# A survey of 4-manifolds through the eyes of surgery

@article{Kirby1998ASO,
title={A survey of 4-manifolds through the eyes of surgery},
author={R. Kirby and L. Taylor},
journal={arXiv: Geometric Topology},
year={1998}
}
• Published 1998
• Mathematics
• arXiv: Geometric Topology
Surgery theory is a method for constructing manifolds satisfying a given collection of homotopy conditions. It is usually combined with the s{cobordism theorem which constructs homeomorphisms or diieomorphisms between two similar looking manifolds. Building on work of Sullivan, Wall applied these two techniques to the problem of computing structure sets. While this is not the only use of surgery theory, it is the aspect on which we will concentrate in this survey. In dimension 4, there are two… Expand
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#### References

SHOWING 1-10 OF 99 REFERENCES
Surgery and duality
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 ofExpand
• Mathematics
• 1996
In this paper we investigate the relationship between isotopy classes of knots and links in S and the dieomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initialExpand
The topology of four-dimensional manifolds
0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie inExpand
On the homotopy theory of simply connected four manifolds
• Mathematics
• 1990
THIS PAPER is concerned with the homotopy theory of l-connected 4-manifolds. Our principal results are explicit characterizations and computations of the groups HE(X), HE + (X), HE,(X) and n4( X)Expand
On 4-dimensional $s$-cobordisms
• Mathematics
• 1985
The idea that topological problems can be converted into questions of algebra and homotopy theory underlies much of modern higher-dimensional topology of manifolds. The s-cobordism theorem, alsoExpand
Isotopy of 4-manifolds
The principal result of this paper is that the group of homeomorphisms mod isotopy (the "homeotopy" group) of a closed simply-connected 4-manifold is equal to the automorphism group of the quadraticExpand
Some new four-manifolds
• Mathematics
• 1976
The quotient space Q = X'/Z, of the involution is a smooth manifold of the (simple) homotopy type of real projective 4-space P', but not diffeomorphic or even piecewise linear (PL) homeomorphic toExpand
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that everyExpand
TORSION IN H-SPACES
A topological space with a continuous multiplication with unit is called an H-space. The topological properties of these spaces have been investigated by many authors, in particular the homology andExpand
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