Mathematics

Let G be a finite group, X be a compact G -space. In this note we study the ( Z + ×Z/2Z) -graded algebra F q G (X)= ⨁ n≥0 q n ⋅ K G≀ S n ( X n )⊗C, defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra. More specifically, let H be another finite group and Y be a compact H -space, we give a decomposition of F q G×H (X×Y) in terms of F q G (X) and F q H (Y) . For this, we need to study the representation theory of pullbacks of groups. We discuss also some applications of the above result to equivariant connective K-homology.

Higson-Kapsparov-Trout introduced an infinite-dimensional Clifford algebra of a Hilbert space, and verified Bott periodicity on K-theory. To develop algebraic topology of maps between Hilbert spaces, in this paper we introduce an induced Hilbert Clifford algebra, and construct an induced map between K-theory of the Higson-Kasparov-Trout Clifford algebra and the induced Clifford algebra. We also compute its K-group for some concrete case.

We construct, study, and apply a characteristic map from the relative periodic cyclic homology of the quotient map for a group action to the periodic Hopf-cyclic homology with coefficients associated with inertia of the action. This result admits, and in fact, comes from, its noncommutative-geometric, or quantized, counterpart. The crucial ingredient is the construction of the appropriate quantization of the cyclic nerve of the action groupoid, the cyclic object related to inertia, as the Connes-cyclic dual of a Hopf-cyclic object with coefficients in some stable anti-Yetter--Drinfeld module quantizing the Brylinski space. For the Hopf-Galois quantization of the case of trivial inertia, we find a non-trivial identification of our characteristic map with a well-known isomorphism of Jara--Ştefan. In presence of nontrivial inertia, for an analytic ramified Galois double cover, we show that the cokernel of our characteristic map, which can be interpreted as an invariant of inertia modulo topology of the maximal free action, is supported on the branch locus of the quotient map. Finally, we use our inertial Hopf-cyclic object to construct a new invariant of finite-dimensional algebras.

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more general, groups with finite decomposition complexity.

There are two Z 2 orbifolds of the Podle? quantum two-sphere, one being the quantum two-disc D q and other the quantum two-dimensional real projective space R P 2 q . In this article we calculate the Hochschild and cyclic homology and cohomology groups of these orbifolds and also the corresponding Chern-Connes indices.

In this note, we establish isomorphisms up to bounded torsion between relative K 0 -groups and Chow groups with modulus as defined by Binda-Saito.

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A=R[x_1,...,x_n]/I over R, the map H^1_Nis(A,G) -> H^1_Nis(R,G) induced by evaluation at x_1=...=x_n=0, is a bijection. If k has characteristic 0, then, moreover, the map H^1_et(A,G) -> H^1_et(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is >=2, and A is square-free, then K_1^G(A)=K_1^G(R), where K_1^G(R)=G(R)/E(R) is the corresponding non-stable K_1-functor, also called the Whitehead group of G. The corresponging statements for G=GL_n were previously proved by Ton Vorst.

For an extension of associative algebras B⊂A over a field and an A -bimodule X , we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of A and B , and the relative Hochschild homology, all of them with coefficients in X . This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness.

We generalize to higher algebraic K -theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.

We present a quick approach to computing the K -theory of the category of locally compact modules over any order in a semisimple Q -algebra. We obtain the K -theory by first quotienting out the compact modules and subsequently the vector modules. Our proof exploits the fact that the pair (vector modules plus compact modules, discrete modules) becomes a torsion theory after we quotient out the finite modules. Treating these quotients as exact categories is possible due to a recent localization formalism.