A TOPOLOGICAL CHERN-WEIL THEORY 3
ory of characteristic classes, we work with specific cocycles, where the
influence of local geometry on the global topology may be discerned.
In comparison to the classical Chern-Weil theory, our work is carried
out at the small scale rather than the infinitesimal one; this allows it a
wider domain of application. From this point of view we hope that our
approach to the geometry of principal bundles will benefit the teaching
of differential geometry. As the theory of difference equations is more
elementary than that of differential equations, so the notions of con-
nection, curvature etc. are, we believe, easier to comprehend in a local-
geometric theory than are their infinitesimal counterparts in standard
differential geometry. In fact our work could be viewed as a systematic
extension of the observation one usually makes when teaching a first
course in differential geometry, that curvature is the infinitesimal form
of the defect in parallel translation around a rectangle.
Now curvature is an essentially Lie-algebraic concept: the curvature
tensor incorporates the fact that there is a finite dimensional vector
space, with a (multi)-linear product operation, which gives a faithful
infinitesimal picture of the group operation near the identity. This is
where the differentiability of a Lie group comes into play. In our con-
struction, finite-dimensionality becomes the requirement that the group
G under consideration be homologically finite in the sense mentioned
above. Then for the Lie algebra we can substitute a (finite) system of
s.h.m. representatives for the generators of the cohomology of G .
Here is an outline of the rest of this work. Our method of calculat-
ing characteristic classes, like the algebraic-topological and differential-
geometric ones, also has a universal and a particular part.
Universal: The basic object we work with is C/*, the singular complex
of G. The group acts on its singular complex, and we take g* = Q*/G as
the topological Lie algebra of G. Milgram () defines a model of the
universal G-bundle; on its total space we construct geometrically a cell
complex £* which turns out to be the acyclic bar construction on Q*.
The group acts here also, and we take Tg* = £*/G as the topological
tensor algebra of G. Then following a construction of Stasheff's ()
we define the invariant subalgebra /(g*); this is where we use the system
of s.h.m. representatives of the generators of the cohomology of G.
Stasheff's work guarantees that /(g*) ^ H*(BG]H).
Particular: Let £ = (riE —• X) be a principal G-bundle; we take
as local geometric data for £ a triangulation A of X , a local ordering o
of the vertices of A, and a parallel transport function V for £ defined