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John W. Milnor’s Work on the Classification of Differentiable Manifolds

  • L. C. SiebenmannEmail author
Chapter
Part of the The Abel Prize book series (AP)

Abstract

Jack Milnor has recently given this account of his unexpected encounter with exoticity:

…I was trying to study 3-connected 8-manifolds. The case H4=0 seemed too hard. For \(H_{4} = \mathbb{Z}\), one can assume that the 4-skeleton is a 4-sphere. To build an 8-manifold, one can try to fatten it up by taking a tubular normal bundle neighborhood, and then adjoin an 8-cell. This worked so beautifully that I came up with many manifolds which couldn’t possibly exist…

This article begins with a detailed elementary exposition of his revolutionary 1956 article that resolved the mentioned paradoxes, by unveiling exotic smooth structures on the 7-sphere. It continues by extensively describing Milnor’s introduction of ‘surgery’ to classify smooth structures on spheres. It concludes by sketching (all too briefly) the subsequent evolution of surgery theory—always confined to dimensions ≥5. In the hands of several following generations of topologists, it has become and remains the dominant tool for classification of compact, smooth (or piecewise linear or topological) manifolds.

Keywords

Smooth Manifold Homotopy Class Morse Function Topological Manifold Pontrjagin Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supplementary material

A lecture by Prof. Ghys in connection with the Abel Prize 2011 to John Milnor (MP4 399 MB)

A lecture by Prof. Hopkins in connection with the Abel Prize 2011 to John Milnor (MP4 306 MB)

A lecture by Prof. McMullen in connection with the Abel Prize 2011 to John Milnor (MP4 362 MB)

The Abel Lecture by John Milnor, the Abel Laureate 2011 (MP4 379 MB)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire de MathématiqueUniversité de Paris-SudOrsayFrance

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