The Development of Topological 4-manifold Theory

2021 ◽  
pp. 295-330
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Smooth Structures ◽  
Normal Bundles ◽  
Exotic Smooth Structures ◽  
Poincare Conjecture ◽  
Manifold Theory ◽  
Simply Connected

The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.

Context for the Disc Embedding Theorem

2021 ◽  
pp. 1-26
Author(s):  
Stefan Behrens ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Euclidean Space ◽  
Embedding Theorem ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Four Dimensions ◽  
Surgery Theory ◽  
Smooth Structures ◽  
Exotic Smooth Structures ◽  
Central Result ◽  
Manifold Theory

‘Context for the Disc Embedding Theorem’ explains why the theorem is the central result in the study of topological 4-manifolds. After recalling surgery theory and the proof of the s-cobordism theorem for high-dimensional manifolds, the chapter explains what goes wrong when trying to apply the same techniques in four dimensions and how to start overcoming these problems. The complete statement of the disc embedding theorem is provided. Finally the most important consequences to manifold theory are listed, including a proof of why Alexander polynomial one knots are topologically slice and the existence of exotic smooth structures on 4-dimensional Euclidean space.

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.

scholarly journals NEW MEAN VALUES FOR HOMOGENEOUS SPATIAL TESSELLATIONS THAT ARE STABLE UNDER ITERATION

2010 ◽  
Vol 29 (3) ◽  
pp. 143 ◽  
Author(s):  
Christoph Thäle ◽  
Viola Weiss
Keyword(s):  
Euclidean Space ◽  
Convex Geometry ◽  
Dimensional Euclidean Space ◽  
Mean Values ◽  
3 Dimensional ◽  
Extremum Problems ◽  
Random Tessellations ◽  
Typical Cell

Homogeneous random tessellations in the 3-dimensional Euclidean space are considered that are stable under iteration – STIT tessellations. A classification of vertices, segments and flats is introduced and a couple of new metric and topological mean values for them and for the typical cell are calculated. They are illustrated by two examples, the isotropic and the cuboid case. Several extremum problems for these mean values are solved with the help of techniques from convex geometry by introducing an associated zonoid for STIT tessellations.

scholarly journals BIHARMONIC CURVES INTO QUADRICS

2014 ◽  
Vol 57 (1) ◽  
pp. 131-141 ◽  
Author(s):  
S. MONTALDO ◽  
A. RATTO
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Algebraic Method ◽  
Polynomial Equation ◽  
Complete Classification ◽  
Dimensional Euclidean Space ◽  
Implicit Surface ◽  
Biharmonic Curves ◽  
Real Quadrics

AbstractWe develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves in surfaces defined by a polynomial equation: In particular, we use it to give a complete classification of biharmonic curves in real quadrics of the three-dimensional Euclidean space.

scholarly journals CLASSIFICATION OF YANG–MILLS FIELDS ADMITTING INTEGRALS OF MOTION FOR THE WONG EQUATIONS

2020 ◽  
pp. 14-24
Author(s):  
M. N. Boldyreva ◽  
A. A. Magazev ◽  
I. V. Shirokov
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
First Integrals ◽  
Gauge Fields ◽  
Dimensional Euclidean Space ◽  
Integrals Of Motion ◽  
Yang Mills ◽  
Linear Integral

In the paper, we investigate the gauge fields that are characterized by the existence of non-trivial integrals of motion for the Wong equations. For the gauge group 𝑆𝑈(2), the class of fields admitting only the isospin first integrals is described in detail. All gauge non-equivalent Yang–Mills fields admitting a linear integral of motion for the Wong equations are classified in the three-dimensional Euclidean space

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