Equivariant orientations and Thom isomorphisms
J.P. May
University of Chicago
may@math.uchicago.edu
Despite a great deal of work, notably by Costenoble and Waner,
there is still not a fully satisfactory theory of orientations
and Thom isomorphisms of equivariant bundles. Working in a given
RO(G)-graded cohomology theory, one wants somehow to grade Thom
classes on fiber representations. A G-space B is G-connected if
each of its fixed point subspaces B^H is non-empty and connected.
Over such base spaces, one can fix the same fiber representation
for all fibers, and work of Lewis and myself gives a satisfactory
theory. In any approach to more general base spaces, one must
parametrize changes of fiber representation as one moves around B
on the equivariant fundamental groupoid pi(B), which depends on all
components of all fixed point spaces and all paths connecting them.
Costenoble and Waner package the complexity in a generalization
of RO(G)-graded cohomology. I propose an alternative. Giving up
the idea that an orientation should be a single cohomology class,
I propose that orientations should be compatible collections of
cohomology classes in the cohomologies of the Thom H-spaces of the
pullbacks of the given bundle to the ``H-connected covers'' of B.
This allows one to quote rather than generalize the theory of Lewis
and myself. The H-connected covers introduced for this purpose
should have other uses. As in the case of orientations, they provide
a substitute in the equivariant world for the standard first step in
so many nonequivariant arguments, namely: ``We may assume without
loss of generality that X is connected''.