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Equivariant Surgery Theories and Their Periodicity by Karl Heinz Dovermann

By Karl Heinz Dovermann

The concept of surgical procedure on manifolds has been generalized to different types of manifolds with crew activities in numerous alternative ways. This booklet discusses a few easy homes that such theories have in universal. unique 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 staff activities. The contents of the publication presuppose a few familiarity with the elemental principles of surgical procedure thought and transformation teams, yet no earlier wisdom of equivariant surgical procedure is thought. The e-book is designed to serve both as an advent to equivariant surgical procedure concept for complicated graduate scholars and researchers in comparable components, or as an account of the authors' formerly unpublished paintings on periodicity for experts in surgical procedure thought or transformation groups.

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3. In this ease one obtains a homomorphism such that c~Z7z is reduction mod B. " Results of M. 5 when SA satisfies the Gap Hypothesis but S2A does not. Namely, the group in question is the Bak group Wdim ~(Z[I~],/-'~r~(X~); wfl) introduced in [Ba], where I"Wz(Xz) is a set defined as on page 467 of [Mor], and w~ is a first Stiefel-Whitney class. Frequently this group is isomorphic to the corresponding Wall group. In particular, this is true if G has odd order or dim(a) _= 0 or 1 mod 4 and the orientation homomorphism is trivial.

This condition played a central role in Section 4. In fact, the stepwise surgery question can be rephrased to ask whether a E - a d j u s t e d m a p can be modified to a N ' - a d j u s t e d map, where E' -- E U G{fl} and fl is minimal in 7r(M) - E. 2. If E = ~r(M), t h e n a Z - a d j u s t e d map in the category of G - CW complexes is an equivariant homotopy equivalence (as noted in Section 4, this follows from considerations involving pushouts and the equivariant Whitehead Theorem). 3. It is possible to take 2 = 0 , and for this choice all maps are N-adjusted.

Math. Helv. 46 (1971), 381-407. [CS] S. Cappell and J. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. 99 (1974), 277-348. [CF] P. E. Conner and E. E. Floyd, Maps of odd period, Ann. of Math. 84 (1966), 132-156. [CV] F. Connolly and V. Vijums, G-normal maps and equivariant homotopy theory, preprint, University of Notre Dame, 1979. [CMY] E. H. Connell, D. Montgomery, and C. T. Yang, Compact groups in E n, Ann. of Math. 80 (1964), 94-103; correction, Ann.

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