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

Morris W. Hirsch, Barry Mazur (Autoren)

eBook Download: PDF

2016
140 Seiten
Princeton University Press (Verlag)
978-1-4008-8168-0 (ISBN)

Infos

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 "e;solved"e; 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.

Produkt-Details

• Erscheint lt. Verlag: 2.3.2016
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: affine transformation • Approximation • associative property • bijection • Bundle map • Classification theorem • Codimension • coefficient • cohomology • commutative property • Computation • Convex cone • convolution • corollary • counterexample • diffeomorphism • Differentiable function • differentiable manifold • Differential structure • Dimension • Direct proof • Division by zero • Embedding • empty set • equivalence class • equivalence relation • Euclidean space • existential quantification • Exponential map (Lie theory) • fiber bundle • fibration • functor • Grassmannian • Homeomorphism • Homotopy • H-space • Integral curve • inverse problem • isomorphism class • K0 • Linearization • manifold • Mathematical Induction • Milnor conjecture • Natural transformation • Neighbourhood (mathematics) • Normal bundle • Obstruction theory • Open set • partition of unity • Piecewise linear • polyhedron • Reflexive relation • Regular map (graph theory) • Sheaf (mathematics) • smoothing • Smoothness • Special case • submanifold • tangent bundle • Tangent vector • Theorem • Topological Manifold • Topological space • Topology • Transition function • Transitive relation • vector bundle • Vector field
• ISBN-10: 1-4008-8168-4 / 1400881684
• ISBN-13: 978-1-4008-8168-0 / 9781400881680