A topological Chern-Weil theory / Anthony V. Phillips, David A. Stone.

Format:

Book

Author/Creator:

Phillips, Anthony V. (Anthony Valiant), 1938- author.

Stone, David A. (David Aurel), author.

Series:

Memoirs of the American Mathematical Society ; Volume 105, Number 504.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 105, Number 504

Language:

English

Subjects (All):

Characteristic classes.

Fiber bundles (Mathematics).

Topological groups.

Physical Description:

1 online resource (90 p.)

Edition:

1st ed.

Other Title:

Chern-Weil theory.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1993.

Language Note:

English

Summary:

This work develops a topological analogue of the classical Chern-Weil theory as a method for computing the characteristic classes of principal bundles whose structural group is not necessarily a Lie group, but only a cohomologically finite topological group. Substitutes for the tools of differential geometry, such as the connection and curvature forms, are taken from algebraic topology, using work of Adams, Brown, Eilenberg-Moore, Milgram, Milnor, and Stasheff. The result is a synthesis of the algebraic-topological and differential-geometric approaches to characteristic classes. In contrast to the first approach, specific cocycles are used, so as to highlight the influence of local geometry on global topology. In contrast to the second, calculations are carried out at the small scale rather than the infinitesimal; in fact, this work may be viewed as a systematic extension of the observation that curvature is the infinitesimal form of the defect in parallel translation around a rectangle. This book could be used as a text for an advanced graduate course in algebraic topology.

Contents:

Intro

Contents

Introduction

1 Combinatorial preliminaries

2 The universal side of the problem: the topological Lie algebra, tensor algebra and invariant polynomials

3 Parallel transport functions and principal bundles

4 The complex C[sub(*)], the twisting cochain of a parallel transportfunction, and the algebraic classifying map S[sub(*)]:C[sub(*)] ε[sub(*)]

5 Cochains on C[sub(*)] with values in Tg[sub(*)]

6 The main theorem

Appendix. The cobar construction, holonomy, and parallel transport functions

Bibliography.

Notes:

"September 1993, Volume 105, Number 504 (fiifth of 6 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0081-2