Equivariant Morse relations

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 $π_G(X)$. 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 $γ$ of $π_G(M)$, then the Morse relations are satisfied by the “$γ$th homology groups”. For semi-free $G$-actions, we characterise the Morse functions which naturally give rise to such representations $γ$ of $π_G(M)$. We also show that corresponding to any equivariant Morse function on a $Z_2$-manifold, it is always possible to define virtual representations $γ$ so that the Morse relation is satisfied by the “$γ$th homology groups”. In particular, the Morse relation is satisfied by Bredon homology.

Keywords

equivariant Morse function, $RO(π_G(X))$ graded homology, Morse relation

2010 Mathematics Subject Classification

55N25, 55N91, 57R70, 57R91

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Published 1 January 2007