Remark

Let $E$ be a complex, $h : E \to C$ and $h' : E \to C'$ be homotopy equivalences with $C$ and $C'$ bounded of type $\mathscr{C}$. Then $\chi(C) = \chi(C')$. Indeed, if $h_1$ is an inverse of $h$ up to homotopy, $h' \circ h_1$ is a homotopy equivalence, hence a homology isomorphism from $C$ to $C'$, and one can apply Prop. 10. Consequently, one can extend Definition 8 by setting $\chi(E) = \chi(C)$ whenever there exists a homotopy equivalence from $E$ to a complex $C$ bounded of type $\mathscr{C}$. Propositions 10, 11, 12 and their corollaries generalise in this setting.

Application:

\* Let X be a topological space admitting a finite cellular decomposition (cf. No. 3). a) Let K and K' be two fields, put $b_i = \dim_K(H_i(X, K))$ and $b'_i = \dim_{K'}(H_i(X, K'))$. It is not necessarily the case that $b_i = b'_i$, but one has $\Sigma (-1)^i b_i = \Sigma (-1)^i b'_i$. b) Let $(X_n)$ and $(X'_n)$ be two finite cellular decompositions of X, and denote by $c_n$ and $c'_n$ the number of cells of dimension n in these two decompositions. One has

c) With the notations of a) and b), one has $\Sigma (-1)^i c_i = \Sigma (--1)^i b_i$. The properties a) and b) result from c), and c) results from prop. 11 applied to the complex $\Gamma$ described in No. 3, taking for $\mathscr{C}$ the class of finite-dimensional K-vector spaces and for $\varphi$ the function defined by $\varphi([M]) = \dim_K(M)$ (X, p. 40, example 1).