TL;DR: In this paper, the Atiyah-Singer $G$-signature theorem is extended to the case of simple Poincare embeddings with simple homotopy.
Abstract: Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important special case The even-dimensional case The odd-dimensional case The bounded odd-dimensional case The bounded even-dimensional case Completion of the proof Patterns of application: Manifold structures on Poincare complexes Applications to submanifolds Submanifolds: Other techniques Separating submanifolds Two-sided submanifolds One-sided submanifolds Calculations and applications: Calculations: Surgery obstruction groups Calculations: The surgery obstructions Applications: Free actions on spheres General remarks An extension of the Atiyah-Singer $G$-signature theorem Free actions of $S^1$ Fake projective spaces (real) Fake lens spaces Applications: Free uniform actions on euclidean space Fake tori Polycyclic groups Applications to 4-manifolds Postscript: Further ideas and suggestions: Recent work Function space methods Topological manifolds Poincare embeddings Homotopy and simple homotopy Further calculations Sullivan's results Reformulations of the algebra Rational surgery References Index.
TL;DR: Exact Sequences in the Algebraic Theory of Surgery as mentioned in this paper is a recent work on algebraic theory of surgery that deals with exact sequences in the algebraic model of surgery.
Abstract: The Description for this book, Exact Sequences in the Algebraic Theory of Surgery. (MN-26): , will be forthcoming.
TL;DR: The surgery classification of manifolds is illustrated with a model based on the model derived from the inequality of the discrete-time involution of the EMTs.
Abstract: Preface 1. The surgery classification of manifolds 2. Manifolds 3. Homotopy and homology 4. Poincare duality 5. Bundles 6. Cobordism theory 7. Embeddings, immersions and singularities 8. Whitehead torsion 9. Poincare complexes and spherical fibrations 10. Surgery on maps 11. The even-dimensional surgery obstruction 12. The odd-dimensional surgery obstruction 13. The structure set References Index
TL;DR: In this paper, Madsen et al. constructed an abelian group W (G; Y ) and an element (f) 2 W(G, Y ), called the surgery obstruction of f such that the vanishing of f in W (g, Y ) guarantees that f can be converted by G-surgery to a homotopy equivalence.
Abstract: Let G be a nite group. Let f : X ! Y be a k-connected, degree 1, G-framed map of simply connected, closed, oriented, smooth manifolds X and Y of dimension 2k = 6. Under the assumption that the dimension of the singular set of the action of G on X is at most k, we construct an abelian group W (G; Y ) and an element (f) 2 W (G; Y ), called the surgery obstruction of f such that the vanishing of (f) in W (G; Y ) guarantees that f can converted by G-surgery to a homotopy equivalence. Acknowledgement. We wish to express our gratitude to I. Madsen for stressing the impor- tance of the problem above. We are also grateful to E. Laitinen and K. Pawa lowski for their suggestions concerning brushing up the manuscript. The second author gratefully acknowl- edges support by the Japan Association for Mathematical Sciences and a Grant-in-Aid for Scientic Research from the Ministry of Education, Science and Culture, Japan.