Mathematics

If a differential operator D on a smooth Hermitian vector bundle S over a compact manifold M is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If D is also elliptic, then the Hilbert space of square integrable sections of S with the canonical left C(M) -action and the operator χ(D) for χ a normalizing function is a Fredholm module, and its K -homology class is independent of χ . In this expository article, we provide a detailed proof of this fact following the outline in the book "Analytic K-homology" by Higson and Roe.

We give a necessary and sufficient condition for the existence of an enhancement of a finite triangulated category. Moreover, we show that enhancements are unique when they exist, up to Morita equivalence.

In this article, we establish a motivic analog of an enumeration result of James-Thomas on non-stable vector bundles in topological setting. Using this, we obtain results on enumeration of projective modules of rank d over a smooth affine k -algebra A of dimension d , recovering in particular results of Suslin and Bhatwadekar on cancellation of such vector bundles. Admitting a conjecture of Asok and Fasel, we prove cancellation of such vector bundles of rank d−1 if the base field k is algebraically closed. We also explore the cancellation properties of symplectic vector bundles.

We compute the equivariant K -homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological K -theory of the reduced C ∗ -algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated K -theory groups are torsion-free. As a result we can promote previous rational computations to integral compu- tations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.

We give a detailed and unified survey of equivariant KK -theory over locally compact, second countable, locally Hausdorff groupoids. We indicate precisely how the "classical" proofs relating to the Kasparov product can be used almost word-for-word in this setting, and give proofs for several results which do not currently appear in the literature.

We prove a {\Gamma}-equivariant version of the algebraic index theorem, where {\Gamma} is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypoelliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over C ∗ -algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the K -theory of a group C ∗ -algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the K -theory of a group C ∗ -algebra.

We study the Equivariant Cartan Homotopy Formula for the DG -algebra obtained by a finite group action.

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type corresponding to the open stratum and also in an iterated sense, with connecting equivariant fibrations over the boundary hypersurfaces covering the resolutions of the other strata. This structure descends to a resolution of the quotient as a stratified space. For an Abelian group action the equivariant K-theory can then be described in terms of bundles over the bases `dressed' by the representations of the isotropy types with morphisms covering the connecting maps. A similar model is given here covering the non-Abelian case. Now the reduced objects are torsion-twisted bundles over finite covers of the bases, corresponding to the projective action of the normalizers on the representations of the isotropy groups, again with morphisms over all the boundaries. This leads to a closely related iterated deRham model for equivariant cohomology and, now with values in forms twisted by flat bundles of representation rings over the bases, for delocalized equivariant cohomology. We show, as envisioned by Baum, Brylinksi and MacPherson, that the usual equivariant Chern character, mapping to equivariant cohomology, factors through a natural Chern character from equivariant K-theory to delocalized equivariant cohomology with the latter giving an Atiyah-Hirzebruch isomorphism.

We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra. As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic K -homology. Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.