Mathematics

We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (ie. non-classical) twist of K -theory over G=SU(n) . This twist is represented by a Fell bundle E→G , which reduces to the basic gerbe for the top exterior power functor. The groupoid G comes equipped with a G -action and an augmentation map G→G , that is an equivariant equivalence. The C ∗ -algebra C ∗ (E) associated to E is stably isomorphic to the section algebra of a locally trivial bundle with stabilised strongly self-absorbing fibres. Using a version of the Mayer-Vietoris spectral sequence we compute the equivariant higher twisted K -groups K G ∗ ( C ∗ (E)) for arbitrary exponential functor twists over SU(2) , and also over SU(3) after rationalisation.

By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic K-theory. We give a new and direct proof of Suslin's result based on an exact sequence of categories of perfect modules. In fact, we prove a more general descent result for a pullback square of ring spectra and any localizing invariant. Besides Suslin's result, this also contains Nisnevich descent of algebraic K-theory for affine schemes as a special case. Moreover, the role of the Tor-unitality condition becomes very transparent.

This paper provides a generalization of excision theorems in controlled algebra in the context of equivariant G-theory with fibred control and families of bounded actions. It also states and proves several characteristic features of this theory such as existence of the fibred assembly and the fibrewise trivialization.

In this paper we introduce exotic twisted T -equivariant K-theory of loop space LZ depending on the (typically non-flat) holonomy line bundle L B on LZ induced from a gerbe with connection B on Z . We also define exotic twisted T -equivariant Chern character that maps the exotic twisted T -equivariant K-theory of LZ into the exotic twisted T -equivariant cohomology as defined in an earlier paper of ours, and which localises to twisted cohomology of Z .

We show that certain expanders are counterexamples to the coarse p -Baum-Connes conjecture.

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This approach could be called {\Gamma}-homological algebra. The abstract kernel of non-abelian extensions of groups, its relation with the obstruction to the existence of non-abelian extensions and with the second group cohomology are extended to the case of non-abelian {\Gamma}-extensions of {\Gamma}-groups. We compute the rational {\Gamma}-equivariant (co)homology groups of finite cyclic {\Gamma}-groups. The isomorphism of the group of n-fold {\Gamma}-equivariant extensions of a {\Gamma}-group G by a G o {\Gamma}-module A with the (n+1)th {\Gamma}-equivariant group cohomology of G with coefficients in A is proven.We define the {\Gamma}-equivariant Hochschild homology as the homology of the {\Gamma}- Hochschild complex involving the cyclic homology when the basic ring contains rational numbers and generalizing the {\Gamma}equivariant(co)homology of {\Gamma}-groups when the action of the group {\Gamma} on the Hochschild complex is induced by its action on the basic ring. Important properties of the {\Gamma}-equivariant Hochschild homology related to Kahler differentials, Morita equivalence and derived functors are established. Group (co)homology and {\Gamma}-equivariant group (co)homology of crossed {\Gamma}-modules are introduced and investigated by using relevant derived functors Finally, applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of groups are given.

We compute the Ext group of the (filtered) Ogus category over a number field K . In particular we prove that the filtered Ogus realisation of mixed motives is not fully faithful.

We construct a Spanier-Whitehead type duality functor relating finite C -spectra to finite C op -spectra and prove that every C -homology theory is given by taking the homotopy groups of a balanced smash product with a fixed C op -spectrum. We use this to construct Chern characters for certain rational C -homology theories.

We prove that the action of a generalized braid group on an enhanced triangulated categories, generated by spherical twist functors along an ADE-configuration of ω -spherical objects, is faithful for any integer ω≠1 .

In this paper, we give a new proof of a well-known theorem due to tom Dieck that the fat realization and Segal's classifying space of an internal category in the category of topological spaces are homotopy equivalent.