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 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).
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