Equivariant isotopies and submersions (Q762809)

In equivariant smoothing theory, one would like to make use of equivariant versions of the topological immersion, resp. submersion theorems due to \textit{J. A. Lees} [Bull. Am. Math. Soc. 75, 529-534 (1969; Zbl 0176.217)] and \textit{D. Gauld} [Trans. Am. Math. Soc. 149, 539-560 (1970; Zbl 0197.204)]. A G-immersion theorem is developed and used by the author and \textit{M. Rothenberg} in [Proc. Symp. Pure Math. 32, Part 1, 211-266 (1978; Zbl 0407.57018)]. The aim of this paper is the proof of an analogue G-immersion theorem: Let G be a finite group, N and Q be topological G-manifolds such that N satisfies the Bierstone condition - which already occurs in the smooth setting [see \textit{E. Bierstone}, Topology 13, 327-345 (1974; Zbl 0297.58005)]. Then the natural map from the simplicial set of G-submersions between N and Q into that of G- microbundle surjections of their tangent microbundles is a homotopy equivalence. The main ingredients of the proof are a G-isotropy extension theorem and a G-lifting theorem for submersions. These, in turn, are proved using the techniques of \textit{L. C. Siebenmann} [Comment. Math. Helv. 47, 123-163 (1972; Zbl 0252.57012)].