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Register a new userTopological coHochschild Homology and the Homology of Free Loop Spaces
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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.
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In recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBokstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.
In this paper, we build up a scaled homology theory, $lc$-homology, for metric spaces such that every metric space can be visually regarded as locally contractible with this newly-built homology. We check that $lc$-homology satisfies all Eilenberg-Steenrod axioms except exactness axiom whereas its corresponding $lc$-cohomology satisfies all axioms for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first $lc$-homology group.
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth--Willerton magnitude homology of graphs, and detects geometric information such as convexity.
Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of interval posets. We give several applications. First, we show the vanishing of higher magnitude homology groups for convex subsets of the Euclidean space. Second, magnitude homology groups carry the information about the diameter of a hole. Third, we construct a finite graph whose $3$rd magnitude homology group has torsion.
We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows, topological Hochschild homology, and Morita invariance in bicategorical generality.
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