Equivariant Surgery Theory: Construction of Equivariant Normal Maps

This procedure is one of the important ideas of equivariant surgery theory, and has enabled us to construct various exotic actions (see e.g. [BMol-2], [LaMo], [LaMoPa], [Mo 1-3], [MoU], [Pel-3], [PeR]). A method for (Step I) was presented by T. Petrie in [Pel-3], which we call the equivariant transversality construction. Roughly speaking, it is as follows: Let G be a finite group, and let Y be a compact, smooth G-manifold. If F is a real G-module and a : Y x F > 7 x F i s a proper G-map then a is properly G-homotopic to /?: Y x V -> Y x V such that j8 is transversal to Y x {0}. Then we obtain a G-normal map f:X-+Y, where X = p~(Yx {0}) and /= 0\x: X-+ Y. (Step II) is to convert f\X-*Y to a G-map f':X'^>Y belonging to a prescribed class of maps, e.g. of G-homotopy equivalences, of homotopy equivalences, of Zp-homology equivalences, etc. If some properties of X' are specified before the construction then it is a key to find an adequate real G-module V and an appropriate G-map a. Modified equivariant transversality construction has been employed in [BMol-2], [LaMoPa], [LaMo], [Mol-3], [MoU]. For