1. The Component Structure of a Smooth G Manifold This section develops the combinatorial structure relevant to G surgery which arises from the discrete invariants of a smooth G manifold and of an equivariant map between smooth G manifolds. The first part is mainly des- criptive and sets up the notation for the entire paper. The second part deals with subtle technical properties of the combinatorial structure of a smooth G manifold, as well as the properties of a G map. The main technical con- cept is the notion of a G poset being "p connected (1.21). The authors thank Bruce Williams for his aid and insights in this section. Let G be a finite group and •'(G) the set of subgroups of G. Here G acts on •'(G) via conjugation and a partial order is defined by (caution1.) H £. K if and only if H 2 K. For any G set T T and a c ir we define G = {g c G| gar = a] to be the isotropy group of a, AG poset is a pair (ir, p) where IT is a partially ordered G set, and ptir -* *^(G) is a map of partially ordered G sets. Most of the time the map p will be understood and we speak about the G poset -rr. A G poset map is a G map between two G posets which preserves partial order. It need not commute with p. To each G space X we associate G posets i (X) C ir(X) C FI(X) and extract the com- binatorial properties of G posets which are relevant to G surgery. In par- ticular, tr(X) is geometric, H(X) is complete, and the first is dense in the Received by the editor March 24th, 1981. Supported by NSF grant MCS 7905036 and MCS 8100751 for K.H.Dovermann. Supported by NSF grant MCS 7903234 for T.Petrie. 1
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