Homology Theory Research Articles

Splittings and calculational techniques for higher THH

Tensoring finite pointed simplicial sets with commutative ring spectra yields important homology theories such as (higher) topological Hochschild homology and torus homology. We prove several structural properties of these constructions relating $X \otimes (-)$ to $\Sigma X \otimes (-)$ and we establish splitting results. This allows us, among other important examples, to determine $THH^{[n]}_*(\mathbb{Z}/p^m; \mathbb{Z}/p)$ for all $n \geq 1$ and for all $m \geq 2$.

On the axiomatic systems of singular cohomology theory

On the category of pairs of topological spaces having a homotopy type of CW complexes the singular (co)homology theory was axiomatically studied by J. Milnor [8]. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomophic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support [3]. In the paper [6], L. Mdzinarishvili proposed partially compact support property for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory. In this paper, we will give additional different axiomatic characterizations of the singular cohomology theory. Moreover, we will study connections of the mentioned axiomatic systems (cf. [2]).

Homotopy algebras in higher spin theory

Motivated by string field theory, we explore various algebraic aspects of higher spin theory and Vasiliev equation in terms of homotopy algebras. We present a systematic study of unfolded formulation developed for the higher spin equation in terms of the Maurer-Cartan equation associated to differential forms valued in L-infinity algebras. The elimination of auxiliary variables of Vasiliev equation is analyzed through homological perturbation theory. This leads to a closed combinatorial graph formula for all the vertices of higher spin equations in the unfolded formulation. We also discover a topological quantum mechanics model whose correlation functions give deformed higher spin vertices at first order.

HOMOLOGY THEORIES FOR COMPLEXES BASED ON FLATS

AbstractIn this paper, we introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Gorenstein flat precovers. We compare the Gorenstein relative homology to the Tate/unbounded homology and get some results that improve the known ones.

Pair component categories for directed spaces

The notion of a homotopy flow on a directed space was introduced in Raussen (Appl Categ Struct 15(4):355–386, 2007) as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all directed maps along such a 1-parameter deformation preserve the homotopy types of path spaces, such a flow and the parameter maps are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points to which a functor associates the space of directed paths between them. The monoid of all inessential maps acts on such a category by endofunctors leaving the associated path spaces invariant up to homotopy. We construct a pair component category as quotient category: it has as objects pair components along which the homotopy type is invariant—for a coherent and transparent reason. This paper follows up Fajstrup et al. (J Homotopy Relat Struct 12(1):81–108, 2004); Goubault and Haucourt (Appl Categ Struct 15(4):387–414, 2007); Raussen (Appl Categ Struct 15(4):355–386, 2007) and removes some of the restrictions for their applicability. At least in several examples, it gives reasonable results for spaces with non-trivial directed loops. If one uses homology equivalence instead of homotopy equivalence as the basic relation, it yields an alternative to computable versions of “natural homology” introduced in Dubut et al (in: Automata, languages, and programming. 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015. Proceedings. Part II, 171–183, Springer, Berlin 2015) and elaborated in Dubut (Directed homotopy and homology theories for geometric models of true concurrency. Ecole normale superieure Paris-Saclay 2017). It refines, for good and for evil, the stable components introduced and investigated in Ziemianski (Appl Categ Struct 27:217–244, 2019).

Equivariant Khovanov Homology of Periodic Links

The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov homology of torus links $T(n,2)$.

Generalizations of Intersection Homology and Perverse Sheaves with Duality over the Integers

We provide a generalization of the Deligne sheaf construction of intersection homology theory and a corresponding generalization of Poincaré duality on pseudomanifolds such that the Goresky–MacPherson, Goresky–Siegel, and Cappell–Shaneson duality theorems all arise as particular cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky–Siegel IP spaces on which upper-middle perversity intersection homology is self-dual. We also examine torsion-sensitive t-structures and categories of perverse sheaves that contain our torsion-sensitive Deligne sheaves as intermediate extensions.

Fredholm Property of Operators from 2D String Field Theory

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ-instanton equation. Solutions of ζ-instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining the Fredholm property and constructing relevant homology theory. In this article, we study the Fredholm property of a sort of differential operators with degenerate asymptotics. As an application, we verify certain Fredholm property of the linearized operator of ζ-instanton equations.

Split Injectivity ofA-Theoretic Assembly Maps

AbstractWe construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly maps. On the way, we prove that the fundamental structural theorems for Waldhausen’s algebraic $K$-theory functor carry over to its nonconnective counterpart defined by Blumberg–Gepner–Tabuada.

Coarse homology theories and finite decomposition complexity

Using the language of coarse homology theories, we provide an axiomatic account of vanishing results for the fibres of forget-control maps associated to spaces with equivariant finite decomposition complexity.

Hodge theory for intersection space cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincar\'e duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.

Tilting classes over commutative rings

Abstract We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri, Pospíšil, ŠÅ¥ovíček and Trlifaj (2014). We show that the n-tilting classes can equivalently be expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: Tor * ⁡ ( R / I , - ) {\operatorname{Tor}_{*}(R/I,-)} , Koszul homology, Čech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n-cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of Šťovíček, Trlifaj and Herbera (2014). Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n-cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

The Godbillon–Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional manifold are satisfied. The magnetic Godbillon–Vey helicity invariant in magnetohydrodynamics (MHD) is a higher-order helicity invariant that occurs for flows in which the magnetic helicity density$h_{m}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}=\boldsymbol{A}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{A})=0$, where$\boldsymbol{A}$is the magnetic vector potential and$\boldsymbol{B}$is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon–Vey field$\unicode[STIX]{x1D6C8}=\boldsymbol{A}\times \boldsymbol{B}/|\boldsymbol{A}|^{2}$and the Godbillon–Vey helicity density$h_{\text{gv}}=\unicode[STIX]{x1D6C8}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6C8})$in general MHD flows in which either$h_{m}=0$or$h_{m}\neq 0$. A conservation law for$h_{\text{gv}}$occurs in flows for which$h_{m}=0$. For$h_{m}\neq 0$the evolution equation for$h_{\text{gv}}$contains a source term in which$h_{m}$is coupled to$h_{\text{gv}}$via the shear tensor of the background flow. The transport equation for$h_{\text{gv}}$also depends on the electric field potential$\unicode[STIX]{x1D713}$, which is related to the gauge for$\boldsymbol{A}$, which takes its simplest form for the advected$\boldsymbol{A}$gauge in which$\unicode[STIX]{x1D713}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$where$\boldsymbol{u}$is the fluid velocity. An application of the Godbillon–Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon–Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection, are discussed.

Topology Based Object Tracking

A model for tracking objects whose topological properties change over time is proposed. Such changes include the splitting of an object into multiple objects or the merging of multiple objects into a single object. The proposed model employs a novel formulation of the tracking problem in terms of homology theory whereby 0-dimensional homology classes, which correspond to connected components, are tracked. A generalisation of this model for tracking spatially close objects lying in an ambient metric space is also proposed. This generalisation is particularly suitable for tracking spatial-temporal phenomena such as rain clouds. The utility of the proposed model is demonstrated with respect to tracking communities in a social network and tracking rain clouds in radar imagery.

Homology of path complexes and hypergraphs

The path complex and its homology were defined in previous papers of the authors. The notion of a path complex is a natural discrete generalization of the notion of a simplicial complex. The theory of path complexes contains homotopy invariant homology theory of digraphs and (nondirected) graphs.In the paper we study the homology theory of path complexes. In particular, we describe functorial properties of paths complexes, introduce the notion of homotopy for path complexes and prove the homotopy invariance of path homology groups. We prove also several theorems that are similar to the results of classical homology theory of simplicial complexes. Then we apply this approach to construct homology theories on various categories of hypergraphs. We describe basic properties of these homology theories and relations between them. As a particular case, these results give new homology theories on the category of simplicial complexes.

Application of Homotopy to the Ageing Process of Human Body Within the Framework of Algebraic Topology

This paper presents the ageing process of the human body from the homotopy concept in the algebraic topology. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. The complex connected cylindrical shape of the body x=S^1*I, described by the functions f,h: x→x is a Cartesian product of the vertical interval I=[0,β] and a circle S^1. The initial state of the body x=S^1*I is the topological shape of the infant. The ageing process called the homotopy is the family of continuous functions f_t(x) on the interval I=[0,α] where α>β and t≥β. It is an increasing sequence of the function f_t(x) of the body x=S^1*I. The homotopy relates the topological shape of the infant to the topological shape of the adult. The homology theory assigns to the human body (x) a sequence of abelian groups H_n(x) for n=0,1,2…and to the continuous function a sequence of homomorphism. The homology group characterizes the number and continuity of the compact surface of human body. The study excludes genetic, hormonal, environmental and other factors in the ageing process and recognizes the fact that ageing continues throughout life, through childhood, and adolescence into adulthood. Topologically the infant is equal to the adult since the infant continuously grows into the adult. The study found an algebraic way of relating homotopy to the process of ageing of human body. It was also established that this could offer other topologists useful topological tips in the application of homotopy to other physical continuous processes.
 
  
 
  

Connectivity Based k-Coverage Hole Detection in Wireless Sensor Networks

A connectivity based k-coverage hole detection algorithm is proposed in this paper. We adopt Rips complex in homology theory to model a wireless sensor network (WSN). We firstly present a simplicial complex reduction algorithm to simplify the network topology by vertex and edge deletion, while keeping the homology intact. Then a connectivity-based algorithm is proposed for discovering boundary cycles of non-triangular k-coverage holes. The algorithm consists of two parts, one part is 1-coverage hole detection and the other part is coverage degree reduction. In the 1-coverage hole detection part, boundary cycles of 1-coverage holes are found. In the coverage degree reduction part, an independent covering subset of nodes for the covered region is found and these nodes are set to dormant state to decrease the coverage degree of the target region by one. The k-coverage hole detection algorithm is an iterative process of the two parts. Computation complexity of the algorithm is analyzed and simulation results show that more than 95% of non-triangular k-coverage holes can be accurately detected by our algorithm.

Homology and cohomology of functional spaces

Let {Xα} be an inverse system of compact spaces Xα and Y be an ANR. Consider a direct system {F(Xα,Y)} of topological spaces F(Xα,Y), where F(Xα,Y) is the space of all continuous maps f:Xα→Y, given the compact-open topology. In [5], S. Mardešić proved that for the singular homology there is an isomorphism(1)lim⟶H⁎s(F(Xα,Y))⟶∼H⁎s(F(X,Y)), where X=lim⟵Xα.In the present paper we prove that for the singular cohomology there is a finite exact sequence(2)0⟶lim⟵Hs1(2n−3)Fα⟶⋯⟶lim⟵Hsn−1(1)Fα⟶HsnF⟶⋯⟶lim⟵HsnFα⟶lim⟵Hsn−1(2)Fα⟶⋯⟶lim⟵Hs1(2n−2)Fα⟶0, where HsqFα=Hsq(F(Xα,Y),G), HsqF=Hsq(F(X,Y),G), X=lim⟵Xα, G is an abelian group.Let X be a compact space and S={Sm,σm} be the spherical spectrum. Consider the functional spectrum F(X,S)={Fm(X)}, where Fm(X)=F(X,Sm) is the space of all continuous maps f:X→Sm, given the compact-open topology. The functional spectrum F(X,S) induces the direct system {Cm−⁎(Fm(X))} of integral singular chains. The direct limit C⁎(X)=lim⟶mCm−⁎(Fm(X)) is a free cochain complex, where Cq(X)=lim⟶mCm−q(Fm(X)), q≥0. The Milnor homology H‾⁎(X,G) of the compact space X over the coefficients group G is defined as the homology of the chain complex C⁎(X)=Hom(C⁎(X),G)[8].In the present work for the compact space X and Milnor homology H‾⁎ we prove that there is an exact sequence(3)0⟶lim⟵Hsm−q−1(1)(Fm(X),G)⟶H‾q(X,G)⟶lim⟵Hsm−q(Fm(X),G)⟶0 and that the Milnor homology on the category Ac of compact pairs is a homology theory in the Berikashvili sense.

C∗–algebraic drawings of dendroidal sets

In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article we introduce the concept of a $C^*$-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on $C^*$-algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable $\infty$-categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant K-theory. Finally, a method to analyse graph algebras in terms of trees is sketched.

Intersection space cohomology of three-strata pseudomanifolds

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.