Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted cartesian product.
As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields.
Some examples showing our implementation of these constructions in both \sagemath\ and \kenzo\ are shown, together with an approach to compute the homology of the universal cover when the group is abelian even in some cases where there is no effective homology, using the twisted homology of the space.
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.
In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.
In this paper, we study the generalized (co)homology Hopf algebras of the loop spaces on the infinite classical groups, generalizing the work due to Kono-Kozima and Clarke. We shall give a description of these Hopf algebras in terms of symmetric functions. Based on topological considerations in the first half of this paper, we then introduce a universal analogue of the factorial Schur $P$- and $Q$-functions due to Ivanov and Ikeda-Naruse. We investigate various properties of these functions such as the cancellation property, which we call the $\mathbb{L}$-supersymmetric property, the factoriza…
We adopt the viewpoint that topological Andé-Quillen theory for commutative $S$-algebras should provide usable (co)homology theories for doing calculations in the sense traditional within Algebraic Topology. Our main emphasis is on homotopical properties of universal derivations, especially their behaviour in multiplicative homology theories. There are algebraic derivation properties, but also deeper properties arising from the homotopical structure of the free algebra functor $\mathbb{P}_R$ and its relationship with extended powers of spectra. In the connective case in ordinary $\bmod{\,p}$ h…
In the present paper we study hyperbolic manifolds that are rational homology 3-spheres obtained by colouring of right--angled polytopes. We study the existence (or absence) of different kinds of symmetries of rational homology spheres that preserve the tessellation of the manifold into polytopes. We also describe how to create colourings with given symmetries.
The Nerve Theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an $\varepsilon$-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of…
In the paper the Čech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of Čech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces.
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.
In this article we further the study of non-commutative motives. Our main result is the construction of a simple model, given in terms of infinite matrices, for the suspension in the triangulated category of non-commutative motives. As a consequence, this simple model holds in all the classical invariants such as Hochschild homology, cyclic homology and its variants (periodic, negative, ...), algebraic K-theory, topological Hochschild homology, topological cyclic homology, ...
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…
In this work we investigate the parallel computation of homology using the Mayer-Vietoris principle. We present a two stage approach for parallelizing persistence. In the first stage, we produce a cover of the input cell complex by overlapping subspaces. In the second stage, we use this cover to build the Mayer-Vietoris blowup complex, a topological space, which organizes the various subspaces needed for employing the Mayer-Vietoris principle. Next, we compute the homology of each subspace in the blowup complex in parallel and then glue these results together in serial. We show how to use the …
We study oriented closed manifolds M^n possessing the following Universal Realisation of Cycles (URC) Property: For each topological space X and each integral homology class z of it, there exist a finite-sheeted covering \hM^n of M^n and a continuous mapping f of \hM^n to X such that f takes the fundamental class [\hM^n] to kz for a non-zero integer k. We find wide class of examples of such manifolds M^n among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we prove that for each 4-dimensional …
We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$-local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.
We provide a general method for finding all natural operations on the Hochschild complex of E-algebras, where E is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative or A_infty-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with th…
The "zero in the spectrum conjecture" asserted (in its strongest form) that for any manifold M zero should be in the l2-spectrum of the Laplacian (on forms) of the universal covering of M, i.e. that at least one (unreduced) L2-cohomology group of (the universal covering of) M is non-zero. Farber and Weinberger gave the first counterexamples to this conjecture. In this paper, using their fundamental idea to show the following stronger version of this result: Let G be a finitely presented group and suppose that the homology groups H_k(G,\ell^2(G)) are zero for k=0,1,2. For every dimension n\ge 6…
Given a non-compact Riemannian manifold M and a submanifold N of codimension q, we will construct under certain assumptions on both M and N a wrong way map in uniformly finite homology. Using an equivariant version of the construction and applying it to universal covers, we will construct wrong way maps in homology of groups. As applications we discuss obstructions against positive scalar curvature metrics and against inessentialness.
Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz \cite{F} and was further developed in the work of S. Ault \cite{Au1, Au2}. In this paper, we show that, for algebras defined over a field of characteristic $0$, the symmetric homology theory is naturally equivalent to the (one-dimensional) representation homology theory introduced by the authors (jointly with G. Khachatryan) in \cite{BKR}. Using known results on representa…
Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate the relationship between their integral homology. The Cheeger-Mueller Theorem implies that a certain product of orders of torsion homology and of regulators for M_1 agrees with that for M_2. We exhibit a connection between the torsion in the integral homology of M_1 and M_2 on th…
