Approximation by equivariant homeomorphisms. I

Abstract

Locally linear (= locally smoothable) actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other. For these actions, an equivariant o-approximation theorem is proved using engulfing techniques. As corollaries are obtained equivariant ';fibrations are bundles and controlled h-cobordism theorems, as well as an equivariant version of Edwards' cell-like mapping theorem and the vanishing of the set of transfer-invariant G-homotopy topological structures, rel bound- ary, on Tn x Dp (when Tn is the n-torus with trivial G action and Dp is a representation disc). Here we consider locally linear (locally smooth (Br)) PL and topological actions of a finite group G on n-manifolds. In addition, we require that a G-manifold, M, have gaps of codimension > 3 (i.e., for H c G, if MH is a component of the H-fixed point set, MH, and M>H c MH is the subspace of points x for which the isotropy subgroup, Gx, strictly contains H, then either M>H = MH or M>H has codimension > 3 in MH). We shall use the term necessarily locally linear when we wish to drop the local linearity hypothesis but retain the property that if MH c M: but MH + M: then MH is a locally flat (inequivariantly) submanifold of MK Of codimension at least three. This, while self-contained, is the second in a series of papers (SW2,...,8,S) in which we analyze the extent of failure of the topological invariance of equivari- ant Whitehead torsion and the consequent failure of subgroups of the equivariant Whitehead group WhGL(M) of Illman (Ill) (cf. Rothenberg (R)) to classify PL or smooth G-h-cobordisms up to topological equivalence (under our gap hypotheses the arguments of Browder and Quinn (BQ) and Rothenberg (R) show that G-h- cobordisms are classified up to PL or smooth equivalence by such subgroups). We eventually conclude in (Sw7 S) (cf. (SW2)) that the topological equivalence classes of G-h-cobordisms on a G-manifold M that are products over the union of fixed point components of M having dimension < 5 are classiSed by a group WhG°P(M) to which WhGL(M) maps homomorphically. The image in WhG°P(M) of the per- tinent subgroup of WhGL(M) classifies up to equivariant homeomorphism, rel M, those G-h-cobordisms on M that admit equivariant handlebody decompositions relative to M. (This homomorphism is not generally surjective and there are G-h- cobordisms that do not admit such handle decompositions (cf. (SW5, S)). In order