Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA spectral sequence for the homology of a finite algebraic delooping
193
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvilis Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kahler differentials.
rate research
Read More
Operadic tangent cohomology generalizes the existing theories of Harrison cohomology, Chevalley--Eilenberg cohomology and Hochschild cohomology. These are usually non-trivial to compute. We complement the existing computational techniques by producing a spectral sequence that converges to the operadic tangent cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams--Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas--Sullivan loop product. Finally, we consider relative Sullivan--de Rham models of a fibration $p$, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber homotopy equivalences of $p$.
In this note, we use Curtiss algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $mathbb{R}P^infty$ with the aid of a computer, which gives us its Adams $E_2$-page in the range of $t<72$. We also compute the transfer map on the Adams $E_2$-pages. These data are used in our computations of the stable homotopy groups of $mathbb{R}P^infty$ in [6] and of the stable homotopy groups of spheres in [7].
An exact computation of the persistent Betti numbers of a submanifold $X$ of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of $X$ is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of $X$ from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.
In this article we develop the cotangent complex and (co)homology theories for spectral categories. Along the way, we reproduce standard model structures on spectral categories. As applications, we show that the invariants to descend to stable $infty$-categories and we prove a stabilization result for spectral categories.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
