On the Euler-Poincaré characteristics of a simply connected rationally elliptic CW-complex (Q2139562)

For a simply connected rationally elliptic CW-complex \(X\), we have two known numerical invariants namely the cohomology Euler characteristic \(\chi _H\) and the homotopy Euler characteristic \(\chi_\pi\) defined by \[ \chi _H=\sum_{i\geq 0}(-1)^i{\dim}\ H^i(X; \mathbb{Q}), \chi_\pi=\sum_{i\geq 0}(-1)^i {\dim}\ \pi_{i}(X)\otimes \mathbb{Q}. \] In the paper under review, the author introduces two new numerical homotopy invariants namely \[ \rho_X=1+\sum_{i\geq 2}(-1)^{i}{\dim}\ H^i(X^{[i-2]}; \mathbb{Q}), \eta_{X}=1+\sum_{i\geq 2}(-1)^i{\dim}\ \Gamma_i(X), \] where \(X^{[i]}\) denotes the \(i\)th Postnikov section of \(X\), \(X^i\) denotes its \(i\)th skeleton and \(\Gamma_i(X)={\ker}(\pi_{i}(X^i)\otimes \mathbb{Q}\rightarrow \pi_{i}(X^i, X^{i-1})\otimes \mathbb{Q})\) for \(i\geq 2\). The author shows that \(\rho_X, \eta_{X}\) are related to the cohomology and the homotopy Euler-Poincaré characteristics. By virtue of the Whitehead exact sequences associated respectively to the Sullivan model and the Quillen model of \(X\), the author shows the following results: If \(X\) is a rational elliptic CW-complex, then \begin{itemize} \item[(i)] \(H^{i}(X^{[i-2]}; \mathbb{Q})\cong \Gamma_{i-1}(X)\) for all \(i\geq 2\); \item[(ii)] \(\rho_X=\eta_{X}>0\); \item[(iii)] Either \(\rho_X=\chi_H\) or \(\rho_X=-\chi_\pi\); \item[(iv)] If \(X\) is \(2\)-connected, then \(\Gamma_i(X)=\pi_{i}(X^i)\otimes \mathbb{Q}\) for all \(i\geq 2\). \end{itemize}