Manifolds and 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.

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The Development of Topological 4-manifold Theory

10.1093/oso/9780198841319.003.0021 ◽

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.

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Smooth structures, normalized Ricci flows, and finite cyclic groups

Annals of Global Analysis and Geometry ◽

10.1007/s10455-008-9136-6 ◽

2008 ◽

Vol 35 (3) ◽

pp. 267-275 ◽

Cited By ~ 2

Author(s):

Masashi Ishida ◽

Ioana Şuvaina

Keyword(s):

Cyclic Groups ◽

Smooth Structures ◽

Ricci Flows

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Smooth structures and Einstein metrics on

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004109002527 ◽

2009 ◽

Vol 147 (2) ◽

pp. 409-417 ◽

Cited By ~ 2

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.

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