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 e a 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;

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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.

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