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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