Examples of cohomology manifolds which are not homologically locally connected

Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n?3.     

Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.     

A homology theory for framed links in I-bundles using embedded surfaces

This paper introduces a homology theory for framed links in I-bundles over an orientable surface. The theory is unique in that the elements of the chain groups are embedded surfaces instead of diagrams. It is then shown this theory recaptures the homology theory constructed by Asaeda, Przytycki and Sikora.     

Hochschild and cyclic (co)homology of superadditive categories

We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.     

Leibniz and Hochschild homology

We construct and study the map from Leibniz homology HL_?(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH_??1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.     

无限箭图的同调群

对任意箭图Q,我们研究路代数A=kQ的Hochschild同调群H_n(A)和同调群Tor_n^Ae(A,A),其中A^e是代数A的包络代数。在本文中,我们具体地给出了各次同调群和Hochschild同调群。     

Measure homology and singular homology are isometrically isomorphic

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the ℓ¹-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology – as already claimed by Thurston. For example, measure homology can be used to prove Gromov's proportionality principle of simplicial volume.     

一个Cluster-Tilted代数的Hochschild同调与循环同调

基于Furuya构造的一个Cluster-Tilted代数的极小投射双模分解,用组合的方法计算了Cluster-Tilted代数的Hochschild同调空间的维数与基.当基础域的特征为零时,也计算了代数的循环同调群的维数.     

Homology Stability for Unitary Groups II

In this paper, the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied. We obtain a much better range of homology stability compared to the results existing in the literature. An application of our results is given.     

一个自入射Koszul代数的Hochschild同调与循环同调

代数的Hochschild同调群与其对应的Gabriel箭图的循环圈有着紧密的联系.本文基于Furuya构造的一个四点自入射Koszul代数的极小投射双模分解,用组合的方法计算了该代数的Hochschild同调空间的维数,并用循环圈的语言给出该代数的Hochschild同调空间的一组k-基.进一步,当基础域k的特征为零时,我们也得到了该代数的循环同调群的维数.     

Some computations of stable twisted homology for mapping class groups

Abstract

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper.

Communicated by Jason P. Bell     

Vanishing of Tate Homology — An Application of Stable Homology for Complexes

It has been proved that the vanishing of Tate homology is a sufficient condition for the derived depth formula to hold in [J. Pure Appl. Algebra, 219, 464–481(2015)]. In this paper, we investigate when Tate homology vanishes by studying the stable homology theory for complexes. Properties such as the balancedness and vanishing of stable homology for complexes are studied. Our results show that the vanishing of this homology can detect finiteness of homological dimensions of complexes and regularness of rings.     

Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C*–algebras given by nonstable E–theory are the universal such theories. By specializing to Abelian C*–algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an –spectrum which represents the theory.     

On a generalised Connes-Hochschild-Kostant-Rosenberg theorem

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character.     

The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.     

On semilocally simply connected spaces

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces.While the condition “semilocally simply connected” plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition “homotopically Hausdorff” instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.     

Piunikhin-Salamon-Schwarz isomorphisms for Lagrangian intersections

We study the intersections of gradient trajectories and holomorphic discs with Lagrangian boundary conditions in cotangent bundles, and give a construction of Piunikhin-Salamon-Schwarz isomorphisms in Lagrangian intersections Floer homology.     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

The plus-construction, Bousfield localization, and derived completion

We define a plus-construction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plus-construction is related to both Bousfield localization and Carlsson’s derived completion.