Homotopy classification of finite group actions on aspherical spaces

The author proposes a method for investigating actions of finite groups on aspherical spaces. Complete homotopy classification of free actions of finite groups on aspherical spaces is obtained. Also there are some results about non-free actions. For example a relation between the cohomology of finite groups and the lattice structure of its subgroups is obtained by the proposed method. This relation is formulated in terms of spectral sequences.

💡 Research Summary

The paper investigates actions of finite groups on aspherical spaces (i.e., K(π,1) spaces) from a homotopy‑theoretic viewpoint and provides a complete classification of free actions, together with several results concerning non‑free actions. The central idea is that a free action of a finite group G on an aspherical space X yields a regular covering X → X/G, and consequently an exact sequence of fundamental groups

 1 → π₁(X) → π₁(X/G) → G → 1.

This sequence is interpreted as a group extension of G by π₁(X). The author shows that two free actions are homotopy‑equivalent if and only if the associated extensions are equivalent in the sense of group‑extension theory (i.e., they lie in the same element of Ext(G,π₁(X))). Conversely, given any extension class