Equivariant Morse relations

Abstract

For a finite group G, Costenoble and Waner defined a cellular (co-)homology theory for G-spaces X, which is graded on virtual representations of the equivariant fundamental groupoid …G(X). Using this homology, we associate an infinite (Morse) series with an equivariant Morse function f defined on a closed Riemannian G-manifold M. Wasserman has shown that when the critical locus of f is a disjoint union of orbits, M has a canonical decomposition into disc bundles. We show that if this decomposition ‘corresponds’ to a virtual representation ∞ of …G(M), then the Morse relations are satisfied by the ‘∞th homology groups’. For semi-free G-actions, we characterise the Morse functions which naturally give rise to such representations ∞ of …G(M). We also show that corresponding to any equivariant Morse function on a Z2-manifold, it is always possible to define virtual representations ∞ so that the Morse relation is satisfied by the ‘∞th homology groups’. In particular, the Morse relation is satisfied by Bredon homology.