What question did this study set out to answer?

To show the openness and density of equivariant definable Morse functions with certain critical orbit conditions.

June 4, 2026Open Access

Equivariant definable Morse functions in definably complete structures

Key Points

To show the openness and density of equivariant definable Morse functions with certain critical orbit conditions.
Considered a compact definable C r group G and a compact affine definable C r G manifold X.
Analyzed the set of G invariant definable C r functions and their topology.
Established the conditions for the loci of Morse functions being finite unions of nondegenerate critical orbits.
Demonstrated that the set of equivariant definable Morse functions is open within G invariant definable C r functions.
Confirmed that this set is also dense in the specified topology.

Abstract

Let G be a compact definable C r group, X a compact affine definable C r G manifold and 2 ≤ r < ∞.We prove that the set of equivariant definable Morse functions on X whose loci are finite unions of nondegenerate critical orbits is open and dense in the set of G invariant definable C r functions with respect to the definable C r topology.

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