By Tammo tom Dieck, Ian Hambleton
These notes have been ready for the DMV-Seminar held in Dusseldorf, Schloss Mickeln from June 28 to July five, 1987. They include components which might be learn independently. The reader is presumed to have a easy schooling in differential and algebraic topology. surgical procedure conception is the elemental software for the research of differential and topological manifolds. a scientific improvement of the speculation is a protracted and hard activity. the aim of those notes is to explain easy examples and whilst to offer an advent to a couple of the systematic elements of the speculation. the 1st half is anxious with examples. they're concerning representations of finite teams and workforce activities on spheres, and are regarded as a generalisation of the round house shape challenge. the second one half experiences the overall environment of surgical procedure concept and experiences at the computation of the surgical procedure abstraction teams. either components current fabric no longer coated in any textbook and in addition provide an advent to the literature and components of analysis. 1. illustration kinds AND HOMOTOPY REPRESENTATIONS. Tammo tom Dieck Mathematical Institute Gottingen college Fed. Rep. of Germany enable G be a (finite) crew. We give some thought to team activities of G on spheres and spherelike spaces.
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Extra info for Surgery Theory and Geometry of Representations
1 Theorem. Let G be cyclic. Then the canonical homo- LZ/m morphisms are isomorphisms. Proof. We first show that JO(G) of dimension functions. 4 whenever ~ \' d~k we obtain LZ/m is two-dimensional and one-dimensional otherwise. 3 Lemma. A function Dim kid LZ/d. dimV(d). u(d)n(LZ/kd). An irreducible real representation of if 2: C(G) n E C(G) is contained in the image of if and only if u(d)n(LZ/kd) _ 0 mod 2 'm/k I > 2. Proof of the Le~~. 2) and the remarks thereafter the con- gruences are necessary.
Then the following holds. 11 Theorem. X has the G-homotopy type of a finite complex if and only if for every element n(H) H ( G such that is a pt-1_ th power Proof. TOM DIECK . G, G/H is non-cyclic the mod pt = [G/HI. 31 5. Cyclic groups. 1 Theorem. Let G be cyclic. Then the canonical homo- LZ/m morphisms are isomorphisms. Proof. We first show that JO(G) of dimension functions. 4 whenever ~ \' d~k we obtain LZ/m is two-dimensional and one-dimensional otherwise. 3 Lemma. A function Dim kid LZ/d.
S n Thus if there exists a homotopy equivalence, then the stated congruence must hold. ) jO(G,f) 0 be the jO group of G-representations which are direct sums of faithful irreducible representations. :1 } , ~ j 0 (G , f) By lifting representations of m = I Gr. 9 Proposition. s = ~ jO(G). (cHIH ( G) jO(G/H,f) -7 jO(G) H~G is an isomorphism. bH:jO(G) -7 jO(G/H). Proof. Taking H--fixed paints defines a map bHc K = 0 jO(G/K,f) We have equals if H ~ K; -p jO(G/K) 7 and for H ( K jO(G/H). this composition Clearly s is surjec- tive.