Homology Classes Research Articles

On the Homological Classification of Semigroups with Local Units

In this paper, we study homological classification of semigroups with local units depending on properties around projectivity and flatness of acts over them, the results form a new progress in this research area. First, we show that Hom functors and tensor functors in the category of acts over semigroups are left and right exact, respectively. Then we characterize semigroups with local units over which all acts have some property with respect to (principal) weak flatness and torsion freeness of acts. Finally, we establish relationships between several different properties and Rees short exact sequence of acts.

An introduction to persistent homology for time series

AbstractTopological data analysis (TDA) uses information from topological structures in complex data for statistical analysis and learning. This paper discusses persistent homology, a part of computational (algorithmic) topology that converts data into simplicial complexes and elicits information about the persistence of homology classes in the data. It computes and outputs the birth and death of such topologies via a persistence diagram. Data inputs for persistent homology are usually represented as point clouds or as functions, while the outputs depend on the nature of the analysis and commonly consist of either a persistence diagram, or persistence landscapes. This paper gives an introductory level tutorial on computing these summaries for time series using R, followed by an overview on using these approaches for time series classification and clustering.This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Applications of Computational Statistics > Computational Mathematics

Selective Photoelectrocatalytic Removal for Group-Targets of Phthalic Esters

The development of a selective removal method for group-targets of pollutants under the inhibition of nontoxic organic interferents is of great importance in environmental science. A novel TiO2 photoelectrode functionalized with dummy templates (PS-PAE-TiO2) is designed, exhibiting group-targeting selectivity for nine phthalate ester (PAE) analogs. In total, 90-99% of PAEs were removed from 30 μg L-1 in actual wastewater (chemical oxygen demand, 14.5 mg O2 L-1). The selectivity for PAEs originated from preferential enrichment close to the PS-PAE-TiO2 surface result in a twofold improvement in the apparent kinetic constant. The specific sites can be attributed to phenyl rings and o-ester carbonyl groups through the molecular recognition process. The intermediates were analyzed quantitatively, and a degradation pathway with lower toxicity was proposed, excluding ring-hydroxylated phthalates. Almost 100% of the estrogenic activity and acute aquatic toxicity were eliminated and the genotoxicity was reduced by 92.5%, which was about 40% higher than that at the nonselective photoanode. An enhanced removal at the PS-PAE-TiO2 photoanode with better economic benefits was confirmed, saving energy consumption by 2.5 kWh m-3 per order than that at the nonselective anode. The advanced removal method with group-targeting selective capability can provide a propagable strategy for the removal of a class of homologues from complex aqueous systems.

Fukaya categories of surfaces, spherical objects and mapping class groups

Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows

We prove that, for a $C^\infty$-generic contact form $\lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d\lambda$. This is a quantitative refinement of the $C^\infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss a question about generic behavior of periodic Reeb orbits representing ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.

A note on the complexity of h–cobordisms

We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from X to X. As an application, we give results restricting the minimal number of double points of immersed spheres in manifolds homeomorphic to rational surfaces.

Morse index of multiplicity one min-max minimal hypersurfaces

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program ([31], [34], [36], [43]) proposed by the authors with the goal of developing a Morse theory for the area functional.

Topological character of three-dimensional nexus triple point degeneracies

Recently a generic class of three-dimensional band structures was identified that host two-fold line degeneracies meeting at three-fold or triple point degeneracies, which resist the usual topological characterization of isolated point degeneracies as in Dirac/Weyl semimetals. For these so-called ``Nexus" fermions which lie beyond Dirac/Weyl fermions, we lay out several concepts to characterize the wavefunction geometry and spell out its topology. Our approach is based on an understanding of the analyticity properties of Nexus wavefunctions building on a two-dimensional analogue studied recently by us. We use this to write down a homological classification of various Nexus triple point degeneracies in three dimensions.

On the 2-Systole of Stretched Enough Positive Scalar Curvature Metrics on S<sup>2</sup> times S<sup>2</sup>

We use recent developments by Gromov and Zhu to derive an upper bound for the 2-systole of the homology class of S 2 × { * } in a S 2 × S 2 with a positive scalar curvature metric such that the set of spheres homologous to S 2 × { * } is wide enough in some sense.

Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes

We study the homology of Riemannian manifolds of finite volume that are covered by an r-fold product ({mathbb {H}}^2)^r = {mathbb {H}}^2 times cdots times {mathbb {H}}^2 of hyperbolic planes. Using a variation of a method developed by Avramidi and Nguyen-Phan, we show that any such manifold M possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic r-dimensional submanifolds whose fundamental classes are linearly independent in the homology group H_r(M;{mathbb {Z}}).

Bridge trisections in ℂℙ2 and the Thom conjecture

In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $\mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques.

Isotopies of surfaces in 4–manifolds via banded unlink diagrams

In this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected $4$-manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in $\mathbb{C}P^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $\mathbb{C}P^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.

Exact bosonization in arbitrary dimensions

We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2d and 3d to arbitrary dimensions. This bosonization map gives a duality between any fermionic system in arbitrary $n$ spatial dimensions and a new class of $(n-1)$-form $\mathbb{Z}_2$ gauge theories in $n$ dimensions with a modified Gauss's law. This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold. A new formula for Stiefel-Whitney homology classes on lattices is derived. In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological "Steenrod square" term to the spacetime action.

Homology supported in Lagrangian submanifolds in mirror quintic threefolds

AbstractIn this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $\mathbb {Z}$ can be determined by these orbits with coefficients in $\mathbb {Z}_p$ .

Rotation Numbers and the Euler Class in Open Books

This paper introduces techniques for computing a variety of numerical invariants associated with a Legendrian knot in a contact manifold presented by an open book with a Morse structure. Such a Legendrian knot admits a front projection to the boundary of a regular neighborhood of the binding. From this front projection, we compute the rotation number for any null-homologous Legendrian knot as a count of oriented cusps and linking or intersection numbers; in the case that the manifold has nontrivial second homology, we can recover the rotation number with respect to a Seifert surface in any homology class. We also provide explicit formulas for computing the necessary intersection numbers from the front projection, and we compute the Euler class of the contact structure supported by the open book. Finally, we introduce a notion of Lagrangian projection and compute the classical invariants of a null-homologous Legendrian knot from its projection to a fixed page.

Periodic Points and Topological Restriction Homology

Abstract We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $n$-periodic points from a self-map $f$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.

Instantons and Hilbert functions

We study superpotentials from worldsheet instantons in heterotic Calabi-Yau compactifications for vector bundles constructed from line bundle sums, monads and extensions. Within a certain class of manifolds and for certain second homology classes, we derive simple necessary conditions for a non-vanishing instanton superpotential. These show that non-vanishing instanton superpotentials are rare and require a specific pattern for the bundle construction. For the class of monad and extension bundles with this pattern, we derive a sufficient criterion for non-vanishing instanton superpotentials based on an affine Hilbert function. This criterion shows that a non-zero instanton superpotential is common within this class. The criterion can be checked using commutative algebra methods only and depends on the topological data defining the Calabi-Yau X and the vector bundle V.

The spectrum of simplicial volume

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in mathbb {R}_{ge 0}. In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the l^1-semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.

Transverse link invariants from the deformations of Khovanov 𝔰𝔩3–homology

In this paper we will make use of the Mackaay-Vaz approach to the universal $\mathfrak{sl}_3$-homology to define a family of cycles (called $\beta_3$-invariants) which are transverse braid invariants. This family includes Wu's $\psi_{3}$-invariant. Furthermore, we analyse the vanishing of the homology classes of the $\beta_3$-invariants and relate it to the vanishing of Plamenevskaya's and Wu's invariants. Finally, we use the $\beta_3$-invariants to produce some Bennequin-type inequalities.

Sex in Symbiodiniaceae dinoflagellates: genomic evidence for independent loss of the canonical synaptonemal complex

Dinoflagellates of the Symbiodiniaceae family encompass diverse symbionts that are critical to corals and other species living in coral reefs. It is well known that sexual reproduction enhances adaptive evolution in changing environments. Although genes related to meiotic functions were reported in Symbiodiniaceae, cytological evidence of meiosis and fertilisation are however yet to be observed in these taxa. Using transcriptome and genome data from 21 Symbiodiniaceae isolates, we studied genes that encode proteins associated with distinct stages of meiosis and syngamy. We report the absence of genes that encode main components of the synaptonemal complex (SC), a protein structure that mediates homologous chromosomal pairing and class I crossovers. This result suggests an independent loss of canonical SCs in the alveolates, that also includes the SC-lacking ciliates. We hypothesise that this loss was due in part to permanently condensed chromosomes and repeat-rich sequences in Symbiodiniaceae (and other dinoflagellates) which favoured the SC-independent class II crossover pathway. Our results reveal novel insights into evolution of the meiotic molecular machinery in the ecologically important Symbiodiniaceae and in other eukaryotes.