Mathematics

We compute the isomorphism class in KK alg of all noncommutative generalized Weyl algebras $A=\CC[h](\sigma, P)$, where σ(h)=qh+ h 0 is an automorphism of $\CC[h]$, except when q≠1 is a root of unity. In particular, we compute the isomorphism class in KK alg of the quantum Weyl algebra, the primitive factors B λ of U( sl 2 ) and the quantum weighted projective lines O( WP q (k,l)) .

We establish a combinatorial model for the Boardman--Vogt tensor product of several absolutely free operads, that is free symmetric operads that are also free as S -modules. Our results imply that such a tensor product is always a free S -module, in contrast with the results of Kock and Bremner--Madariaga on hidden commutativity for the Boardman--Vogt tensor square of the operad of non-unital associative algebras.

We give a proof of the Bott periodicity theorem for topological K-theory of C*-algebras based on Loring's treatment of Voiculescu's almost commuting matrices and Atiyah's rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu's localization algebra and an associated explicit formula for the pairing between the first K-homology and first K-theory groups of a (separable) C*-algebra.

We use filtered modules over a Noetherian ring and fibred bounded control on homomorphisms to construct a new kind of controlled algebra with applications in geometric topology. The theory here can be thought of as a "pushout" of the bounded K-theory with fibred control and the controlled G-theory constructed and used by the authors. This paper contains the non-equivariant theory including controlled excision theorems crucial for computations.

We derive a lower and an upper bound for the rank of the finite part of operator K -theory groups of maximal and reduced C ??-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group S(M) and the group of positive scalar curvature metrics P(M) for an oriented manifold M . We define a class of groups called "polynomially full groups" for which the upper bound and the lower bound we derive are the same. We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator K -theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups.

We revisit the construction of signature classes in C*-algebra K-theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside of a compact set. As an application, we prove a counterpart for signature classes of a codimension two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).

The cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences between smooth varieties over an infinite prefect field k , chark≠2 , is proved, the isomorphism Ho m DM GW eff ( A ∙ , B ∙ )≃Ho m DM GW eff ( A ∙ (1), B ∙ (1)), for A ∙ , B ∙ ∈ DM GW eff (k) in the category of effective Grothendieck-Witt-motives constructed in \cite{AD_DMGWeff} is obtained (and similarly for Witt-motives). This implies that the canonical functor Σ ∞ G ∧1 m : DM GW eff (k)→ DM GW (k) is fully faithful, where DM GW (k) is the category of non-effective GW-motives (defined by stabilization of DM GW eff (k) along G ∧1 m ) and yields the main property of motives of smooth varieties in the category DM GW (k) : Ho m DM GW (k) ( M GW (X), Σ ∞ G ∧1 m F[i])≃ H i Nis (X,F), for any smooth variety X and homotopy invariant sheave with GW-transfers F (and similarly for DM W (k) ).

We prove cancellation theorems for reciprocity sheaves and cube-invariant modulus sheaves with transfers of Kahn--Saito--Yamazaki, generalizing Voevodsky's cancellation theorem for A 1 -invariant sheaves with transfers. As an application, we get some new formulas for internal hom's of the sheaves Ω i of absolute Kähler differentials.

We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces LX in derived algebraic geometry and the resulting close relationship between S 1 -equivariant quasi-coherent sheaves on LX and D X -modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficients for Hopf-cyclic theories. The cohomology is then obtained as a Hom in this dg-category between a Chern character object associated to an algebra and an arbitrary coefficient mixed anti-Yetter-Drinfeld contramodule.

We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution R such that 1 2 ∈R ; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian K -theory and use it to construct an E ∞ motivic ring spectrum KR alg representing homotopy Hermitian K -theory. From these results, we show that KR alg is stable under base change, and cdh descent for homotopy Hermitian K -theory of rings with involution is a formal consequence.