Mathematics

The purpose of this article is to show a version of dévissage theorem of non-connective K -theory. Our theorem contains Quillen's dévissage theorem, Waldhausen's cell filtration theorem and theorem of heart as special cases. In this sense, we give an affirmative answer to Thomason's problem in Thomason-Trobaugh's paper. We introduce the notions of cell structures and dévissage spaces and our main theorem states a structure of non-connective K -theory of dévissage spaces in terms of non-connective K -theory of heart of cell structures. A specific feature in our proof is 'motivic' in the sense that properties of K -theory which we will utilze to prove the theorem are only categorical homotopy invariance, localization and co-continuity. On the other hands, it is well-known that the analogue of the dévissage theorem for K -theory does not hold for Hochschild homology theory. In this point of view, we could say that dévissage theorem is not 'motivic' over dg-categories. To overcome this dilemma, the notion of dévissage spaces should not be expressed by the language of dg-categories. First three sections are devoted to the foundation of our model of stable (∞,1) -categories which we will play on to give a description of dévissage spaces.

We construct a finitely generated group that does not satisfy the generalized Burghelea conjecture.

We prove that the computation of the Fredholm index for fully elliptic pseudodifferential operators on Lie manifolds can be reduced to the computation of the index of Dirac operators perturbed by smoothing operators. To this end we adapt to our framework ideas coming from Baum-Douglas geometric K-homology and in particular we introduce a notion of geometric cycles that can be classified into a variant of the famous geometric K-homology groups, for the specific situation here. We also define comparison maps between this geometric K-homology theory and relative K-theory.

We describe Universal Coefficient Theorems for the equivariant Kasparov theory for C*-algebras with an action of the group of integers or over a unique path space, using KK-valued invariants. We compare the resulting classification up to equivariant KK-equivalence with the recent classification theorem involving a K-theoretic invariant together with an obstruction class in a certain Ext^2-group and with the classification by filtrated K-theory. This is based on a general theorem that computes these obstruction classes.

This paper explores further the computation of the twisted K-theory and K-homology of compact simple Lie groups, previously studied by Hopkins, Moore, Maldacena-Moore-Seiberg, Braun, and Douglas, with a focus on groups of rank 2. We give a new method of computation based on the Segal spectral sequence which seems to us appreciably simpler than the methods used previously, at least in many key cases. The exposition has been clarified and one mistake in the previous version has been fixed. Also the references have been updated.

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products.

We deduce an analogue of Quillen--Suslin's local-global principle for the transvection subgroups of the general quadratic (Bak's unitary) groups. As an application we revisit the result of Bak--Petrov--Tang on injective stabilization for the K_1-functor of the general quadratic groups.

We collect three observations on the homology for Smale spaces defined by Putnam. The definition of such homology groups involves four complexes. It is shown here that a simple convergence theorem for spectral sequences can be used to prove that all complexes yield the same homology. Furthermore, we introduce a simplicial framework by which the various complexes can be understood as suitable "symmetric" Moore complexes associated to the simplicial structure. The last section discusses projective resolutions in the context of dynamical systems. It is shown that the projective cover of a Smale space is realized by the system of shift spaces and factor maps onto it.

In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group W E (R,I) is defined. In this article we will give the details of this construction. Then we studied the injectivity of the relative Vaserstein symbol V R,I :U m 3 (R,I)/ E 3 (R,I)??W E (R,I) . We established injectivity of this symbol if R is an affine non-singular algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R . It is believed that for a 3 -dimensional affine algebra non-singularity is not necessary for establishing injectivity of the Vaserstein symbol . At the end of the article we will give an example of a singular 3 -dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective.

For an abelian group A , we give a precise homological description of the kernel of the natural map Γ(A)→A ⊗ Z A , γ(a)↦a⊗a , where Γ is whitehead's quadratic functor from the category of abelian groups to itself.