In this paper we examine certain filtrations of topological Hochschild homology and topological cyclic homology. As an example we show how the filtration with respect to a nilpotent ideal gives rise to an analog of a theorem of Goodwillie saying that rationally relative K-theory and relative cyclic homology agree. Our variation says that the p-torsion parts agree in a range of degrees. We use it to compute K_i(Z/p^m) for i < p-2.
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis, the usual Nerve Lemma does not provide isomorphism of persistent homology groups. Our argument involves some homological algebra: the key point being that although the maps produced in the standard proof of the Nerve Lemma do not commute as maps of chain complexes, the maps they induce on homology do.
The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a TQ-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here TQ-homology is short for topological Quillen homology. We also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.
Given a point cloud $P$ in Euclidean space and a positive parameter $t$ we can consider the $t$-neighborhood $P^{t}$ of $P$ consisting of points at distance less than $t$ to $P$. Homology of $P^{t}$ gives information about components, holes, voids etc. in $P^{t}$. The idea of persistent homology is that it may happen that we are interested in some of holes in the spaces $P^t$ that are not detected simultaneously in homology for a single value of $t$, but where each of these holes is detected for $t$ in a wide range. When the dimension of the ambient Euclidean space is small, persistent homolog…
We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower z_k X whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections, and that any nilpotent space for which the space of pointed self-maps map_*(X,X) is "canonically" discrete must be aspher…
The aim of this paper is to present a method for computation of persistent homology that performs well at large filtration values. To this end we introduce the concept of filtered covers. We show that the persistent homology of a bounded metric space obtained from the Čech complex is the persistent homology of the filtered nerve of the filtered Čech cover. Given a parameter $δ$ with $0 < δ\le 1$ we introduce the concept of a $δ$-filtered cover and show that its filtered nerve is interleaved with the Čech complex. Finally, we introduce a particular $δ$-filtered cover, the divisive cover. The sp…
We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity, which controls the impact of removing vertices on persistent homology. We achieve this control through the use of an interleaving type of distance between fitered simplicial complexes. We study the special case of Vietoris-Rips filtrations and show that our bounds offer a significant improvement over the immediate bounds coming from considerations related to the Gromov-Hausdorff distance. Based on these ideas we give an iterative method for the practical simplification of filtered sim…
Let H be a symplectic vector space, let V be a vector space, and consider the nilpotent Lie algebra L_H(V) = H \otimes V + S^2(V) with bracket [(h_1 \otimes v_1;a_1),(h_2 \otimes v_2;a_2)] = (0,<h_1,h_2> v_1 v_2) . In this paper, we calculate the Lie algebra homology of L_H(V) as a polynomial functor of V. This has applications to the analysis of the Leray-Serre spectral sequence of the fibrations of moduli spaces of pointed curves M_{1,n}->M_{1,1} and M_{g,n}->M_g.
This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explic…
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally, (co)sheaves have proven to be powerful instruments in tracking locally defined data across global systems, resulting in innovative applications to network science. In this paper, we combine the topological results of persistent homology and the quantitative data tracking capabilities of cosheaf theory to develop novel techniques in network data flow analysis.…
The classical Künneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove Künneth-type theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. In addition to comparing the two products, we present two applications in the setting of Vietoris-Rips complexes: one towards more efficient algorithms for product metric spaces with the maximum metric, and the other recov…
The aim of this paper is to show that homotopy pro-nilpotent structured ring spectra are TQ-local, where structured ring spectra are described as algebras over a spectral operad O. Here, TQ is short for topological Quillen homology, which is weakly equivalent to O-algebra stabilization. An O-algebra is called homotopy pro-nilpotent if it is equivalent to a limit of nilpotent O-algebras. Our result provides new positive evidence to a conjecture by Francis-Gaisgory on Koszul duality for general operads. As an application, we simultaneously extend the previously known 0-connected and nilpotent TQ…
To any finite simplicial complex X, we associate a natural filtration starting from Chari and Joswig's discrete Morse complex and abutting to the matching complex of X. This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph.
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the c…
Generalizing F-nilpotent completion for a ring spectrum F we first define the notion of completion with respect to a thick subcategory in a monogenic stable homotopy category. Specializing this to the thick subcategory generated by F-injectives gives an injective completion functor. This is the completion functor adapted to the modified Adams spectral sequence, which uses absolute instead of relative injective resolutions. Finally we show, that both constructions coincide for suitable ring spectra.
We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.
Multi-parameter persistent homology naturally arises in applications of persistent topology to data that come with extra information depending on additional parameters, like for example time series data. We introduce the concept of a Vietoris-Rips transformation, a method that reduces the computation of the one-parameter persistent homology of pathwise subcomplexes in multi-filtered flag complexes to the computation of the Vietoris-Rips persistent homology of certain semimetric spaces. The corresponding pathwise persistence barcodes track persistence features of the ambient multi-filtered comp…
We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit expression for the associated graded object of these homologies when X is the matching complex of the Tait graph of a plane graph $G$, in terms of subgraphs determined by certain matchings on the dual of G. We then use one of these homologies, in the case where X is a graph, to define a conjecturally optimal dissimilarity pseudometric for graphs; we prove various re…
This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here presented rely on a novel relationship between discrete Morse theory, matroid theory, and classical matrix factorizations. We provide background, detail the algorithms, and benchmark the software implementation in the Eirene package.
We identify the equivariant structure of the filtered pieces of the motivic filtration defined by Bhatt, Morrow and Scholze on the topological Hochschild cohomology spectrum of polynomial algebras over $\mathbb{F}_p$.
We show existence of a natural rational structure on periodic cyclic homology, conjectured by L. Katzarkov, M. Kontsevich, T. Pantev, for several classes of dg-categories, including proper connective $\mathbb{C}$-dg-algebras and dg-categories of local systems. The main ingredient is derived nilpotent invariance of A. Blanc's semi-topological K-theory, which we establish along the way.
Hesselholt defined a spectrum $\operatorname{TP}(X)$, the periodic topological cyclic homology of a scheme $X$, using topological Hochschild homology and the Tate construction, which is a topological analogue of Connes-Tsygan periodic cyclic homology $\operatorname{HP}$ defined by Hochschild homology and the Tate construction. Goodwillie proved that for $R$ an algebra over a field of characteristic 0 and $I$ a nilpotent ideal of $R$, the quotient map $R\to R/I$ induces an isomorphism on $\operatorname{HP}$. In this paper, we show that the analogous result for $\operatorname{TP}$ does not hold,…
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.
Motivated by the second author's construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed topological space $(Y,\bullet)$. These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra $C$. We show that the homology of the topological operad of based $Y$-cacti is the linear operad of based $H_*(Y)$-cacti. In addition, we show that for every coalgebra $C$ the operad of based $C$-cacti is Koszul.…
Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic ri…
We prove that a certain bilinear pairing (analagous to the Poincare-Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh-Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain ration…
