Mathematics

For each member A of a family of linear cycle sets whose underlying abelian group is cyclic of order a power of a prime number, we compute all the central extensions of A by an arbitrary abelian group.

We give a new categorical way to construct the central stability homology of Putman and Sam and explain how it can be used in the context of representation stability and homological stability. In contrast to them, we cover categories with infinite automorphism groups. We also connect central stability homology to Randal-Williams and Wahl's work on homological stability. We also develop a criterion that implies that functors that are polynomial in the sense of Randal-Williams and Wahl are centrally stable in the sense of Putman.

Let G,H be groups, ?:G?�H a group morphism, and A a G -graded algebra. The morphism ? induces an H -grading on A , and on any G -graded A -module, which thus becomes an H -graded A -module. Given an injective G -graded A -module, we give bounds for its injective dimension when seen as H -graded A -module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.

The paper combines several fortunate mini miracles to achieve its two objectives. These were woven together in a several year's effort to answer a question raised by Iz Singer a decade ago. Our answer is accessible to the topologist, to the differential geometer and to the analyst who appreciates the statement of the Index theorem of Atiyah,Patodi,Singer for manifolds with boundary. The mini miracles are these: a] The Conner Floyd miracle that complex bordism tensored over the Todd genus and the Bott miracle that stable complex vector bundles respectively satisfy the axioms of a generalized homology theory and of a generalized cohomology theory. b] That these theories, with the covariant and contravariant geometric representations indicated, stably almost complex (SAC) manifolds modulo product relations and stable complex bundles, are not only related by Alexander duality but they are also related by Pontryagin duality. c] The abstract corollary of b] that stable complex bundles have a complete system of numerical invariants and that these can be computed by integrals of chern weil characteristic forms over manifolds with boundary reduced modulo integers, thanks to the APS Index Theorem. d] The adiabatic limit argument of the appendix to the last section showing a direct sum connection on the total space of a riemannian family of Riemannian manifolds with connection is Chern Simons equivalent in the limit to the Levi Civita connection of the direct sum metric. This allows the invariants to be described by the eta invariants of odd SAC manifolds reduced mod integers.

For a unitary representation ϕ of the fundamental group of a compact smooth manifold, Atiyah, Patodi, Singer defined the so called α -invariant of ϕ using the Chern-Simons invariants. In this article using traces on C ∗ -algebras, we give an intrinsically(i.e without using Chern character) define an element in KK with real coefficients theory whose pullback by the representation ϕ is the α -invariant.

Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F. Grunewald, J. Mennicke, and L. Vaserstein for G=SL_n, Sp_2n. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.

For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K-homology to general coarse homology theories. Furthermore, we calculate the domain of the coarse assembly map explicitly in terms of locally finite homology theory.

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by F. Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. We show that coarse regular coherence implies weak regular coherence, a weakening of regular coherence by G. Carlsson and the first author. The latter was introduced with the same goal as Waldhausen's, in order to perform computations of algebraic K-theory of group rings. However, all groups known to be weakly regular coherent are also coarsely regular coherent. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Using the language of coarse homology theories, we provide an axiomatic account of vanishing results for the fibres of forget-control maps associated to spaces with equivariant finite decomposition complexity.