Group actions on algebraic cell complexes
It is well known that a \(G\)-map \(f:X\to Y\) between \(G\)-complexes which induces a homotopy equivalence \(X^H\to Y^H\) between the \(H\)-fixed point sets for every subgroup \(H\leq G\), is a \(G\)-homotopy equivalence. In this article, an algebraic version of that result is proved. Given a \(G\)-set \(\Delta\) and a ring \(S\), \(S\Delta\) denotes the free \(S\)-module on \(\Delta\). An \(SG\)-map \(S\Delta \to S\Delta '\) is called admissible if it maps \(S[\Delta ^H]\) into \(S[\Delta '{}^H]\) for every \(H\leq G\). A special \(SG\)-complex consists of a sequence \(\Delta _0, \Delta _1, \dots\) of \(G\)-sets together with admissible maps \(S\Delta _i \to S\Delta _{i-1}\) which define a chain complex. The main result of the paper states that an admissible map between special \(SG\)-complexes is an \(SG\)-chain homotopy equivalence (between the augmented chain complexes) provided that it induces chain homotopy equivalences between the complexes of free \(S\)-modules on the \(H\)-fixed point sets of the \(G\)-sets. As a consequence, the authors obtain a generalization of a result of \textit{S. Bouc} [J. Algebra 220, 415--436 (1999; Zbl 0940.55020)] originally formulated for simplicial complexes, which claims that if \(G\) is finite, \(S\) does not have \(|G|\)-torsion and \(X\) is a finite-dimensional \(S\)-acyclic special \(SG\)-complex, then (the augmented chain complex of) \(X\) is \(SG\)-split. Some interesting alternative proofs of known results are shown as an application of this version of Bouc's theorem.
scientific article; zbMATH DE number 5849325