EQUIVARIANT DEFINABLE MORSE FUNCTIONS ON DEFINABLE MANIFOLDS

Tomohiro Kawakami (Japan)

Received September 17, 2007

Abstract

Let G be a compact definable group, X be a compact affine definable manifold and f be an equivariant definable Morse function on X. We prove that if f has no critical value in then is definably diffeomorphic to Moreover we prove that the set of equivariant definable Morse functions on X whose critical loci are finite unions of nondegenerate critical orbits is dense in the set of G invariant functions on X with respect to the Whitney topology. We also prove that if G is a compact definable group and X is a definable G manifold, then X is definably G homeomorphic to an open definable G CW complex.

Keywords and phrases: o-minimal, equivariant Morse theory, definable groups, equivariant definable Morse functions, critical points, critical values, open definable G CW complexes.