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.

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