then the general problem is to calculate H*(BG;IV). The particular
problem is to determine a map f:X—* BG that classifies £, and then
to compute f*(H*(BG] R)) C H*{X\ R); these are the R-characteristic
classes of £.
When G is a compact, connected Lie group and X is a differentiate
manifold, the differential geometer has what appears to be a different
approach to the same calculation, by means of the Chern-Weil theory.
This too has a general and a particular aspect. Let g be the Lie algebra
of G, and 7*(g) the ring of G-invariant, symmetric polynomials on g.
The calculation of 7*(g) is the general problem. To solve the particular
one, we must choose a connection u; in £. Let SI be the curvature of
LO. Then, for each P G ^*(g), there is a unique differential form ap on
X such that 7r*ap = P(£2). Moreover each ap is closed, and the de
Rham map carries {ap: P /*(g)} onto the R- characteristic classes of
£. Dupont [11] has shown how to generalize this differential-geometric
approach to the case that G is as above, and X is any simplicial space
(a concept more general than that of a simplicial complex).
The fundamental unity of these two approaches is well understood
when G is a Lie group as above. The proof that /*(g) ^ H*(BG; R)
lies at the heart of Chern-Weil theory (see, for example [4]). It is more-
over possible to choose for £ a universal Cr-bundle-with-connection in
such a way that a connection LJ in a given £ corresponds to a particular
classifying map f:X * BG [28]. Thus the Chern-Weil theory may be
regarded as a refinement of the topological theory for this case: a partic-
ular classifying map is required, rather than just its homotopy class; in
return, characteristic classes are specified by particular representative
differential forms.
Our goal is to devise a Chern-Weil-type construction for compact,
connected topological groups so as to refine, in just this sense, the
algebraic-topological approach. This will require (and this is our main
task) finding topological substitutes for all of the items mentioned above
from the differential geometer's toolbox. As will be seen, we find the
substitutes we need in constructions from the post-classical period of
algebraic topology, occurring in works of Adams, Brown, Eilenberg-
Moore, Milnor, Milgram, and Stasheff. It turns out that these substi-
tutes are in many ways true homologues of their differential-geometric
counterparts. Thus we have come some way towards finding a synthe-
sis of the two approaches to characteristic classes, algebraic-topological
and differential-geometric. Compared to the algebraic-topological the-
Previous Page Next Page