scholarly journals A Survey of Bounded Surgery Theory and Applications

1994 ◽  
pp. 57-80 ◽  
Author(s):  
Steven C. Ferry ◽  
Ian Hambleton ◽  
Erik K. Pedersen
Keyword(s):  
Surgery Theory

scholarly journals Contact structures and reducible surgeries

2015 ◽  
Vol 152 (1) ◽  
pp. 152-186 ◽  
Author(s):  
Tye Lidman ◽  
Steven Sivek
Keyword(s):  
Contact Structures ◽  
Surgery Theory ◽  
Contact Topology ◽  
Exceptional Surgery ◽  
Genus 2 ◽  
Cabling Conjecture

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus$g$must have slope$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

Functional Endoscopic Sinus Surgery: Theory and Diagnostic Evaluation

1985 ◽  
Vol 111 (9) ◽  
pp. 576-582 ◽  
Author(s):  
D. W. Kennedy ◽  
S. J. Zinreich ◽  
A. E. Rosenbaum ◽  
M. E. Johns
Keyword(s):  
Endoscopic Sinus Surgery ◽  
Functional Endoscopic Sinus Surgery ◽  
Diagnostic Evaluation ◽  
Sinus Surgery ◽  
Surgery Theory

Near-Infrared Spectroscopy Monitoring in Cardiac Surgery: Theory, Practice, and Utility

2010 ◽  
pp. 113-129 ◽  
Author(s):  
John M. Murkin ◽  
Miguel F. Arango ◽  
Alain Deschamps ◽  
André Y. Denault
Keyword(s):  
Cardiac Surgery ◽  
Infrared Spectroscopy ◽  
Near Infrared Spectroscopy ◽  
Near Infrared ◽  
Surgery Theory

The Disc Embedding Theorem

2021 ◽  
Keyword(s):  
Graduate Students ◽  
Euclidean Space ◽  
Iterative Process ◽  
Embedding Theorem ◽  
Space Theory ◽  
Topological Category ◽  
Surgery Theory ◽  
Smooth Structures ◽  
Decomposition Space ◽  
Exotic Smooth Structures

The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

Surgery Theory and the Classification of Closed, Simply Connected 4-manifolds

2021 ◽  
pp. 331-352
Author(s):  
Patrick Orson ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Embedding Theorem ◽  
Surgery Theory ◽  
Simply Connected

Surgery theory and the classification of simply connected 4-manifolds comprise two key consequences of the disc embedding theorem. The chapter begins with an introduction to surgery theory from the perspective of 4-manifolds. In particular, the terms and maps in the surgery sequence are defined, and an explanation is given as to how the sphere embedding theorem, with the added ingredient of topological transversality, can be used to define the maps in the surgery sequence and show that it is exact. The surgery sequence is applied to classify simply connected closed 4-manifolds up to homeomorphism. The chapter closes with a survey of related classification results.

Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets

2008 ◽  
Vol 2 (3) ◽  
pp. 507-600 ◽  
Author(s):  
Anthony Bak ◽  
Masaharu Morimoto
Keyword(s):  
Intersection Theory ◽  
Finite Family ◽  
Dimensional Sphere ◽  
Singular Set ◽  
Intersection Form ◽  
Surgery Theory ◽  
Equivariant Surgery ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Simply Connected

AbstractLet G denote a finite group and n = 2k 6 an even integer. Let X denote a simply connected, compact, oriented, smooth G-manifold of dimension n. Let L denote a union of connected, compact, neat submanifolds in X of dimension k. We invoke the hypothesis that L is a G-subcomplex of a G-equivariant smooth triangulation of X and contains the singular set of the action of G on X. If the dimension of the G-singular set is also k then the ordinary equivariant self-intersection form is not well defined, although the equivariant intersection form is well defined. The first goal of the paper is to eliminate the deficiency above by constructing a new, well defined, equivariant, self-intersection form, called the generalized (or doubly parametrized) equivariant self-intersection form. Its value at a given element agrees with that of the ordinary equivariant self-intersection form when the latter value is well defined. Let denote a finite family of immersions withtrivial normal bundle of k-dimensional, connected, closed, orientable, smooth manifolds into X. Assume that the integral (and mod 2) intersection forms applied to members of and to orientable (and nonorientable) k-dimensional members of L are trivial. Then the vanishing of the equivariant intersection form on × and the generalized equivariant self-intersection form on is a necessary and sufficient condition that is regularly homotopic to a family of disjoint embeddings, each of which is disjoint from L. This property, when is a finite family of immersions of the k-dimensional sphere Sk into X, is just what is needed for constructing an equivariant surgery theory for G-manifolds X as above whose G-singular set has dimension less than or equal to k. What is new for surgery theory is that the equivariant surgery obstruction is defined for an almost arbitrary singular set of dimension k and in particular, the k-dimensional components of the singular set can be nonorientable.

Geometric properties of a spectral sequence in surgery theory

2002 ◽  
Vol 57 (6) ◽  
pp. 1238-1239
Author(s):  
Yu V Muranov ◽  
D Repovs
Keyword(s):  
Spectral Sequence ◽  
Geometric Properties ◽  
Surgery Theory

scholarly journals On a construction in bordism theory

1986 ◽  
Vol 29 (3) ◽  
pp. 413-422 ◽  
Author(s):  
Nigel Ray
Keyword(s):  
Line Bundles ◽  
Complex Line ◽  
Surgery Theory ◽  
Normal Bundles ◽  
Bordism Ring ◽  
Complex Line Bundles

In [2], R. Arthan and S. Bullett pose the problem of representing generators of the complex bordism ring MU* by manifolds which are totally normally split; i.e. whose stable normal bundles are split into a sum of complex line bundles. This has recently been solved by Ochanine and Schwartz [8] who use a mixture of J-theory and surgery theory to establish several results, including the following.

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