Morse theory on G-manifolds

Datta, Mahuya; Pandey, Neeta

doi:10.1016/s0166-8641(01)00199-7

Cited by 2 publications

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“…In our earlier paper [5], we tried to establish Morse relations for equivariant Morse functions on M using Bredon cohomology. The Bredon homology and cohomology theories do not support the above type of cells.…”

Section: Introductionmentioning

confidence: 99%

Equivariant Morse relations

Datta

Pandey

2007

Homology, Homotopy and Applications

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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 'γth 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 'γth homology groups'. In particular, the Morse relation is satisfied by Bredon homology.

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Section: Introductionmentioning

confidence: 99%