A Topological Chern-Weil Theory

90 Pages·1993·7.348 MB·English

by  Anthony V. Phillips, David A., M.D. Stone

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MEMOIRS
-LVA f the
0
American Mathematical Society
Number 504
A Topological Chern-Weil Theory
Anthony V. Phillips
David A. Stone
September 1993 • Volume 105 • Number 504 (fifth of 6 numbers) • ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification.
Primary 53C05, 55R35, 55R40, 57R20, 57R22, 57T30.
Library of Congress Cataloging-in-Publication Data
Phillips, Anthony V. (Anthony Valiant), 1938-
A topological Chern-Weil theory/Anthony V. Phillips, David A. Stone.
p. cm. - (Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 504)
Includes bibliographical references and index.
ISBN 0-8218-2566-6
1. Characteristic classes. 2. Fiber bundles (Mathematics) 3. Topological groups. I. Stone,
David A. II. Title. III. Title: Chern-Weil theory. IV. Series.
QA613.618.P45 1993 93-25081
514'.72—dc20 CIP
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10 9 8 7 6 5 4 3 2 1 98 97 96 95 94 93
Contents
Introduction 1
1 Combinatorial preliminaries 7
2 The universal side of the problem: the topological Lie
algebra, tensor algebra and invariant polynomials 19
3 Parallel transport functions and principal bundles 29
4 The complex C*, the twisting cochain of a parallel trans
port function, and the algebraic classifying map S*:C* —•
e* 33
5 Cochains on C* with values in Tg 47
m
6 The main theorem 63
Appendix, The cobar construction, holonomy, and par
allel transport functions 69
Bibliography 77
v
Abstract
We examine the general problem of computing characteristic invariants of
principal bundles whose structural group G is a topological group. Under the
hypothesis that G has real cohomology finitely generated as an R-module,
we are able to give a completely topological, local method for computing
representative cocycles for real characteristic classes; our method applies,
for example, to the (homologically) 10-dimensional non-Lie group of Hilton-
Roitberg-Stasheff. We work with £*, the singular complex of G, and the
the topological tensor algebra Tg* derived from it. Our cohomological hy
pothesis on G guarantees the existence of a certain family of Stasheff's s.h.m
maps from G into various i^(R, n)'s; from these we define Jg*, an algebra
of cocycles on Tg*. We prove that Jg* ~ H*(BG; R), so this algebra can be
used like the invariant subalgebra of the Lie algebra in the classical case.
We show how to encode a principal G-bundle £ = (TT: E —> X), by data
(A, o, V), where A is a sufficiently fine triangulation of X, o is a local ordering
of the vertices of A, and V is a parallel transport function (p.t.f.) defined
on A. A p.t.f., closely related to a "twisting cochain" of Brown, is a kind of
intermediate object between a connection in the classical sense and a lattice
gauge field; in particular, it can be constructed from a finite amount of data,
but from it we may reconstruct £ up to isomorphism. From V we define a
topological connection u and its curvature fi, which appear as Tg*-valued
cochains. We then use 7g* to prove that the real characteristic classes of £
are represented by cocycles on A which are defined in terms of $7, and are
thus calculated completely as functions of the data.
In an Appendix we indicate how our theory is related to the cobar con
struction of Adams.
Key words: topological group, characteristic classes, parallel transport func
tion, connection, curvature, bar construction, s.h.m. maps, twisting cochain.
vi
Introduction
This work grew out of our earlier research in the topology of lattice
gauge fields [29], [30], and mainly out of the computational aspect of
that research. There we gave algorithms which can be used to compute
the characteristic classes of U(l)- and 5[/(2)-bundles from their repre
sentations as lattice gauge fields (in particular, from a finite amount of
data). Here we examine the general problem of computing characteris
tic invariants of principal bundles whose structural group is a topolog
ical group, but not necessarily a Lie group, so that we must make do
without integration. We are able to give a completely topological, lo
cal method for computing representative cocycles for real characteristic
classes, under the hypothesis that the structural group has real coho-
mology finitely generated as an R-module; so our theory applies, for ex
ample, to the (homologically) 10-dimensional non-Lie group of Hilton-
Roitberg-Stasheff [35]. As a by-product, for bundles whose structural
group G is a Lie subgroup of G£(p, C), we obtain new methods for
locally calculating characteristic cocycles without integrals, methods
which should be applicable to extending the computations mentioned
above. These have been explained separately in [31].
As a setting for our problem, let us examine two methods of cal
culating real characteristic classes: the algebraic-topological and the
differential-geometric.
Let G b ea topological group, and £ = (mE —• X) a principal G-
bundle. Given the problem of calculating the R-characteristic classes
of £, the algebraic topologist divides it into two parts, a general and
a particular one. Let £ = (n: EG —> BG) be a universal G-bundle;
1 Received by the Editors July 25, 1991; in final form March 19, 1992.
The first author was partially supported by NSF grants DMS-8607168 and DMS-
8907753; the second author was partially supported by a grant from PSC-CUNY
and by NSF grant DMS 8805485.
1
2 ANTHONY V. PHILLIPS and DAVID A. STONE
then the general problem is to calculate H*(BG;IV). The particular
problem is to determine a map f:X—* BG that classifies £, and then
to compute f*(H*(BG] R)) C H*{X\ R); these are the R-characteristic
classes of £.
When G is a compact, connected Lie group and X is a differentiate
manifold, the differential geometer has what appears to be a different
approach to the same calculation, by means of the Chern-Weil theory.
This too has a general and a particular aspect. Let g be the Lie algebra
of G, and 7*(g) the ring of G-invariant, symmetric polynomials on g.
The calculation of 7*(g) is the general problem. To solve the particular
one, we must choose a connection u; in £. Let SI be the curvature of
LO. Then, for each P G ^*(g), there is a unique differential form ap on
X such that 7r*ap = P(£2). Moreover each ap is closed, and the de
Rham map carries {ap: P € /*(g)} onto the R- characteristic classes of
£. Dupont [11] has shown how to generalize this differential-geometric
approach to the case that G is as above, and X is any simplicial space
(a concept more general than that of a simplicial complex).
The fundamental unity of these two approaches is well understood
when G is a Lie group as above. The proof that /*(g) ^ H*(BG; R)
lies at the heart of Chern-Weil theory (see, for example [4]). It is more
over possible to choose for £ a universal Cr-bundle-with-connection in
such a way that a connection LJ in a given £ corresponds to a particular
classifying map f:X —* BG [28]. Thus the Chern-Weil theory may be
regarded as a refinement of the topological theory for this case: a partic
ular classifying map is required, rather than just its homotopy class; in
return, characteristic classes are specified by particular representative
differential forms.
Our goal is to devise a Chern-Weil-type construction for compact,
connected topological groups so as to refine, in just this sense, the
algebraic-topological approach. This will require (and this is our main
task) finding topological substitutes for all of the items mentioned above
from the differential geometer's toolbox. As will be seen, we find the
substitutes we need in constructions from the post-classical period of
algebraic topology, occurring in works of Adams, Brown, Eilenberg-
Moore, Milnor, Milgram, and Stasheff. It turns out that these substi
tutes are in many ways true homologues of their differential-geometric
counterparts. Thus we have come some way towards finding a synthe
sis of the two approaches to characteristic classes, algebraic-topological
and differential-geometric. Compared to the algebraic-topological the-
A TOPOLOGICAL CHERN-WEIL THEORY 3
ory of characteristic classes, we work with specific cocycles, where the
influence of local geometry on the global topology may be discerned.
In comparison to the classical Chern-Weil theory, our work is carried
out at the small scale rather than the infinitesimal one; this allows it a
wider domain of application. From this point of view we hope that our
approach to the geometry of principal bundles will benefit the teaching
of differential geometry. As the theory of difference equations is more
elementary than that of differential equations, so the notions of con
nection, curvature etc. are, we believe, easier to comprehend in a local-
geometric theory than are their infinitesimal counterparts in standard
differential geometry. In fact our work could be viewed as a systematic
extension of the observation one usually makes when teaching a first
course in differential geometry, that curvature is the infinitesimal form
of the defect in parallel translation around a rectangle.
Now curvature is an essentially Lie-algebraic concept: the curvature
tensor incorporates the fact that there is a finite dimensional vector
space, with a (multi)-linear product operation, which gives a faithful
infinitesimal picture of the group operation near the identity. This is
where the differentiability of a Lie group comes into play. In our con
struction, finite-dimensionality becomes the requirement that the group
G under consideration be homologically finite in the sense mentioned
above. Then for the Lie algebra we can substitute a (finite) system of
s.h.m. representatives for the generators of the cohomology of G [36].
Here is an outline of the rest of this work. Our method of calculat
ing characteristic classes, like the algebraic-topological and differential-
geometric ones, also has a universal and a particular part.
Universal: The basic object we work with is C/*, the singular complex
of G. The group acts on its singular complex, and we take g* = Q*/G as
the topological Lie algebra of G. Milgram ([23]) defines a model of the
universal G-bundle; on its total space we construct geometrically a cell
complex £* which turns out to be the acyclic bar construction on Q*.
The group acts here also, and we take Tg* = £*/G as the topological
tensor algebra of G. Then following a construction of Stasheff's ([33])
we define the invariant subalgebra /(g*); this is where we use the system
of s.h.m. representatives of the generators of the cohomology of G.
Stasheff's work guarantees that /(g*) ^ H*(BG]H).
Particular: Let £ = (riE —• X) be a principal G-bundle; we take
as local geometric data for £ a triangulation A of X, a local ordering o
of the vertices of A, and a parallel transport function V for £ defined