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In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.

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A reference for an equivariant Morse Lemma

Check Wasserman, Arthur G. Equivariant differential topology. Topology 8 1969 127--150. MR0250324 (40 #3563). In particular see Lemma 4.1 for an equivariant Morse lemma.
Allan Edmonds's user avatar
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Equivariant handle decompositions

This foundational paper: Arthur G. Wasserman, Equivariant Differential Topology, Topology, 8(1967), 127-150, has section 4 dealing with equivariant Morse theory for manifolds with a smooth action of a …
Allan Edmonds's user avatar