Equivariant homology decompositions for cyclic group actions on definite 4-manifolds

TL;DR

This paper investigates the equivariant homology of cyclic group actions on certain 4-manifolds, providing a splitting result under specific divisibility conditions, with a focus on actions of prime power order.

Contribution

It introduces a method to decompose equivariant homology for cyclic group actions on 4-manifolds using cellular filtrations derived from unitary representations.

Findings

Cellular filtrations constructed on connected sums of complex projective planes.

Splitting of equivariant homology achieved under divisibility hypotheses.

Results applicable to cyclic groups of prime power order.

Abstract

In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular filtrations on the connected sum using spheres inside unitary representations. A judicious choice of filtration implies a splitting on equivariant homology for general cyclic groups under a divisibility hypothesis, and in all cases for those of prime power order.