Open Access

Eric Schlarmann

• For a finite group G, a G-vector bundle is the equivariant analogue of an ordinary vector bundle. By applying the usual Grothendieck group construction to the abelian monoid of isomorphism classes of G-vector bundles with direct sum, one arrives at an equivariant version of the K-theory functor, which was already studied by Atiyah and Segal. With the correct setup, there is also a theory of characteristic classes, and an equivariant Chern character homomorphism, which, just like the ordinary Chern character, is a rational isomorphism. Additionally, one has a Chern–Weil homomorphism, leading to a differential refinement of the equivariant characteristic classes. We construct models of the classifying spaces of even and odd equivariant K-theory that are infinite-dimensional Banach manifolds. These are given by restricted versions of the usual Grassmannian and the unitary group of an infinite-dimensional Hilbert space. We show that they carry natural odd and even Chern forms that can beFor a finite group G, a G-vector bundle is the equivariant analogue of an ordinary vector bundle. By applying the usual Grothendieck group construction to the abelian monoid of isomorphism classes of G-vector bundles with direct sum, one arrives at an equivariant version of the K-theory functor, which was already studied by Atiyah and Segal. With the correct setup, there is also a theory of characteristic classes, and an equivariant Chern character homomorphism, which, just like the ordinary Chern character, is a rational isomorphism. Additionally, one has a Chern–Weil homomorphism, leading to a differential refinement of the equivariant characteristic classes. We construct models of the classifying spaces of even and odd equivariant K-theory that are infinite-dimensional Banach manifolds. These are given by restricted versions of the usual Grassmannian and the unitary group of an infinite-dimensional Hilbert space. We show that they carry natural odd and even Chern forms that can be adapted to give (delocalized) equivariant differential forms that refine the universal equivariant Chern character. Using this refinement, we construct a model of differential equivariant K-theory based on smooth classifying spaces, together with natural addition and inversion operations given by geometric operations directly on these spaces. We then show that the abelian group structure is induced by these operations. The regularity and explicitness of these maps allows us to work completely on the level of classifying spaces, and we do not require a compactness assumption on our manifolds that is present in many other descriptions of differential refinements. We therefore define the theory on the full category of smooth G-manifolds. One of the key features of K-theory is that one can, at least in the compact case, find vector bundles as geometric representatives for any class. This also remains true in the differential refinement, where one has to consider vector bundles with the additional datum of a connection. We investigate the possibility of such a cycle description in the equivariant setting and find that a key role is played by an equivariant version of the Venice Lemma by J. Simons. We show that our model is the unique differential equivariant extension that admits both an odd and an even degree differential cycle map.…