Mathematics

The algebraic K-theory of Lawvere theories provides a context for the systematic study of the stable homology of the automorphism groups of algebraic structures, such as the symmetric groups, the general linear groups, the automorphism groups of free groups, and many, many more. We develop this theory and present a wealth of old and new examples to compare our non-linear setting to the theories of modules over rings via assembly maps. For instance, a new computation included here is that of the algebraic K-theory of the Lawvere theory of Boolean algebras and all theories Morita equivalent to it, in terms of the stable homotopy groups of spheres. We give a comprehensive discussion of Morita invariance: The higher algebraic K-theory of Lawvere theories is invariant under passage to matrix theories, but, in general, not under idempotent modifications. We also prove that algebraic K-theory is a monoidal functor on the category of Lawvere theories with the Kronecker product as its monoidal product. This result enables us to embed the classical assembly maps in algebraic K-theory into our framework and discuss many other examples and extensions.

By a diagonal embedding of U(1) in S U q (m) , we prolongate the diagonal circle action on the Vaksman-Soibelman quantum sphere S 2n+1 q to the S U q (m) -action on the prolongated bundle. Then we prove that the noncommutative vector bundles associated via the fundamental representation of S U q (m) , for m∈{2,…,n} , yield generators of the even K-theory group of the C*-algebra of the Vaksman-Soibelman quantum complex projective space C P n q .

We define asymptotic transfers in bounded K-theory together with a context where this can be done in great generality. Controlled algebra plays a central role in many advances in geometric topology, including recent work on Novikov, Borel, and Farrell-Jones conjectures. One of the features that appears in various manifestations throughout the subject, starting with the original work of Farrell and Jones, is an asymptotic transfer whose meaning and construction depend on the geometric circumstances. We first develop a general framework that allows us to construct a version of asymptotic transfer maps for any finite aspherical complex. This framework is the equivariant parametrized K-theory with fibred control. We also include several fibrewise excision theorems for its computation and a discussion of where the standard tools break down and which tools replace them.

We prove the Banach strong Novikov conjecture for groups having polynomially bounded higher-order combinatorial functions. This includes all automatic groups.

For any Kumjian-Pask algebra K P R (Λ) defined over a k -graph Λ of a special kind (a "standard k -graph"), we obtain an R -basis.

We prove that Hochschild cohomology with coefficients in A ∗ =Ho m k (A,k) together with an A -structural map ψ: A ∗ ⊗ A A ∗ → A ∗ is a Batalin-Vilkovisky algebra. This applies to symmetric, Frobenius and monomial path algebras.

The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and compute its action on K-theory (of tori) described geometrically in terms of Baum-Douglas cocycles, showing that the Fourier-Mukai transform maps the class of a subtorus to the class of a suitably defined dual torus. We deduce the Fourier-Mukai inversion formula. We use these results to give a purely geometric description of the Baum-Connes assembly map for free abelian groups.

We calculate suitably localized Hochschild homologies of various quantum groups and Podleś spheres after realizing them as generalized Weyl algebras (GWAs). We use the fact that every GWA is birationally equivalent to a smash product with a 1-torus. We also address and solve the birational equivalence problem and the birational smoothness problem for GWAs.

This is a survey on Kasparov's bivariant KK -theory in connection with the Baum-Connes conjecture on the K -theory of crossed products A ??r G by actions of a locally compact group G on a C*-algebra A . In particular we shall discuss Kasparov's Dirac dual-Dirac method as well as the permanence properties of the conjecture and the "Going-Down principle" for the left hand side of the conjecture, which often allows to reduce K -theory computations for A ??r G to computations for crossed products by compact subgroups of G . We give several applications for this principle including a discussion of a method developed by Cuntz, Li and the author for explicit computations of the K -theory groups of crossed products for certain group actions on totally disconnected spaces. This provides an important tool for the computation of K -theory groups of semi-group C*-algebras.

Let ℓ be a commutative ring with involution ∗ containing an element λ such that λ+ λ ∗ =1 and let Alg ∗ ℓ be the category of ℓ -algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category k k h and a functor j h : Alg ∗ ℓ →k k h that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in k k h . By the universal property of the latter, this implies that any functor H: Alg ∗ ℓ →T with values in a triangulated category which is homotopy invariant, matricially and hermitian stable and excisive satisfies the fundamental theorem. We also prove a bivariant version of Karoubi's 12 -term exact sequence.