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Smoothly equivalent real analytic proper \(G\)-manifolds are subanalytically equivalent. (English) Zbl 0863.57029

The following result is proved:
Theorem A. Let \(G\) be a Lie group and let \(M\) and \(N\) be real analytic proper \(G\)-manifolds. Suppose that there exists a \(G\)-equivariant \(C^r\) diffeomorphism \(f:M \to N\), where \(1\leq r< \infty\). Then there exists a subanalytic \(G\)-equivariant \(C^r\) diffeomorphism \(h:M \to N\). Furthermore we can choose \(h\) to be \(G\)-homotopic to \(f\).
We recall here that a map \(h:M \to N\) between real analytic manifolds is subanalytic if it is continuous and its graph is a subanalytic subset of \(M \times N\). Thus if \(h:M \to N\) is subanalytic and a topological homeomorphism then also \(h^{-1}: N\to M\) is subanalytic. The main part of the conclusion of Theorem A is the very fact that \(h: M \to N\) is a \(G\)-equivariant subanalytic isomorphism. However, for the actual proof of Theorem A it is crucial that one maintains the \(C^r\) diffeomorphism, \(1\leq r< \infty\), property throughout the various stages of the proof. The additional fact that one can choose \(h\) to be \(G\)-homotopic to the given map \(f\) is important for applications of Theorem A to equivariant simple-homotopy theory and equivariant Whitehead torsion, and these questions will be discussed in another paper.
An immediate applications of Theorem A is its role in establishing the uniqueness part in the equivariant triangulation theorem for smooth proper \(G\)-manifolds. Let \(M\) be a smooth proper \(G\)-manifold. We first note that by [the author, Every proper smooth action of a Lie group is equivalent to a real analytic action: a contribution to Hilbert’s fifth problem, Ann. Math. Stud. 138, 189-220 (1995)] there exists a real analytic structure \(\beta\) on \(M\), compatible with the given smooth structure on \(M\), such that the action of \(G\) on \(M_\beta\) is real analytic. Furthermore, by [the author, Subanalytic equivariant triangulation of real analytic proper \(G\)-manifolds for \(G\) a Lie group, Preprint, Dept. Math., Princeton Univ. 1992] we know that the real analytic proper \(G\)-manifold \(M_\beta\) can be given a subanalytic equivariant triangulation and any two subanalytic equivariant triangulations of \(M_\beta\) have a common subanalytic equivariant subdivision. Now by Theorem A, the real analytic \(G\)-manifold \(M_\beta\) is uniquely determined up to \(G\)-equivariant subanalytic isomorphism, by the given smooth proper \(G\)-manifold \(M\). Thus we obtain, by the above procedure, for every smooth proper \(G\)-manifold \(M\) a class of equivariant triangulations of \(M\) with the property that any two equivariant triangulations in this class have equivariant subdivisions that are isomorphic.

MSC:

57S20 Noncompact Lie groups of transformations
32C05 Real-analytic manifolds, real-analytic spaces
32B20 Semi-analytic sets, subanalytic sets, and generalizations
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References:

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