Equivariant Surgery Theories and Their Periodicity by Karl Heinz Dovermann

By Karl Heinz Dovermann

The thought of surgical procedure on manifolds has been generalized to different types of manifolds with crew activities in different alternative ways. This publication discusses a few easy houses that such theories have in universal. exact emphasis is put on analogs of the fourfold periodicity theorems in traditional surgical procedure and the jobs of ordinary basic place hypotheses at the strata of manifolds with crew activities. The contents of the e-book presuppose a few familiarity with the fundamental rules of surgical procedure concept and transformation teams, yet no past wisdom of equivariant surgical procedure is believed. The publication is designed to serve both as an advent to equivariant surgical procedure concept for complex graduate scholars and researchers in similar components, or as an account of the authors' formerly unpublished paintings on periodicity for experts in surgical procedure thought or transformation groups.

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Finally, the proof that o'~ is onto proceeds as follows: Take a G-manifold V such that Av ~ S - l < and let ~ C L~imAZ[WA,w). As in [WL, Sections 5-6] or [DR] we can add handles to (V~ x I)/W~ away from the singular set to obtain an equivariant normal map F~ : U~ ---+ V~ x I that is a W~-equivalence on the boundary and near the singular set, and such that the relative stepwise surgery obstruction for the top closed substratum is a (here relative surgery means surgery away from the boundary). By construction U~ is formed from V~ × [0, c] by a finite sequence of equivariant surgeries.

Then there is an abelian group homomorphism I~(G;XxI;EC_E') o i~(g;x;E,) such that I~(G; X × I; E) g, I~(G; X × I, ~ c_ E') o I~(G; X; E') , f> Ia(G; X; ~) is also exact. 0) depends upon such identifications, the existence of such isomorphisms can be viewed as an additional axiom for a theory of equivariant surgery obstruction groups. Unfortunately, there is no fully satisfactory way of formalizing this, but in each case it is not diificult to show that there is an isomorphism of the desired type.

6, p. 34). For this theory the analog of a homotopy equivalence is a transverse linear isovariant homotopy equivalence, the analog of a simple homotopy equivalence is the concept described in the preceding paragraph, and the analogs of the Wall groups L,(~rl(X), c ~ j. w x ) (c = h or s) are the groups ~,, r BQ'c~X ~ In this section we have only discussed those portions of [BQ] dealing with equivariant surgery groups, and we have not attempted to explain the relevance of these groups to the Stratified Surgery Exact Sequence of [BQ, Thms.

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