Grassmannians and Gauss Maps in Piecewise-Linear and Piecewise-Differential Topology

Norman Levitt3540507566, 9783540507567

The book explores the possibility of extending the notions of “Grassmannian” and “Gauss map” to the PL category. They are distinguished from “classifying space” and “classifying map” which are essentially homotopy-theoretic notions. The analogs of Grassmannian and Gauss map defined incorporate geometric and combinatorial information. Principal applications involve characteristic class theory, smoothing theory, and the existence of immersion satifying certain geometric criteria, e.g. curvature conditions. The book assumes knowledge of basic differential topology and bundle theory, including Hirsch-Gromov-Phillips theory, as well as the analogous theories for the PL category. The work should be of interest to mathematicians concerned with geometric topology, PL and PD aspects of differential geometry and the geometry of polyhedra.

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Table of contents :
front-matter……Page 1
01Introduction……Page 5
02Local formulae for characteristic classes……Page 15
03Formal links and the PL grassmannian G n,k……Page 47
04Some variations of the G n,k construction……Page 64
05The immersion theorem for subcomplexes of G n,k……Page 74
06Immersions equivariant with respect to orthogonal actions on Rn+k……Page 91
07Immersions into triangulated manifolds (with R. Mladineo)……Page 105
08The grassmannian for piecewise smooth immersions……Page 120
09Some applications to smoothing theory……Page 165
11Equivariant piecewise differentiable immersions……Page 185
12Piecewise differentiable immersions into riemannian manifolds……Page 192
back-matter……Page 202