Submission date: 5 June 2012.

Abstract:

Let G be a definably compact definable C^r group and r a positive integer. Let X be a definable C^r G submanifold of a representation of G and Y a definable C^r submanifold of R^n. We prove that every G invariant surjective submersive definable C^r map $f:X \to Y$ is piecewise definably C^r G trivial.

Mathematics Subject Classification: 14P10, 14P20, 57R35, 58A05, 03C64.

Keywords and phrases: O-minimal, real closed fields, piecewise definable C^r G triviality, definable C^r G compactification.

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