Grassmannians and Gauss Maps in Piecewise-linear Topology
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 Grassma
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1366 Norman Levitt
Grassmannians and Gauss Maps in Piecewise-linear Topology
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1366 Norman Levitt
Grassmannians and Gauss Maps in Piecewise-linear Topology
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Norman Levitt Department of Mathematics Rutgers, The State University New Brunswick, NJ 08903, USA
Mathematics Subject Classification (1980): 57035,57050,57091, 57R20 ISBN 3-540-50756·6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50756-6 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
To my Parents
CONTENTS
CHAPTER
0
Introduction
CHAPTER
1
Local Formulae for Characteristic classes
11
CHAPTER
2
Formal Links and the PL Grassmannian
43
CHAPTER
3
Some Variations of the
CHAPTER
4
The Immersion Theorem for Subcomplexes of
CHAPTER
5
1
.-J!:ln, k
60
. . . .
70
Immersions Equivariant with Respect to Orthogonal Actions on Rn+k . . . . . .
. . . .
87
CHAPTER
6
Immersions into Triangulated Manifolds
101
CHAPTER
7
The Grassmannian for Piecewise Smooth Immersions
116
CHAPTER
8
Some Applications to Smoothing Theory
161
CHAPTER
9
Equivariant Piecewise Differentiable Immersions
181
CHAPTER 10
Piecewise Differentiable Immersions into Riemannian Manifolds
APPENDIX:
. . . . . 188
Glossary of Important Definitions and Constructions . . . . . . . . . .
REFERENCES
198
. . . . . . . . . . . . . . . 202
v
0.1
O.
Introduction
This monograph brings together a number of results centered on an attempt to import into the study of PL manifolds some geometric ideas which take their inspiration from the origins of differential topology and differential geometry, ideas from which many important aspects of fiber-bundle theory have developed.
The reader is
presumed to be familiar with the central role that the theory of fiber bundles has played in the study of differentiable manifolds for the past four decades.
The central theme here has been that a wide
class of geometric problems can be reformulated as bundle-theoretic problems.
Typical results flOWing from this approach have been the
Cairns-Hirsch Smoothing Theorem; The Hirsch Immersion Theorem, together with its generalization, the Gromov-Phillips Theorem, and much of the import
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