---
address: 'Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan'
author:
- Toshiyuki Akita
title: |
     A formula for the Euler characteristics of\
    even dimensional triangulated manifolds
---

[^1]

A finite simplicial complex $K$ is called an [*Eulerian manifold*]{} (or a [*semi-Eulerian complex*]{} in the literature) if all of maximal faces have the same dimension and, for every nonempty face $\sigma\in K$, $$\chi({\operatorname{Lk}}\sigma)=\chi(S^{\dim K-\dim\sigma-1})$$ holds, where ${\operatorname{Lk}}\sigma$ is the link of $\sigma$ in $K$ and $S^n$ is the $n$-dimensional sphere. Note that $K$ is not necessary connected. Any triangulation of a closed manifold is an Eulerian manifold. More generally, a triangulation of a homology manifold without boundary provides an Eulerian manifold. The purpose of this short note is to prove the following alternative formula for the Euler characteristics of even dimensional Eulerian manifolds.

\[main\] Let $K$ be a $2m$-dimensional Eulerian manifold. Then $$\label{eq-main}
\chi(K)=\sum_{i=0}^{2m}\left(-\frac{1}{2}\right)^i f_i(K)$$ holds, where $f_i(K)$ is the number of $i$-simplices of $K$.

A finite simplicial complex $L$ is called a [*flag complex*]{} if every collection of vertices of $L$ which are pairwise adjacent spans a simplex of $L$. The formula was proved in [@akita] under the additional assumptions that $K$ is a PL-triangulation of a closed $2m$-manifold and is a flag complex. M. W. Davis pointed out that the formula follows from a result in [@davis], provided $K$ is a flag complex (see [*Note added in proof*]{} in [@akita]). Both results follow from the considerations of the Euler characteristics of Coxeter groups. In this note, we deduce the formula from the generalized Dehn-Sommerville equations proved by Klee [@klee].

Let $K$ be a finite $(d-1)$-dimensional simplicial complex and $f_i=f_i(K)$ the number of $i$-simplices of $K$ as before. The $d$-tuple $(f_0,f_1,\dots,f_{d-1})$ is called the [*$f$-vector*]{} of $K$. The [*$f$-polynomial*]{} $f_K(t)$ of $K$ is defined by $$f_K(t)=t^d+f_0t^{d-1}+\cdots+f_{d-2}t+f_{d-1}.$$ Define the [*$h$-polynomial*]{} $h_K(t)$ of $K$, $$h_K(t)=h_0t^d+h_1t^{d-1}+\cdots+h_{d-1}t+h_d,$$ by the rule $h_K(t)=f_K(t-1)$. The $(d+1)$-tuple $(h_0,h_1,\dots,h_d)$ is called the [*$h$-vector*]{} of $K$. The $h$-vector of $K$ satisfies the generalized Dehn-Sommerville equations, as stated below in Theorem \[DS\].

\[DS\] Let $K$ be a $(d-1)$-dimensional Eulerian manifold. Then $$h_{d-i}-h_i=(-1)^i\binom{d}{i}(\chi(K)-\chi(S^{d-1}))$$ holds for all $i$ $(0\leq i\leq d)$.

Klee stated the generalized Dehn-Sommerville equations in terms of the $f$-vector rather than the $h$-vector. The formulae quoted in Theorem \[DS\] are equivalent to those in [@klee] and can be found in [@ed]. Theorem \[DS\] was also proved in [@panov] by a quite different method, provided that $K$ is a triangulation of a closed manifold.

Now we prove Theorem \[main\]. We have $$h_K(-1)=\sum_{i=0}^{2m+1}(-1)^{2m+1-i}h_i
=\sum_{i=0}^{m} (-1)^i (h_{2m+1-i}-h_i).$$ Now Theorem \[DS\] asserts that $$h_{2m+1-i}-h_i=(-1)^i\binom{2m+1}{i}(\chi(K)-2).$$ Hence we obtain $$\label{h-poly}
h_K(-1)=(\chi(K)-2)\sum_{i=0}^m\binom{2m+1}{i}
=2^{2m}(\chi(K)-2).$$ On the other hand, we have $$\label{f-poly}
f_K(-2)=(-2)^{2m+1}+\sum_{i=0}^{2m}(-2)^{2m-i}f_i
=2^{2m}\left( -2+\sum_{i=0}^{2m}\left(-\frac{1}{2}\right)^if_i
\right).$$ Since $h_K(-1)=f_K(-2)$ by the definition of the $h$-polynomial $h_K(t)$, Theorem \[main\] follows from and .

[1]{}

T. Akita, Euler characteristics of Coxeter groups, PL-triangulations of closed manifolds, and cohomology of subgroups of Artin groups, J. London Math. Soc. (2) 61 (2000), 721–736.

V. M. Buchstaber, T. E. Panov, [*Torus actions and their applications in topology and combinatorics*]{}, University Lecture Series 24, American Mathematical Society, Providence, 2002.

R. Charney, M. W. Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991), 373–378.

V. Klee, A combinatorial analogue of Poincaré’s duality theorem, Canad. J. Math. 16 (1964), 517–531.

E. Swartz, From spheres to manifolds, preprint (2005).

[^1]: Partially supported by the Grant-in-Aid for Scientific Research (C) (No.17560054) from the Japan Society for Promotion of Sciences.
