Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe homology of permutation racks
96
0
0.0
(
0
)
Authors
Victoria Lebed
Ask ChatGPT about the research
No Arabic abstract
Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, there are still very few examples of racks whose homology has been fully calculated. In this paper, we compute the entire integral homology of all permutation racks. Our method of choice involves homotopical algebra, which was brought to bear on the homology of racks only recently. For our main result, we establish a spectral sequence, which reduces the problem to one in equivariant homology, and for which we show that it always degenerates. The blueprint given in this paper demonstrates the high potential for further exploitation of these techniques.
rate research
Read More
We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBokstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.
This paper investigates the homology of the Brauer algebras, interpreted as appropriate Tor-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when the parameter is not invertible, this isomorphism still holds in a range of degrees that increases with n.
We introduce the notion of covering homology of a commutative ring spectrum with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bokstedt, Hsiang and Madsens topological cyclic homology. In fact covering homology with respect to the family of orientation preserving isogenies of the circle is equal to topological cyclic homology. Our basic tool for the analysis of covering homology is a cofibration sequence involving homotopy orbits and a restriction map similar to the restriction map used in Bokstedt, Hsiang and Madsens construction of topological cyclic homology. Covering homology with respect to families of isogenies of a torus is constructed from iterated topological Hochschild homology. It receives a trace map from iterated algebraic K-theory and the hope is that the rich structure, and the calculability of covering homology will make covering homology useful in the exploration of J. Rognes ``red shift conjecture.
This paper studies the homology and cohomology of the Temperley-Lieb algebra TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a=v+v^{-1} for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n-2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even. Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of planar injective words that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL_n(a) is not flat over TL_m(a) for m<n, so that Shapiros lemma is unavailable. We resolve this difficulty by constructing what we call inductive resolutions of the relevant modules. Vanishing results for the homology and cohomology of Temperley-Lieb algebras can also be obtained from existence of the Jones-Wenzl projector. Our own vanishing results are in general far stronger than these, but in a restricted case we are able to obtain additional vanishing results via existence of the Jones-Wenzl projector. We believe that these results, together with the second authors work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.
We give a new proof of a result of Lazarev, that the dual of the circle $S^1_+$ in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of $DS^1$ and rule out a Koszul-dual reformulation of the Novikov conjecture.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
