, they have a least
A U B .
AO, A I, ..., A m
of elements of a complex is called a chain
i = O, i, ..., m-l , one of the two relations
A i cAi+ 1
or
Ai+ 1 c A i
holds. If
&
and
A'
are two complexes,
the direct product of their underlying
sets, ordered by
(A,A') C (B,B ')
is a complex
A * A'
1.2.
A C A'
and
B C B'
,
, called the loin of the two complexes.
Let
called the incidence. by inclusion,
iff
V
be a set endowed with a reflexive and symmetric relation,
The subsets of
form a complex
F(V)
V
whose points are pairwise incident,
whose vertices are the sets reduced to one point.
A complex is called a flag complex if it is isomorphic to such an
A morphism
Let
~ , A'
A~A
~
be two complexes and let
V , V'
F(V)
V ~V'
to be the restriction
.
be their sets of vertices.
is completely characterized by its restriction to
condition for a mapping
ordered
of a morphism
V . A necessary A~'
is that
it maps pairs of distinct incident vertices onto pairs of distinct incident vertices; if
~' is a flag complex,
this condition is also sufficient.
1.2.1.
PROPOSITION.
A complex
set of palrwlse incident elements of
A
~
is a fla~ complex if and only if every
has an upper bound. I f
~
has finite rank,
it is a fla~ complex if and only if every triplet of pairwise incident elements has an upper bound. The proof is immediate.
1.3. A complex
A
is called a chamber complex if every element is
contained in a maximal element and if, given two maximal elements
C, C', there exists
a finite sequence
(i)
C = C O , C I , ..., C m = C'
such that
(2)
for all
codlmCi_l(Ci_ 1 N
i = i, ..., m . The maximal elements of
now on, the letter
&
contained in a subset Cham L
C i) = codimci(Ci. 1 0
A
C i) C 1
are then called chambers. From
will always denote a chamber complex. The set of all chambers L
of a chamber complex is denoted by
Cham~ L , or simply by
when no confusion can arise.
An element Indeed, let
A e A
C, C' e C h a m &
above, and let by induction on
B = CO O C1N
has the same codimension in all chambers containing it. be such that
A C C N C'
'
let (Ci)
i=O,
...,
m
be as
... O C m O A . Then, it follows immediately from (2),
i , that
codimc. B = codim C B j so that l ~nd, in view of the fact that codim A B < ~ , we can write
codim C B = codimc. B ,
codim C A = codim C B - codim A B =
= codimc, B - codlinA B = codlinc, A
The common value of all codimenslon of
A (in & )
If chambers,
codlinC A , for
C c Chain ~
and denoted by
and
C DA
codim A .
C, C' c Chain & , all elements of a sequence
(1) satisfying (2) are
as is immediately deduced from (2), by induction on
have just said of the codimension. and extremities
A
i
and using what we
Such a sequence is called a gallery of len6th
C, C' (or joining
is said to be stretched from
is called the
C
to
and A'
C' ).
For
(or between
m
A, A' c & , the gallery (1) A
and
A' )
if
A C C ,
A' C C' and if there is no gallery of strictly smaller length with the same properties; the length
m
is then called the distance of
A
and
A'
, and denoted by
dist AA'
The gallery (I) is called minimal if it is stretched between its extremities, if
m = dist CC' . By definition,
i c [0, I, ..., m-l] stammer.
such that
The diameter of
that is
a gallery (i) stammers if there is an Ci+ I = C i ; in particular,
a minimal gallery cannot
& ~ denoted by diem A , is defined as the
sup (dist CC' I C, C' ~ Chain A} . The fundamental property of chamber complexes, any two chambers
that
can be joined by a gallery, will be referred to as the connectedness
property.
Two chambers
C, C ~
are called adjacent if
dist CC' = i , that is, if
codim (C N C') = i . Thus a finite sequence of chambers is a gallery if and only if any two consecutive
elements of the sequence are identical or adjacent.
A chamber complex is called thick (resp. thin) if every element of codimension 1 is contained in at least three (resp.
exactly two) chambers.
A morphism of a chamber complex into another is called a morphism of chamber complexes
if it maps the chambers onto chambers.
galleries
onto galleries,
and diminishes
Such a morphism obviously maps the
the distance of chambers.
we shall deal almost exclusively with chamber complexes. specified,
In these notes,
Except when otherwise
all mor~hisms which we shall consider will be mor!~hisms
of chamber complexes.
I.~. a subgroup of
Let
G
be a group, let
G , and for every subset
Gi
In the set
A e ~i
F =
and
=
-_L/__ ~/G i iCI = B e ~/Gi
'!i
I
be a set, for every
~ C I
, we set
A C B
iff
! Di
G i ) , contains the subset
are different, this simply means that we provide
is the opposite of the set-theoretical a complex which will be denoted by
[G{i } : G%/] = 2
% __
is the index
for all
and if
A , viewed as a subset
in the set-theoretical P
Together with this relation,
(i C I) F
sense; if all
with the order relation which
C(G;(Gi)i C I ) ' and on which
C(%;(Gi) i C l ) ; in particular, if Gj
B
inclusion.
The star of the element
chambers containing
be
, we introduce the following order relation: for
G.
complex
Gi
Gi
G (a coset of
translations.
let
set
of I
i 6 I
in
C(G;(Gi))
G
F
is
operates by left
clearly is the
is a chamber complex, the number of
[G~ : G~/] , and the complex is thin iff
i e I .
It is an easy exercise to show that, for three subgroups
X, Y, Z
of a
group, the following three properties are equivalent
(1)
(x.Y) n ( x . z ) = x . ( Y n z)
(2)
(x n ~ ) . ( x n z) = x n (Y.z)
;
(3)
If three cosets
have pairwise a non-empty
intersection,
1.4.1. finite, the complex
G
then
xX A yY D z Z / % / .
PROPOSITION. C(G;(Gi))
Gk (! , .J , _k subsets of In any case,
xX, yY, zZ
C(G;(Gi) )
I )
;
Le__~t I, G, (Gi) i C I
be as above. Then, if
is a flag complex iff any three subgroups possess the equivalent properties
is a chamber complex iff the subgroups
I
is
G i , O~ ,
(i), (2) and (3). G{i ) (i e I)
5enerate
The first assert~ion follows from 1.2.1. The second is immediate.
1.5.
In a chamber complex
A , a set
L
of chambers is called convex
if every minimal gallery whose extremities belong to A chamber subcomplex
Zk'
of
convex subcomplex hull of
Z~'
containing
is
~
of convex is called
L
L . The set
of
A
is defined as the smallest
Cham A'
is then called the convex
L . It is easily seen that the convex hull of a set of chambers is the
smallest convex set of chambers
1.6.
PROPOSITION.
element of codimension
~I Cham~ and
Let
of
A
~
Z~ , •'
into
is in~ective and that
be two chamber complexes
~
A' and
, and let ~
~ I Chain ~
i.s. n~
inoective.
~
is sur~ective and
Suppose that
~
and
G = (A = AO, AI, integer such that
and
~
and
~
b_~e
B , then
dist (q~(A) ,
Assume further that
do not coincide on all faces of and let
i
do not coincide on all faces of
~(Ai) O ~(Ai_ I 0 A i) = ~(Ai_ I 0 A i) . Therefore, the second possibility would clearly imply that
A .
~
is thin.
$ I Chain Z~ is distance preservin6.
..., A m = B) be a minimal gallery, $
le__~t $
coincide on the set of all faces of
is also thin,
~
such that every
A, B e Chain A . Assume that
do not coincide on the set of all faces of
< dist AB , and ~'
containing it.
1 is contained in at most two chambers~
two (chamber) morphisms
Ai
L .
of convex chamber subcomplexes.
The full convex hull of a subset
Then,
Chain A
it is itself convex. An arbitrary subcomplex of
convex if it is an intersection
r
is called convex if
It is readily seen that if a chamber subcomplex is an intersection
chamber subcomplexes,
I__f ~
has all its terms in
~ (i. e. a subcomplex which is a chamber complex and
such that the inclusion is a chamber morphism) convex.
L
be the smallest A i . Then,
~(Ai) = ~(Ai_l) ~
(because they already coincide on all faces of
and
~
B , let
or
~(Ai)
. But
coincide on all faces of
Ai_ 1 N A i ). Therefore,
, a n d the gallery
~(Ai) = ~(Ai_l) = ~(Ai_l)
~(G)
stammers,
which proves the first
part of the proposition.
We now assume that
(i)
If
C'
A
is thin and first show that
, D' c Chain A'
C c Chain A , then adjacent to
Indeed, let = C' A D' ~(D) of
~ and let
contains
D
C' N D'
~ , so that
E
are adjacent,
(p-l(D') = [D) , where
= ChamA'
be the face of codimension
be the chamber containing and is different
; the surjectivity of
are adjacent.
the fact that
D is a chamber
from
C'
E
i of
C
such that
and different from
~(E) = C 9 Then
, because of the injectivity
~(D) = D' .
Two chambers $(D)
with
C .
From (I) and the connectedness ~(ChamA)
C' = ~(C)
and if
@
C, D ~ C h a m A
of ~
&'
follows in~nediately that
ensues.
are adjacent
if and only if
~(C)
and
The "only if" is clear; the "if" is consequence of (i). Now,
preserves the distance of chambers and the thinness of
A
follows
readily.
1.7.
COROLLARY.
If an endomorphism of a thin chamber complex is
in~eetive on the set of chambers
and leaves invariant all faces of a 6iven chamber,
it is the identity.
1.8,
An endomorphism
(p : a -~ A
of a thin chamber complex is called
a foldin~ if it is idempotent and if every chamber image of exactly two chambers of
a
by
C
belonging to
$(~
is the
~ . One of these two chambers is necessarily
C
itself;
we set
we denote the other by
two adjacent
IZMMA.
chambers
Let C , belongs
to
Then,
~(D')
then also
~(~)
D'
C
is a chamber not belonging
be a foldins.
to
e(20
,
. Let
A
A 6 @(A)
so that
I.i0.
PROPOSITION.
(i)
be the face of
D' 2#
contains
~(C)
Let
~
~(C) A
: A ~ &
whose image by and is different
with
, so that
A = ~(A) = C N O , and
There exists
~(&)
~
of
We may assume that at least one of them,
C A D , so that it coincides because
Then , the imases by
or adjacent.
be the chamber which
$(D') = D
them belong s to
~ : ~ ~ &
be the two chambers.
contains
we must have
Let
are identical
C , D
C N D , and let
then
. If
~(C) = C
1.9.
say
~(C)
C
. If
~(C) = D = ~(O)
is
from
D . If
D' = ~(D)
~(C)
D' ~ ~ ( ~ D' e ~(~)
Both the set
[C,C']
(iii)
If
~ (ChamA)
dist C~
- dist CD = i
or
(iv) which maps
C'
onto
C .
-1
is such a pair, with
and its complement
If
are as in (i) and if
a.ccordin~ as
C , C'
D
,
ehm~__ers such that one of C e ~(2~
in
Cham A
are convex.
C , C'
,
.
~(C') = C .
(ii)
9
be a foldin 6.
a p a i r of adjacent
and the other not; if
or
~
belongs
D
tO
are as in (i), then
is any chamber,
~(A)
~
or not.
is the only folding
,
We first show that
(i)
If
C , C'
are two adjacent chambers and if
C' ~ $(A) , then
Indeed, since C N C'
$(C') = C
C N C' e $(&)
G
and
~(C) = C'
, it is invariant by
and must coincide with
Let now
and
C e $(~
~
so that
$(C')
contains
C .
be any minimal gallery having elements both in
$(~
and
in its complement. Such a gallery clearly possesses two consecutive elements such that one of them, say
C , belongs to
~(A)
already proves (i). From (I), it follows that the galleries
~(G)
and
~(G)
stammer
consequently, the extremities of
G
otherwise the stammering gallery
~(G)
as
C ,
C'
while the other does not. This
$(C') = C
(~(G)
and
~(C) = C'
, so that
is a gallery in view of lemma 1.9);
cannot both belong to (resp.
~(G) )
~(2~
(resp. to
C~(2~
would have the same extremities
G , which is impossible. This establishes (ii).
Let for
G
joins
C , C' , D
be as in (iii) and assume first that
a minimal gallery joining D
and
D
and
C'
D e $(2~ 9 Take
. Then, the stammering gallery
$(G)
C , and one has
dist DC ~ dist DC' - i
But the opposite inequality also holds, obviously, so that we have equality. If D ~/~(Z~) , one proves in a similar way, taking now for and
C
and using
~(G)
instead of
$(G) , that
There remains to prove (iv). Let and let us denote by
~(Cham &) = ~(Cham 2~)
~(&)
and that
G
a gallery joining
D
dist DC ~ = dist DC - i .
be any folding such that
~(Z~) the smallest subcomplex of
~(Cham &) ; this set being convex, by (ii), it follows that
$
)
&
~(C') = C
containing the set
is a chsmber complex. From (iii) ~(Cham Z~) = ~(Cham 2~ 9 Applying
i0
the proposition 1.6 to the restrictions of we see that ~=~
~ i M(s
= 9 I~(A )
. Similarly
COROLLARY.
adjacent chambers such that that
$' (C) = C'
. Then
contain any chamber, and magpings
to
~(&)
and the chamber
C
9 Consequently,
go I~(A ) = ~ I ~(A)
M'
and
~
L.et ~ : & ~ Z ~
s
U ~' (Z~) = & , the intersection
M'
g0" =
~'
M
o_nn M' (Z~) and with
M(A)U~'(&)=~
C ~ e M"(Z~] ,
$'
~(2r N ~' (~)
such
does not
and M"
M'
o__nn ~(2~ 9
Chain (M(A) N ~ ' ( A ) ) = ~
are
is any folding satisfying these two
C ~q0"(A) 9 thus
M"(C) = C'
by i.i0 (it ~ and
by i. I 0 ( i v ) .
If an element of and
$' ; from the relation
I $(A)
and
, therefore (1.7)
2 p
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