Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80

Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80

Author: Morris W. Hirsch

Publisher: Princeton University Press

ISBN: 9781400881680

Category: Mathematics

Page: 140

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The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology. Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology. The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.
Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80
Language: en
Pages: 140
Authors: Morris W. Hirsch, Barry Mazur
Categories: Mathematics
Type: BOOK - Published: 2016-03-02 - Publisher: Princeton University Press

The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology. Thus the book attacks the problem of
Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88), Volume 88
Language: en
Pages: 368
Authors: Robion C. Kirby, Laurence C. Siebenmann
Categories: Mathematics
Type: BOOK - Published: 2016-03-02 - Publisher: Princeton University Press

Since Poincaré's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung
Surveys on Surgery Theory (AM-149), Volume 2
Language: en
Pages: 380
Authors: Sylvain Cappell, Andrew Ranicki, Jonathan Rosenberg
Categories: Mathematics
Type: BOOK - Published: 2014-09-08 - Publisher: Princeton University Press

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as
Classifying Spaces for Surgery and Corbordism of Manifolds. (AM-92), Volume 92
Language: en
Pages: 296
Authors: Ib Madsen, R. James Milgram
Categories: Mathematics
Type: BOOK - Published: 2016-03-02 - Publisher: Princeton University Press

Beginning with a general discussion of bordism, Professors Madsen and Milgram present the homotopy theory of the surgery classifying spaces and the classifying spaces for the various required bundle theories. The next part covers more recent work on the maps between these spaces and the properties of the PL and
Surveys on Surgery Theory (AM-145), Volume 1
Language: en
Pages: 448
Authors: Sylvain Cappell, Andrew Ranicki, Jonathan Rosenberg
Categories: Mathematics
Type: BOOK - Published: 2014-09-08 - Publisher: Princeton University Press

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. There have been some extraordinary accomplishments in that time, which have led to enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a