### Equivariant orientation theory, by S. R. Costenoble, J. P. May, and S. Waner

We give a long overdue theory of orientations of G-vector
bundles, topological G-bundles, and spherical G-fibrations,
where G is a compact Lie group. The notion of equivariant
orientability is clear and unambiguous, but it is surprisingly
difficult to obtain a satisfactory notion of an equivariant
orientation such that every orientable G-vector bundle admits
an orientation. Our focus here is on the geometric and homotopical
aspects, rather than the cohomological aspects, of orientation
theory. Orientations are described in terms of functors defined
on equivariant fundamental groupoids of base G-spaces, and the
essence of the theory is to construct an appropriate universal
target category of G-vector bundles over orbit spaces G/H.
The theory requires new categorical concepts and constructions
that should be of interest in other subjects where analogous
structures arise.

S. R. Costenoble <Steven.R.Costenoble@Hofstra.edu>

J. P. May <may@uchicago.edu>

S. Waner <matszw@hofstra.edu>