Smooth Structures on S p × S q

1969 ◽  
Vol 90 (1) ◽  
pp. 187 ◽  
Author(s):  
Reinhard E. Schultz
Keyword(s):  
Smooth Structures

Manifolds and Smooth Structures

2004 ◽  
pp. 1-35
Author(s):  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
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.

scholarly journals Smooth structures and Einstein metrics on

2009 ◽  
Vol 147 (2) ◽  
pp. 409-417 ◽  
Author(s):  
RAREŞ RǍSDEACONU ◽  
IOANA ŞUVAINA
Keyword(s):  
Scalar Curvature ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Smooth Structure ◽  
Smooth Structures ◽  
Higher Dimensional

AbstractWe show that each of the topological 4-manifolds $\bcp^2\# k\overline{\bcp^2}$, for k = 5, 6, 7, 8 admits a smooth structure which has an Einstein metric of scalar curvature s > 0, a smooth structure which carries an Einstein metric with s < 0 and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We also exhibit new examples of higher dimensional manifolds carrying Einstein metrics of both positive and negative scalar curvature.

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

Construction of a Surface Model and Layered Manufacturing Data From 3D Homogenization Output

1996 ◽  
Vol 118 (3) ◽  
pp. 412-418 ◽  
Author(s):  
A. L. Marsan ◽  
D. Dutta
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Homogenization Method ◽  
Layered Manufacturing ◽  
Dimensional Structure ◽  
Surface Model ◽  
Structural Topology ◽  
Smooth Structures ◽  
Manufacturing Technique ◽  
Traditional Manufacturing

A homogenization method has been recently developed to optimize the topology of a structure. This method will suggest a structural topology, but the results will be in finite element form. Most engineering applications, however, require smooth structures, whether the faces of the structures be planar or curved. Given the topology of a three-dimensional structure as suggested by the homogenization method, an algorithm is developed to interpret the structure and generate a smooth, manufacturable surface representation of the structure. Structures designed by the homogenization method can be quite complex and traditional manufacturing technique may not be well suited for constructing them. Layered manufacturing is adopted for producing such structures and it is shown how to generate the necessary data for this novel manufacturing technique from the surface model of the structure. Some steps of the algorithm require designer inputs. Examples are given which demonstrates this algorithm.

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