166 V.M. Buchstaber
For example,χ(σn)=1andχ(Sn−^1 )=1+(−1)n−^1.
Putf(t)=tn+f 0 tn−^1 +···+fn− 1 andh(t)=h 0 tn+h 1 tn−^1 +···+hn.
Theh-vectorh(K)=(h 0 ,...,hn)ofan(n−1)-dim simplicial complexKis
defined by the identityh(t)=f(t−1). Note thath 0 =1.
Define theDehn–Sommerville characteristicsof a simplicial complexKby
the formula:
DSi(K)=(−1)n−^1 (hn−i−hi),i=0,...,n.The numbers DSiare obviously combinatorial invariants of a simplicial com-
plexK.Wehave
DS 0 (K)=χ(K)−χ(Sn−^1 ).
Two mapsf 1 ,f 2 :X→Y are calledhomotopicif there is a continuous
mapF:X×I−→Y(hereIis the interval [0,1]) such thatF(x,0) =f 1 (x)
andF(x,1) =f 2 (x) for allx∈X.
Fix a basepoint pt∈X. Homotopy classes of mapsφ:I→Xsuch that
φ(0) =φ(1) = pt form a group called thefundamental groupofXand denoted
π 1 (X).
A continuous mapf:X→Yis called ahomotopy equivalenceif there is a
mapg:Y→Xsuch that the two compositesg◦f:X→Xandf◦g:Y→Y
are homotopic to the identity maps idXand idY, respectively.
A characteristica(X)ofaspaceXis called ahomotopy invariantifa(X)=
a(Y) whenever there is a homotopy equivalencef:X→Y.
The Euler characteristicχ(X) is a homotopy invariant, and therefore so
is DS 0 (K).
Fori>0 the characteristic DSi(K) is not homotopy invariant in general.
Remarkably, it becomes a homotopy invariant if we restrict to triangulated
manifolds. More precisely, for any triangulated manifoldKn−^1 the following
generalised Dehn–Sommerville relationshold:
DSi(K)=(−1)i(
χ(Kn−^1 )−χ(Sn−^1 ))
(
n
i)
,i=0, 1 ,...,n.IfKis a simplicial subdivision of the sphereSn−^1 we obtain
DSi(K)=0.The Dehn–Sommerville relations are the most general linear equations satis-
fied by thef-vectors of triangulated spheres.
Example 5.LetKbe a simplicial subdivision of the sphereS^2. Then it follows
from the Dehn–Sommerville relations that
f(K)=(
f 0 ,3(f 0 −2),2(f 0 −2)