ON
THE
OF
FINITE
M.
NUMBER
L.
OF
SIMPLEXES
OF
SUBDIVISIONS
COMPLEXES
Gromov
UDC 513.83
Combinatorial invariants of a finite simplicial complex K a r e considered that a r e functions
of the number ~i(K) of simplexes of dimension i of this complex. The m a i n r e s u l t is
T h e o r e m 2, which gives the n e c e s s a r y and sufficient condition for two complexes K and L
to have subdivisions K' and L ' such that ~ i ( K ' ) = ~ i ( L ' ) for 0 _~i < ~. The theorem yields
a c o r o l l a r y : if the polyhedra I KI and ILl a r e h o m e o m o r p h i c , then there exist subdivisions K' and L' such that ~i(K')=~i(L') for i >_0.
INTRODUCTION
Let K be a finite simplicial complex, and let c~i(K) be the number of i-dimensional simplexes of K.
The problem is which functions of c~i a r e combinatorial invariants of K. It is well known that the Euler
c h a r a c t e r i s t i c X (K) and the dimension dim K a r e such functions.
The p r o b l e m of invariance of functions of the numbers ~i(K) with r e s p e c t to subdivisions has two
a s p e c t s . F i r s t l y , f o r an individual complex K it is possible to study functions of the n u m b e r s ~i(K) that a r e
invariant under subdivisions of K. Secondly, it is of i n t e r e s t to a s c e r t a i n the c a s e s in which we can e s t a b lish with the aid of such functions that two complexes K and L a r e not c o m b i n a t o r i a l l y equivalent. The l a t t e r question can be formuiated as follows: When do two complexes K and L have subdivisions K' and L '
such that ~i(K)=~i(L) for all i _~0?
The f i r s t a s p e c t of the problem is examined in [2], which gives for e v e r y complex K all the linear
functions of ~i(K) that a r e invariant under subdivisions of K.
To the second p a r t of the p r o b l e m belongs the following r e s u l t of Bing [1]: Any two closed t r i a n g u lated three-dimensional manifolds M i and M 2 have subdivisions M~ and M~ such that e~i(M~) = e~i(M~) for
i_~0.
In the p r e s e n t p a p e r we obtain the n e c e s s a r y and sufficient condition for two complexes K and L to
have subdivisions K' and L' such that ~i(K')---~i(L') for all i ~_0.
In Appendix 1 we formulate an analogous result for the case of m o r e than two complexes.
In Appendix 2 we show how it is possible to obtain by our methods the fundamental t h e o r e m of [2].
w 1.
Notations
and
Formulation
of Results
Let K be a finite simplicial complex and ]K[ the c o r r e s p o n d i n g polyhedron. By si we shall denote
the i-dimensional simplexes of K, as well as the c o r r e s p o n d i n g open simplexes of ]KI. By C(K) we shall
denote a space of functions whose a r g u m e n t s a r e simplexes of K, w h e r e a s the values a s s u m e d by them b e long to the group Z of integers. In the following these functions will be called c o - c h a i n s . A support supp(c)
of a c o - c h a i n c ~ C ( K ) is defined as a minimal subcomplex of K such that if 8 ~ s u p p (c) , then c ( s ) = 0. The
dimension dim (c) of a c o - c h a i n c is defined as the dimension of its support.
If s ~ K ,
Let K' be a subdivision of K. We shall define a natural h o m o m o r p h i s m p : C ( K ) - - C(K') as follows:
8' ~ K " , and in the polyhedron [KI we have the inclusion 8' c s , then we shall write p c ( s ' ) = c ( s ) .
If A is a subset of simplexes of K, then @A will denote the c h a r a c t e r i s t i c f u n c t i o n o f the set A.
F o r e v e r y integer i we shall define a h o m o m o r p h i s m Ai: C(K) -* C(K) by the following conditions:
(AI (c)) c supp (c) , and if s -~ sp , then
A. A. Zhdanov Leningrad State University. T r a n s l a t e d f r o m Matematiche~kie Zametki, Vol. 3, No. 5,
pp. 511-522, May, 1968. Original a r t i c l e submitted September 11, 1967.
326
a+,+p (s) = (-1) v+++l+ +,p (s).
R e m a r k 1. Let us c o n s i d e r the boundary Bd(sq) of the complement of the simplex Sq in the complex
K (see [3]). F r o m the definition of the h o m o r p h i s m Ai we obtain by a trivial calculation the formula
A~tpK (Sq) ----- ~1 + (-- t)i§
(1)
-- (-- t)i+q% (Bd (sq)).
In p a r t i c u l a r , if M is an n-dimensional closed h-manifold, then An CM = 0.
R e m a r k 2. Any c o - c h a i n c ~ C (K) can be r e g a r d e d as a function f(k) in the space I K I b y setting
f2{k)=c(s), if k ~ s r ' - K .
If we have two t r i a n g u l a t i o n s K 1 and K ~ of the space [K[, then the c o - c h a i n s
AiCK 1 and AiCK9 coincide as being functions in the space [K[ 9
F o r the p r o o f we m u s t use formula (1) and the invariance of local homology groups.
Let us define the h o m o m o r p h i s m ~: C ( K ) - - Z by the condition X (r
X (r
(-1)q. It is c l e a r that
= X (K}. Let us denote by P the space of polynomials of the f o r m -Tao --l-
i n t e g e r s , and let us define the h o m o m o r p h i s m X0: C ( K ) - - P by the condition X0(r
that
~alz' , where the a i a r e
x) = x q + l .
It is c l e a r
In the group C{K) let us define a subset C * (K) as follows. The z e r o c o - c h a i n belongs to C* (K),
w h e r e a s the c o - c h a i n c of dimension n _~ 0 belongs to the set C* (K) if and only if there exist simplexes
s~,~K
such thate(sln ) > 0, andc(s2n ) < 0 .
R e m a r k 3. Suppose we have two triangulations K 1 and K2 of the space IK I and two co-chains
t h a t c o i n c i d e a s f a n c t i o n s i n [ K l , a n d l e t c i ~ C ' ( K ~) . Then c 2 ~ C * ( K Z ) .
c i ~ _ C ( K l) and c2~_C(K 2)
F o r the p r o o f we m u s t use the topologic invariance of dimension.
The principal r e s u l t Of the p r e s e n t p a p e r is
THEOREM 1. Let K be a finite complex and let c ~ C (K) . F o r a complex K to have a subdivision
K t such that X 0(P (c))= 0, where p : C (K)-* C (KT), it is n e c e s s a r y and sufficient that the following conditions
hold:
c@C*(K), X(c)=O
I
u ec.~u n ~ dim c. mo A . (c) ~ C'(K). J
(*)
F r o m T h e o r e m 1 follows
THEOREM 2. F o r two finite complexes K and L to have subdivisions K v and L t such that ~i(K v) =
a i ( L r) f o r all i, it is n e c e s s a r y and sufficient that the following conditions hold:
dimK=dimL=n'
and in the space
x(K)=x(L)
}
C (K O L) we have the inclusion
(**)
(A~tpK -- A~,L) ~ C" (K tJ L)
P r o o f . Let us apply T h e o r e m 1 to the polyhedron K U L and the c o - c h a i n c ---- (,~
*L)
(K U L) E C 9 With r e s p e c t to the c o - c h a i n c the conditions (* *) a r e equivalent to the conditions (*).
the other hand,
On
oo
XO (C) = 2 ~ _ 0 X i~-1 (Of,i, ( K ) - - o~i ( L ) )
which completes the p r o o f of T h e o r e m 2.
327
COROLLARY 1. Let us c o n s i d e r complexes K, L, and K t, L 1 such that the polyhedron IKI is h o m e o m o r p h i c to the polyhedron I K I I , whereas the polyhedron ILl is h o m e o m o r p h i c to the polyhedron I L l 1 , and
let o~i(K)=cri(L) for all i_~0. Then the c o m p l e x e s K 1 and L 1 will have subdivisions KI and L] such that
r
=~i(L~) for all i _>0.
F o r the p r o o f we m u s t use the r e m a r k s 2 and 3.
COROLLARY 2. Let us c o n s i d e r two n - d i m e n s i o n a l closed h - m a n i f o l d s M 1 and M 2 such that
)((M1) =X (M2). Then they have subdivisions M~ and M~. such that ~i(M~) = ~i(M~) f o r all i m0.
F o r the p r o o f it is sufficient to use R e m a r k 1.
COROLLARY 3. L e t us c o n s i d e r a complex K of dimension n such that h,, (*K) E C" (K) and let L be a
complex such that L - - d i m K and X (L) =X (K). Then there e x i s t subdivisions K' and L ' such that o~i(K')----~ i ( L ' ) f o r all i _>0.
The p r o o f is evident.
R e m a r k . F o r any i n t e g e r s q and n such that n_>0, and if n = 0 , then q > 0, there e x i s t s a c o m p l e x K
such that X (K)=q, dim K = n , and A,~ (~I~)~C*(K) .
Proof. If n = 0 , then we m u s t take K in the f o r m of a complex consisting of q v e r t i c e s . L e t n > 0. By
Kn we shall denote a complex consisting of three n - d i m e n s i o n a l (closed) s i m p l e x e s , linked along the (n-1)dimensional boundary. Let Lq be a o n e - d i m e n s i o n a l complex such that X (Lq) = q - 1 . The complex K will be
taken in the f o r m of the complex K,, U Lq 9 The p r o o f that the complex K---- K, U Lq has the d e s i r e d
p r o p e r t y is trivial.
The l a t t e r r e m a r k and C o r o l l a r y 3 show that (in a c e r t a i n sense) t h e r e do not e x i s t c o m b i n a t o r i a l inv a r i a n t s , a p a r t f r o m dimension and E u l e r ' s c h a r a c t e r i s t i c , that a r e functions of the n u m b e r s ~i(K).
w 2.
Lemmas
about
Subdivisions
F o r e v e r y integral k let us define a h o m o m o r p h i s m dk: P--* P by the f o r m u l a
d~p (x) ~--p (x) -4- (--1)~P ( - - t -- x).
It can be d i r e c t l y v e r i f i e d that
d~+t'd~-----~O, d:,~=dk.
In the group P let us define a subgroup p n by the followingcondition:
dn+tP(X) = O, and the value p(O) is an integer.
LEMMA 1. If p ( x ) ~ p n
(2)
p ( x ) E Pn
if and only if deg p (x)~n,
and p ( 0 ) = 0 , then p ( x ) = x ( x + l ) "q(x), where q ( x ) ~ P n-2
P r o o f . Since dn+ t p ( x ) = 0 , it follows that p ( - 1 ) - - p ( 0 ) = 0 , and t h e r e f o r e (x (x + l)) -I p(x) E P. In this
case
dn-~ ((x (x + t))-t p (x)) ----(z (x § i))-t p (x) + ( - - i F ~ (x (x
+ i))-, p (--t - x) = (x (x + ~))-~ d ~ , ~ (x) ---- 0.
This c o m p l e t e s the p r o o f of the l e m m a .
L e t K be a finite complex. By xI(K) we s h a l l d e n o t e the polynomial i + ~ , ~ x t §
(g) . Let us note
that X 1(K; x)= 1 +)( 0( CK; x). We shall c o n s i d e r the union K 9 L of two c o m p l e x e s . By a d i r e c t calculation
we can v e r i f y the f o r m u l a
X, (K o L) ffi X, (K).xl (L).
(3)
LEMMA 2. L e t us c o n s i d e r a c o m p l e x K, a s i m p l e x s l ~ K , a c o - c h a i n c ~ C ( K ) , a n d a n u m b e r a
such that f o r a n y s i m p l e x s ~ s t ( s i ) we have c ( s ) = a . L e t K' be an e l e m e n t a r y subdivision (see [3]) of the
c o m p l e x K with r e s p e c t to the s i m p l e x s 1, and suppose that the s i m p l e x s~~ K' lies in the s i m p l e x s 1.
L e t p : C (K) --~ C (K'). Then
328
Bd (s~) = Bd (s~)
and
Xo(P (c); x) -----Xo(C; x) q- ax(x A- t) X~(Bd (s~); z).
P r o o f . The f i r s t of the above f o r m u l a s is evident, w h e r e a s the second formula follows from (3).
Let
US
define the h o m o m o r p h i s m X * : C (K) - - P by the formula X * (c; x) = ~ 0 (c; x) + 1 X (c).
LEMMA 3. Let K' be a subdivision of K, let p :
C ( K ) ~ C(K') and let c ~ C (K). Then:
a) dkx * (c) = X * ZXk(c);
b) z~+~ ZXk(e)=0, ZXk(C)=ZXk+2(c);
c) x (e) = • ~ (e));
d) ,U~p (c) =p,:~k(c);
e) if p (c) ~ C" (K'), then also c ~ C" (K).
P r o o f . It is sufficient to v e r i f y the f o r m u l a s (a), (b), (c) and (d) for co-chains c ~ C ( K )
c = Cs, where s ~ K. It is evident that
=
k- ' t
)
/,
n
,~.Jt=oCn+l z
T
Xn+l . ~ (
t ) n+~+l (t § X) ~+1 §
-
X "+I
§ (
--
of the f o r m
t) t'+'.
-
Hence
(~(%~); z)=x~+l-t-
§
+
2
--d~ x-+l+
= ~X (9,.; x)
and thus we have proved formula (a).
From (a) and (2) follows that X * Z k k + l A k ( r
but, on the other hand, if si-<~8 and s~-.~s , then
ZI k +lAk ~s(S I) = A k +lZ~k Cs(S~); therefore we obtain for any st ~ K the relation A k +1Zl k Cs(Si) = 0. Since
the formula A k L-~ k +2 is evident, we thus proved (b).
Formula (c) is a consequence of the invarianee of the Euler characteristic of complexes under subdivisions.
Let us denote by [Dn[ the closure of the simplex sn in the polyhedron [ K I , and let Din be a subdivision of the complex Dn. By fli(s) we shall denote the n u m b e r of simplexes st ~ D ~ such that s~N- s ~ D ' ,
and st ~ 0 (D~) . Since the polyhedron I Dn' I is an n-dimensional manifold with a boundary I a (Dh) 1 we have
Y,,~ ( - i) ~~ (,) = (--i)".
B y applying this f o r m u l a to the case that the subdivision DWn is induced by a subdivision K' of K, we obtain
oo
~ w % . (s) = ~,=0 (-- t)
~:+i+1
~t (s) + p,.~ (s) = (--
t),~+~+l
+ p,,. (s) = p ~ % . (s),
thus p r o v i n g f o r m u l a (d). F o r the p r o o f of (e) it is sufficient to r e f e r to R e m a r k 3.
Remark.
F r o m f o r m u l a (a) of L e m m a 3 and R e m a r k 1 follows that if M is a closed n-dimensional
h-manifold, then 2X" (*M; x) E P"+~ , w h e r e a s by denoting with Sn the n-dimensional s p h e r e and writing
X (M) = X (sn), we obtain Xt (M) ~ p~n 9
LEMMA 4. Let Kn be a triangulation of the sphere Sn. Then there exist in the complex Kn subdivi-
pn+l.
329
Proof. F o r n = 0 , 1 the a s s e r t i o n of the l e m m a is trivial. Now let n_~2 and let Kn-2,.
1
9 , K nZ_2 be
i
subdivisions of the s p h e r e S n-2 such that the polynomials X1 (Kn-2)
g e n e r a t e the group p n - t 9 It is evident
r
that the complex K has a subdivision K n with the following p r o p e r t i e s : T h e r e e x i s t o n e - d i m e n s i o n a l s i m p l e x e s . S~, sl,3..., stt*~~--~,~' such that f o r any i = 1, 2 . . . . .
l the c o m p l e x Bd(s~ +i1) is i s o m o r p h i c to the corn - ~
1
plex Kn_~,
and, m o r e o v e r , st (s~) ~ st (~) = ~ for i ~ j. F o r i > 1, the complex Kn will be taken b y us in the
f o r m of an e l e m e n t a r y subdivision of the complex K~ with r e s p e c t to the s i m p l e x s~. F o r i > 1 we obtain
b y virtue of L e m m a 2
gt (Kin; z) = Xl (K~; x) -{- r
t-1
i) Xt (Kn~;
x).
Since x I ( K ~ ; 0) =1 and by virtue of L e m m a 1 the p o l y n o m i a l s xl(Kni; x) g e n e r a t e f o r 1 < i _< Z +1 the
group p n + l , we thus p r o v e d L e m m a 4.
LEMMA 5. L e t K be a finite complex and let
cEC(K) with
dim c = n . Let
(c) = 0, d~ ~" (c) ~ 0.
(4)
We shall m o r e o v e r a s s u m e that there e x i s t s i m p l e x e s ~ E K
, s =1, 2, such that c(Sln)= v 1 > 0 and c(S2n)=
v 2 < 0, and that all the coefficients of the polynomial X 0(c; x) a r e divisible by the g r e a t e s t c o m m o n d i v i s o r v
of the n u m b e r s v 1 and v 2. Then t h e r e e x i s t s a subdivision K' of K such that X0(P (c); x) = X* (P (c); x ) = 0 ,
w h e r e p : C (K) - - C (K').
Proof. F r o m L e m m a 4 follows that there e x i s t s a subdivision K ~ of K with the following p r o p e r t i e s :
All the nnswW s i m p l e x e s of the complex supp P0(C), where P0: C ( K ) - - C ( K ~
lie in the union s~Us~ ;
t h e r e e x i s t o n e - d i m e n s i o n a l s i m p l e x e s st~'~ ~ K o , w h e r e 1 ~ i <_ l and s = 1, 2 such that s~t, ~ c- e~ ,
with st (s~,')nst(s~,t)=~b f o r i ~ j ;
If BCd(s) denotes the boundary of the c o m p l e m e n t of the s i m p l e x s in the complex (p ~
then XI"
(BCd(s~'~);x)=x1 (BCd(s~'2);x) and the polynomials Xl(BdsliJ; x) for 1 < i < Z g e n e r a t e the group pn-1.
It is c l e a r that A n c ( s n1) =AnC (s~)= 0, and hence
(5)
Xo (po~. (c); x ) = xo (~.(c); x).
By a t r i v i a l calculation with the use of f o r m u l a s (4) and (5) and A s s e r t i o n s (a), (c), and (d) of L e m m a 3 we
obtain the f o r m u l a
d~0
(po (c); x)
(6)
---- 0.
It is a l s o c l e a r that all the coefficients of the polynomial X 0(P 0(e); x) a r e divisible b y v . B y v l r t u e of (6) we obtain
v-'xo
~po (c);
z) ~ Pn+L
(7)
L e t us note that X 0(P o(C); 0) = 0 and let us w r i t e Pi (x) = X I (BCd(si '1); x) = 1t (BCdl(s~'2); x). F r o m (7) and L e m ma 1 follows that there e x i s t nonnegative l n t e g e r s a i and bi where i < i < l , such that
I
X, (PJ (c); z) . . . .
~ t - , (alP, + bivs) x (z -t- I) Pi (x).
(8)
If L is a finite c o m p l e x and s l, s 9-, . . . ,
s k a r e o n e - d i m e n s i o n a l s i m p l e x e s of L such that st (st) {3
st (s:)= ~ f o r i ~ j , and if Pl, P2,. 9 9 , Pk a r e nonnegative i n t e g e r s , then we shall denote b y L(s I, Pl, s2, P~;
. . . , S k P k ) a ~ u b d i v i s i o n of L, defined b y the following conditions: L(s 1, 0; . . . ; s k, 0 ) = L , and the c o m p l e x
L (s 1, ql; .~ 9 ; s , qi; - -" ; sk, qk) is obtained by an e l e m e n t a r y subdivision of the c o m p l e x L (s l, ql; 9 9 9 ; st,
qi; 9 "" ; s , qK) with r e s p e c t to any o n e - d i m e n s i o n a l s i m p l e x contained in the s i m p l e x s i.
L e t us now write
K' ----- K ~ (s~, 1, a,;...
330
;s~, 1, at; s],~, b,;...
;s~, s, bz).
F r o m L e m m a 2 follows
(p (c); x) = Xo(P0 (c); x) ~, ~ - 1 (awl + btv2) x (x -t- l) Pt (x),
(9)
where p : C ( K ) - * C(KI). F r o m f o r m u l a s (8) and (9) follows that X0(P (e); x ) = 0 , but X(P (c))=X (e)=0, and
hence X 0(P (c); x) = X * (P (c); x). This c o m p l e t e s the p r o o f of L e m m a 5.
LEMMA 6. F o r a n y natural n u m b e r v there e x i s t s a h o m o m o r p h i s m rv: P - * P with the following p r o pe t r i e s :
a) d i r v = r v d i ; deg rvP(X)=deg p(x), and if degp(x) > 1, then the coefficient of the leading t e r m of the
polynomial rvp(x ) wiU be divisible by v;
b) f o r a n y finite c o m p l e x K there e x i s t s a subdivision K(v) with a h o m o m o r p h i s m Pv: C(K)-* C(K(v)),
such that X 0 P v = r v x 0 .
Proof.
L e t us define the h o m o m o r p h i s m r v by the following f o r m u l a s :
r~ (a o + atx + a2x9 -~ ao + (ai + (v -- i)a2)x ~ va2x 2,
and if n > 2 , then rv(xn) =xrv((X + l ) n - - x n ) .
If dim K = 0 , then we s e t K(v) =K. Let dim K > 0. We take a o n e - d i m e n s i o n a l skeleton of K and divide
e v e r y o n e - d i m e n s i o n a l s i m p l e x of K into v p a r t s . Then we continue this subdivision o v e r the entire complex
K by induction on the skeletons in the following way: The subdivision of the l - d i m e n s i o n a l s i m p l e x s l is
taken in the f o r m of a cone o v e r a subdivision of the boundary of the s i m p l e x s l with v e r t e x at the c e n t e r of
this s i m p l e x [let us note that K(1)=K and that K(2) is a b a r y c e n t r i c subdivision].
It is p o s s i b l e to d i r e c t l y v e r i f y that the h o m o m o r p h i s m r v and the subdivision K(v), c o n s t r u c t e d by us,
have the d e s i r e d p r o p e r t i e s .
COROLLARY. F o r any natural n u m b e r v and any complex K t h e r e exists a subdivision Kv of K such
that the h o m o m o r p h i s m p : C(K)--~ C(Kv) has the following p r o p e r t i e s : F o r any c o - c h a i n c E C (K) such
that X (e)= 0 and dix * (c)= 0 f o r s o m e i, all the coefficients of the polynomial X 0(P (c); x) will be divisible by
v and dix * (p (c); x) = 0.
P r o o f . L e t us c o n s t r u c t the sequence of subdivisions K i = K ( v ) , and K i + i = K i ( v ) . The quantity Kv is
taken in the f o r m of a subdivision K n, w h e r e n = d i m K. If p : C(K)--*C(Kv), it follows f r o m f o r m u l a (b) of
L e m m a 6 that X0P = (rv)nx0, but since deg X0(P (c); x) _<n and X0(P (c); - 1 ) = XP (c)= X (c) =0, the d e s i r e d
p r o p e r t y of the polynomial• 0(P (c); x ) = (rv)n X 0(e; x) will t r i v i a l l y follow f r o m A s s e r t i o n (a) of L e m m a 6.
w 3.
Proof
of Theorem
1
If X0(P (c); x ) = 0 , then p ( c ) ~ C ' ( K ' )
f r o m (c), (d) and (e) of L e m m a 3.
and X ( p ( e ) ) = 0 , so that the n e c e s s i t y of the conditions (*) follows
Suppose that the c o - c h a i n c, which belongs to the group C(K), s a t i s f i e s the conditions (*), and let dim
c =n and dim An(C)=m. By setting b=An(C) and taking into account that
i
x" (c; 0) = - - x'(c;-- t)---- ~Zx (c) ---- 0,
we obtain
Z (b) = 2X* (b; 0) ---- 2dn X" (c; 0) ---- 0.
(10)
F r o m f o r m u l a (2) and (a) and (b) of L e m m a 3 follows that
dmx* (b) = 0.
(11)
Since b ~ C* (K) , t h e r e e x i s t s i m p l e x e s Slm and S2m such that b(slm) = v 1 > 0 and b(s~m) =v2 > 0. L e t us denote
b y v the g r e a t e s t c o m m o n d i v i s o r of the n u m b e r s v I and v z. To the c o m p l e x K and the n u m b e r v we shall
331
apply the c o r o l l a r y of L e m m a 6 and obtain a subdivision K 1 = K v with a h o m o m o r p h i s m p l: C (K) ~ C (K1).
By virtue of (10) and (11) we can apply L e m m a 5 to the complex K 1 and the c o - c h a i n pl(b); hence t h e r e e x i s t s
a subdivision K 2 with a h o m o m o r p h i s m P2: C(K) -* C(K 2) such that
;r
(b) = o.
F r o m (a) and (d) of L e m m a 3 and f r o m f o r m u l a (12) we obtain
d.x'p~ (c) ---- 0.
By using f o r the c o - c h a i n Og(C) the s a m e r e a s o n i n g a s we used above f o r the c o - c h a i n b, we obtain a subdivision K' with the d e s i r e d p r o p e r t y .
APPENDIX
1
THEOREM 2a. Let K l, K 2. . . .
, Kp be finite c o m p l e x e s . L e t L ---- K~ U K2U ... U Kr, and suppose that
e v e r y c o - c h a i n of the f o r m ~K~--~K E L
s a t i s f i e s condition (*). Then t h e r e e x i s t subdivisions K: of the
c o m p l e x e s Kj such that ~i(K~l)= ~ i ~ )
f o r all i and 1 -<Jl, J~ -<P.
J
The p r o o f of this t h e o r e m is s i m i l a r to the p r o o f of T h e o r e m 1.
APPENDIX
2
L e t us c o n s i d e r a subgroup P~' m c P , consisting of p o l y n o m i a l s such that deg p(x) ~ n , w h e r e a s deg
dn+lP(X) ~ m and p(0) = p ( - 1 ) = 0. Let K be a finite c o m p l e x and let c ~ C (K) . L e t us c o n s i d e r all p o s s i b l e
subdivisions K i of K and h o m o m o r p h i s m s p i: C(K)--~ C(Ki); we shall define the subset P (c) c- p a s the
s e t of all p o l y n o m i a l s of the f o r m X 0(P i(c); x).
THEOREM l a . Let K be a finite complex and suppose that the c o - c h a i n c E C (K) s a t i s f i e s the c o n ditions ( * ) , with dim c = n and dim A n ( c ) = m . Then P (c) c- pn.t, ,n~ and in the group p n + i , m + i t h e r e
e x i s t s a subgroup p n + 1, m +l of finite index and such that p~*l, ~1 ~ p (c) .
The p r o o f of this t h e o r e m can be obtained b y a s i m p l e r e a s o n i n g f r o m T h e o r e m 1 and L e m m a 4.
L e t us denote by II a l i n e a r s p a c e of p o l y n o m i a l s with r e a l coefficients and a z e r o f r e e t e r m . Let K
be a finite complex. All the l i n e a r functions of the n u m b e r s ~i(K), i n v a r i a n t under subdivisions of K, f o r m
a s u b s p a c e I I * (K). of II *, where II * is the s p a c e conjugate to II. The s p a c e II* (K) is an annihilator of the
s u b s p a c e H (K) c l I
g e n e r a t e d by p o l y n o m i a l s of the f o r m pl(x)-p2(x), where pt (x), p ~ ( x ) E P ( ~ K )
In our t e r m i n o l o g y , the p r i n c i p a l r e s u l t of [2] is f o r m u l a t e d a s follows.
L e t K be a finite complex of dimension n, and let dim An@k=m. Then the s p a c e II(K) will coincide
with t h e l i n e a r h u l l spanned o v e r the p o l y n o m i a l s p(x) belonging to the group p n + l , m +l.
This t h e o r e m can be p r o v e d by applying T h e o r e m l a to the c o m p l e x r e p r e s e n t i n g the union of two
s p e c i m e n s of the complex K, and to a c o - c h a i n , equal to 1 on s i m p l e x e s of one s p e c i m e n of the c o m p l e x , and
to - 1 on s i m p l e x e s of the o t h e r s p e c i m e n .
LITERATURE
1.
2.
3.
332
CITED
R. H. Bing, "Some a s p e c t s of the topology of 3 - m a n i f o l d s r e l a t e d to the P o i n c a r ~ c o n j e c t u r e , n L e c t u r e s on Modern M a t h e m a t i c s (edited by T. L. Saaty), Vol. 2 (1964).
C. T. C. Wall, " A r i t h m e t i c i n v a r i a n t s of subdivisions of c o m p l e x e s , " Canad. J . Math., 18, No. 1, 92-96
(1966).
N. Steenrod and S. E i l e n b e r g , Foundations of A l g e b r a i c Topology [Russian t r a n s l a t i o n ] , Moscow (1958).