Surgery and duality

@article{Kreck1999SurgeryAD,
title={Surgery and duality},
author={M. Kreck},
journal={Annals of Mathematics},
year={1999},
volume={149},
pages={707-754}
}
• M. Kreck
• Published 1999
• Mathematics
• Annals of Mathematics
Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension‚ 5. In this paper, a modiflcation of this theory is presented, where instead of flxing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are flxed, where k is at least [n=2… Expand
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References

SHOWING 1-10 OF 70 REFERENCES
Differentiable periodic maps
• Mathematics
• 1964
1. The bordism groups. This note presents an outline of the authors' efforts to apply Thorn's cobordism theory [ó] to the study of differentiable periodic maps. First, however, we shall outline ourExpand
Surgery on Simply-Connected Manifolds
I. Poincare Duality.- 1. Slant Operations, Cup and Cap Products.- 2. Poincare Duality.- 3. Poincare Pairs and Triads Sums of Poincare Pairs and Maps.- 4. The Spivak Normal Fibre Space.- II. The MainExpand
Topological classification of 4-dimensional complete intersections
• Mathematics
• 1996
Let X,,(d) C C P "+r denote a complete intersection, the transversal intersection of r hypersurfaces in C P ~+r defined by r homogeneous polynomials of degrees (d l , . . . ,dr) =: d, with dld2...d,.Expand
Foundational Essays on Topological Manifolds, Smoothings, and Triangulations.
• Mathematics
• 1977
Since Poincare's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when KirbyExpand
Surgery on compact manifolds
Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An importantExpand
Simply connected manifolds of positive scalar curvature
Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold MExpand
On the structure of manifolds with positive scalar curvature
• Mathematics
• 1979
Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which areExpand
Topology of 4-manifolds
• Mathematics
• 1990
One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topologicalExpand
The Mod p Cohomology of BO〈4k 〉
Let B0(4k) denote the 4k -1 connected covering of BO, the classifying space for stable vector bundles. B0(4k) is the 4k-1 connected total space of a fibration 7r: B0(4k)--BO such that 7r*:Expand
Self-homeomorphisms of 4-manifolds with fundamental group Z
• Mathematics
• 2000
Abstract In this paper we study the classification of self-homeomorphisms of closed, connected, oriented 4-manifolds with infinite cyclic fundamental group up to pseudoisotopy, or equivalently up toExpand