Homology Theory Research Articles

Group-Twisted Alexander-Whitney and Eilenberg-Zilber Maps

We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted chain maps allow us to transfer information between resolutions for use in homology theories, for example, in those governing deformation theory. We show how to translate in particular from the default (but often cumbersome) reduced bar resolution to a more convenient twisted product resolution. This provides a more universal approach to some known results classifying PBW deformations.

Cohomological Hall algebra of Higgs sheaves on a curve

We define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the ($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve $X$, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal) cohomology ring $\mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $\mathbb{H}$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_{r,d}]$ of the zero-sections of the map $Higgs_{r,d} \to Coh_{r,d}$, for $r \geq 0, d \in Z$).

Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces

For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra Amathbf{Ha}_C^0 admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an Amathbf{Ha}_C^0-action. These triples can be interpreted as certain sheaves on mathbb {P}_C(omega _Coplus mathcal {O}_C). In particular, we obtain an action of Amathbf{Ha}_C^0 on the cohomology of Hilbert schemes of points on T^*C.

Symmetric Khovanov-Rozansky link homologies

On donne une catégorification de l’évaluation symétrique des toiles 𝔰𝔩 N en utilisant les mousses. On en déduit des théories homologiques d’entrelacs qui catégorifient les invariants quantiques d’entrelacs associés aux puissances symétriques de la représentation standard de 𝔰𝔩 N . Ces théories sont obtenues dans un cadre équivariant. On montre qu’il existe des suites spectrales de l’homologie triplement graduée de Khovanov-Rozansky vers ces homologies symétriques. On donne aussi une interpretation des bimodules de Soergel en terme de mousses.

The loop-stable homotopy category of algebras

Let ℓ be a commutative ring with unit. Garkusha constructed a functor from the category of ℓ-algebras into a triangulated category D, that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D, as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote it by K, whose objects are pairs (A,m) with A an ℓ-algebra and m an integer, and whose Hom-sets can be described in terms of homotopy classes of morphisms. All these computations, however, require a heavy machinery of homotopy theory. In this paper, we give a more explicit construction of the triangulated category K and prove its universal property, avoiding the homotopy-theoretic methods and using instead the ones developed by Cortiñas-Thom for defining kk-theory. Moreover, we give a new description of the composition law in K, mimicking the one in the suspension-stable homotopy category of bornological algebras defined by Cuntz-Meyer-Rosenberg. We also prove that the triangulated structure in K can be defined using either extension or mapping path triangles.

Sutured manifolds and polynomial invariants from higher rank bundles

For each integer $N\geq 2$, Mari\~no and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank $N$. In this paper, Mari\~no and Moore's predictions are confirmed for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that $N=3$. The primary motivation is to study 3-manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants are exploited to define a Floer homology theory for sutured 3-manifolds.

On Steenrod 𝕃-homology, generalized manifolds, and surgery

AbstractThe aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on ageneralized n-manifoldXn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of thenth Steenrod homology group$H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of thegeometricsplitting procedure, the use of which is standard in surgery ontopologicalmanifolds.

Surgery, polygons and SU(N)‐Floer homology

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $SU(N)$-instanton Floer homology with respect to Dehn surgery is studied. In particular, it is shown that there are surgery exact tetragons and pentagons, respectively, for $SU(3)$- and $SU(4)$-instanton Floer homologies. It is also conjectured that $SU(N)$-instanton Floer homology in general admits a surgery exact $(N+1)$-gon. An essential step in the proof is the construction of a family of asymptotically cylindrical metrics on ALE spaces of type $A_n$. This family is parametrized by the $(n-2)$-dimensional associahedron and consists of anti-self-dual metrics with positive scalar curvature. The metrics in the family also admit a torus symmetry.

Equivariant factorization homology of global quotient orbifolds

We introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of ⊗-excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group actions.

Living With Stigma: Spatial and Social Divisions in a Danish City

AbstractInspired by Bourdieu's theory of homology between social, mental and spatial structures, this essay dissects the relationship between spatial and social divisions in the Danish city of Aalborg using varied data from official statistics, surveys, qualitative interviews and field observations. Spatial divisions reflect differences in the objective distribution of economic and cultural capital, and are accompanied by symbolic divisions in residents’ minds. Some nuances are, however, added to Bourdieu's homology argument, as there are discrepancies between objective distributions and mental schemes, such that the latter exaggerate and dramatize the former. The neighbourhood of Aalborg East is subjected to this symbolic exaggeration in the form of spatial stigma or territorial stigmatization. An analysis of residents’ strategies for coping with spatial stigma furthermore provides an illustration of Wacquant's claim that this stigmatization can be met with a range of socially patterned responses ranging from acceptance or indifference to recalcitrance.

ShadowPlay2.5D

An immersive experience brought about by virtual reality can potentially enhance the appreciation of classical Chinese poetry, which is difficult to describe clearly in everyday language or ordinary media. However, making 3-dimensional illustrations for a 360-degree display in virtual reality is usually a labor-intensive and time-consuming procedure and hard to master for non-professional media creators, such as teachers. Motivated by the homology theory of classical Chinese poetry and painting, we propose an image-based approach of building 2.5-dimensional immersive stories to visualize classical Chinese poetry. Specifically, using Chinese shadow play as a metaphor, we have designed and implemented ShadowPlay2.5D , a sketch-based authoring tool to help novices create 360-degree videos of classical Chinese poetry easily. To ensure coverage of the diverse themes in Chinese poetry and preserve the sense of culture, we build a Chinese ink-painting style image repository of essential poetic elements identified via crowdsourcing. To facilitate construction of 2.5-dimensional scenes, we design features that support puppet-like animation, instancing, and camera organization in a 3-dimensional environment. Through two user studies, we show that ShadowPlay2.5D can help novices make a short 360-degree video in about 10--15 minutes, and the 2.5D stylized illustrations created can bring about a better immersive experience for poetry appreciation.

Representation Homology of Topological Spaces

Abstract In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, ${{\mathbb {S}}}^1$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in ${\mathbb {R}}^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in ${\mathbb {R}}^3$.

From positive geometries to a coaction on hypergeometric functions

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally- regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter ϵ. We show that the coaction defined on this class of integral is consistent, upon expansion in ϵ, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric p+1Fp and Appell functions.

Some aspects of cosheaves on diffeological spaces

We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasi-cosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-v{C}ech homology with values in pre-cosheaves is established for diffeological spaces.

The Cech homology theory in the category of soft topological spaces

The Cech homology theory in the category of soft topological spaces

Excision theory in the dihedral and reflexive (co)homology of algebras

In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dih...

Simple sheaves for knot conormals

We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.

Shadow biquandles and local biquandles

Given a shadow biquandle (B,X) composed of a biquandle B and a strongly connected B-set X, we have a local biquandle structure on X. The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented surface-links, using such shadow biquandles coincide with those using the corresponding local biquandles. These results imply that for some cases, the Niebrzydowski's theory in [12–14] for knot-theoretic ternary quasigroups is the same as shadow biquandle theory. We also show that some local biquandle 2- or 3-cocycles and some 1- or 2-cocycles of the Niebrzydowski's (co)homology theory can be induced from Mochizuki's (bi)quandle cocycles.

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]).