scholarly journals Exotic smooth structures on small 4-manifolds with odd signatures

2010 ◽  
Vol 181 (3) ◽  
pp. 577-603 ◽  
Author(s):  
Anar Akhmedov ◽  
B. Doug Park
Keyword(s):  
Smooth Structures ◽  
Exotic Smooth Structures

scholarly journals Surgery along star-shaped plumbings and exotic smooth structures on 4–manifolds

2016 ◽  
Vol 16 (3) ◽  
pp. 1585-1635 ◽  
Author(s):  
Çağri Karakurt ◽  
Laura Starkston
Keyword(s):  
Smooth Structures ◽  
Exotic Smooth Structures

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.

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.

KNOTS AND EXOTIC SMOOTH STRUCTURES ON 4-MANIFOLDS

1993 ◽  
Vol 02 (01) ◽  
pp. 1-10 ◽  
Author(s):  
S. AKBULUT
Keyword(s):  
Smooth Structures ◽  
Short Survey ◽  
Exotic Smooth Structures

In this paper we give a short survey of exotic smooth structures of 4-manifolds. Various methods of constructing exotic structures and their relation to “Shake-Sliceness” and the Zeeman conjecture is discussed.

scholarly journals Exotic smooth structures on topological fiber bundles II

2013 ◽  
Vol 366 (2) ◽  
pp. 791-832 ◽  
Author(s):  
Sebastian Goette ◽  
Kiyoshi Igusa
Keyword(s):  
Fiber Bundles ◽  
Smooth Structures ◽  
Exotic Smooth Structures

scholarly journals NONCOMMUTATIVE MIXMASTER COSMOLOGIES

2012 ◽  
Vol 10 (01) ◽  
pp. 1250086 ◽  
Author(s):  
CHRISTOPHER ESTRADA ◽  
MATILDE MARCOLLI
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Universe Model ◽  
Noncommutative Torus ◽  
Spectral Action ◽  
Smooth Structures ◽  
Slow Roll ◽  
Anisotropic Cosmology ◽  
Exotic Smooth Structures ◽  
Slow Roll Inflation

In this paper we investigate a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3-tori. We consider ways in which the discrete dynamical system describing the mixmaster dynamics can be extended to act on the noncommutative torus moduli, and how the resulting dynamics differs from the classical one, for example, in the appearance of exotic smooth structures. We discuss properties of the spectral action, focussing on how the slow-roll inflation potential determined by the spectral action affects the mixmaster dynamics. We relate the model to other recent results on spectral action computation and we identify other physical contexts in which this model may be relevant.

Sign in / Sign up

Export Citation Format

Share Document