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We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant…
In this note we extend the cyclic homology functor, and in particular the periodic cyclic homology, to the category of DG (= differential graded) coalgebras. We are partly motivated by the question of products and coproducts in the cyclic…
We prove a new kind of stabilisation result, "secondary homological stability", for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the…
We explain how to interpret the complexes arising in the "classical" homology stability argument (e.g. in the framework of Randal-Williams--Wahl) in terms of higher algebra, which leads to a new proof of homological stability in this…
Local Hochschild, cyclic Homology and K-theory were introduced by N. Teleman in [10] with the purpose of unifying different settings of the index theorem. This paper is one of the research topics announced in [10], {\S}10. The definition of…
A classical result of Loday-Quillen and Tsygan states that the Lie algebra homology of the algebra of stable matrices over an associative algebra is isomorphic, as a Hopf algebra, to the exterior algebra of the cyclic homology of the…
Cuntz and Quillen have shown that for algebras over a field $k$ with $char(k)=0$, periodic cyclic homology may be regarded, in some sense, as the derived functor of (non-commutative) de Rham (co-)homology. The purpose of this paper is to…
The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a…
Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have…
Let A be a \C-algebra with an action of a finite group G, let $\natural$ be a 2-cocycle on $G$ and consider the twisted crossed product $A \rtimes \C [G,\natural]$. We determine the Hochschild homology of $A \rtimes \C [G,\natural]$ for two…
In this paper, we will provide constructions of D-module structures on the complex computing the periodic cyclic homology of a stable infinity-category defined over a scheme of characteristic zero. We give two methods. The first one is…
In this paper we introduce the notion of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence associated with Hecke algebras is representation stable.
(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory). We give two general constructions for the passage from unstable to stable homotopy…
Given a smooth one parameter deformation of associative topological algebras, we define Getzler's Gauss-Manin connection on both the periodic cyclic homology and cohomology of the corresponding smooth field of algebras and investigate some…
We propose a way to organise the subject of ``higher-order homological stability'', in the context of a graded $E_2$-algebra $\mathbf{R}$, along the same lines that the chromatic perspective organises stable homotopy theory. From this point…
A pair of graphs $(\Gamma,\Sigma)$ is said to be stable if the full automorphism group of $\Gamma\times\Sigma$ is isomorphic to the product of the full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where…
We introduce a set of combinatorial techniques for studying the simplicial bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We apply these methods to prove several new bounded acyclicity results for…
We study graded connected algebras over a field of characteristic zero and give an explicit formula for the cyclic homology of a tensor algebra. By means of a slightly new definition of David Anick's notion "strongly free" we are able to…
For any scheme which is algebraic over a subfield of the complex numbers we here construct an homological regulator from Suslin homology to period homology and a higher cycle class map from Bloch's higher Chow group to the period…
In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing…