Complexes

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

D e Chem
1,12. opposite to

A

belongs to

~(A) N ~'(2~ , it is invariant by both

$(A) U M' (A) = A, it then follows that the morphisms

r I ~' (Z~) can be glued into an endomorphism

endomorphism is injective on

M

Chmm n , and its

is the identity and

o : A -~ A

This

square leaves invariant each face of p

is an automorphism. Finally, if

M(M'(D)) =p(D(D)) = D , so that

~ (D) = M(D) , and the

D ~ Cham ~' (~)

When it exists, the folding

and denoted by

in 1.8). The automorphism B e &

be two

Chain & . There exists an involutory automorphism

immediate consequences of i. I0 (iii). If relations~ one has

C , C'

is the onl~ foldin6 having these two properties. The

coincide on

The relations

be a foldin 6 and let

q0(C') = C . Assume that there exists a foldin6

: ~ -~ Z~ which coincides with

C

~

9

I.Ii.

(0j

.'p and

~

~'

of corollary 1.11 will be called

(which is consistent with the notation introduced

p is then called the reflection associated with

have codimension i j if there is a reflection

reflection is unique (because if

B = C O C'

, with

0

leaving

B

~ . Let

invariant~ this

C , C' e Chain Z~ 9 one has

ii

p(c)

= c'

onto

C'

p

, and

must be the reflection

), and is called the reflection with resyect to

The image with of

M(A)

and

consisting of all C' / ~

with

Chain A = , 9 z

1.13. complex

~

aS , is b y definition

such that there exist . Two roots

foldings;

or the root associated

~ , ~

adjacent

the subcomplex

chambers

C c

are called opposite if they are

b y i.ii, this means that

r U r = A

and

.

LEMMA.

Let

~ , se___~t A = C O C'

are stretched between of

A c A

A C C O C'

associated with opposite

r n r n

r , denoted b y

C

B .

of a folding is called a root,

~ . The wall of a root A

associated with a folding mapping

A

C , C'

be two adjacent

and let

F

A

invariant

M'(C') = M'(C) = C' . Then, i f

~

and

of the thin

be the set of all galleries

and a chamber of

leavin G all faces of

chambers

~ . Let

M

an__dd ~'

and such that

~'

map

F

G

which

be two endomor~hisms

M(C) = ~(C') = C

in itself,

and

they are opposite

foldin6s.

Let induction on

(i)

If

m = dist A D

C O = C (resp. ~ ~'

(2)

G = (Co, CI, ..., C m) e F

If

C O = C (resp.

For of generality, hypothesis,

(resp. b y

~

and set

C

m

= D . We first show b y

that

C' ) , all faces of (p'

and

C')

, then

~(D)2~D

(resp.

are obvious.

C O = C . Set

leaves invariant

all faces of

~(Cm_ l O D) = Cm_ I N D , so that

$(D) = Cm_ I

excluded b y the assumption made on

are invariant b y

M

and

~' o q );

m = 0 , both statements we m a y assume that

D

q

and

~ = M

qo(D) 2 # D

Let or

m

be

> 0 . Without loss

~ o $' . B y the induction

Cm_ l . The chamber or

) 9

$(D)

contains

D , but the first possibility

(p' . Therefore,

$(D) = D . Since

is

12

also leaves

invariant

and (i) is proved.

all faces of

Now suppose

that

for which it is already proved, follows ~' (D) / D

that

Cm_ I O D , it leaves

invariant

M' (D) = D . Applying

we see that

M'(Cm_I ) = Cm_ 1 ~ which

M'(Cm_ 1 N

contradicts

all faces of

(I) to the gallery

D) = Cm_ 1 0

the induction

D ,

~' (G)

,

D ~ from which

hypothesis.

Therefore,

.

Let now whose last term is all faces of

~(D)

D

be any chamber

D . Applying , so that

of

(i) to

~ , and let ~(G)

M2(B) = $(B)

G

, we see that

for all

be an element of $

leaves

F

invariant

B C D , and the morphism

is idempotent.

From (i) and (2), it follows M(E) = M(@'(E)) and let words,

= E ~M'(E)

E e M-I(D)

9 Then,

~-I(D) = {D, M~(D)}

The same holds for

M'

or

that for every chamber

~'(E) = M'(~(E))

either . Since

= E/M(E)

E = ~(E) = D , or M'(D) 2 # D

9 Now, let

D e Chain M(2~

E = ~'(
~ this shows that

, and the lemma is proved.

E , either

~

. In other

is a folding.

,