4 ANTHONY V. PHILLIPS and DAVID A. STONE

on A. A parallel transport function (the definition will be repeated

from [31]) can be defined from a G-valued lattice gauge field u on A

if the plaquette products of u are sufficiently close to the identity; a

p.t.f. is equivalent to a system of transition functions for £ with domain

the top-dimensional dual cells of A. The p.t.f. V defines a bigraded

cell complex C* on E, which is a ^-module generated by certain maps

Ha\CTa

— E, one for each simplex a £ A, where C£ is a cube of

dimension r = dimcr. (C* is in fact the twisted product Q+ ®^ A* in

the sense of Brown [5], where p is the "twisting cochain" determined

by V.) The translates g • HG, for g 6 G, a 6 A, are called horizontal]

cells which are mapped into a single fibre are called vertical

V also defines a specific classifying map

/ : E - EG (Milgram's model)

i 1

/ : X - BG.

Now / induces a chain map 5*:C* — £* — » Tg*, which is the topolog-

ical equivalent of a Lie-algebra-valued differential form. In particular,

by combining Sm with certain natural projections in C* we define the

topological connection UJ of V and its topological curvature tt. Just

as in differential geometry, u vanishes on horizontal 1-cells of C* and

is (roughly speaking) the identity on vertical ones; while 0 vanishes

except on horizontal 2-cells.

We next define, for Tg*-valued cochains, operations paralleling fa-

miliar ones for differential forms: an exterior derivative d, a covariant

derivative D, and a wedge product A. We prove that UJ and ft satisfy

an Equation of structure and finally the

Main Theorem: Let G be any connected topological group with

real cohomology finitely generated as an R-module; and let a system

of s.h.m. representatives for the generators of H*(G]Tl) be fixed. Let

£ = (TT: E — + X) be a principal G-bundle, and (A, o, V) local geometric

data for £. Then a multiplicative basis for the real characteristic classes

of £ is represented by the set of cocycles y on A determined by

TT*J/

= n P A - ' A O ) , Y e /g

2

*.

k

The cocycles y are calculated completely in terms of the data (A, o, V).