Homology Theory Research Articles

Filtered simplicial homology, graph dissimilarity and überhomology

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 results for this dissimilarity and provide several computations. We further show that, by organising the horizontal homologies of a simplicial complex in the poset of its colourings, we obtain a triply graded homology theory which we call überhomology. This latter homology is not a homotopy invariant, but nonetheless encodes both combinatorial and topological information on X. For example, we prove that if X is a subdivision, the überhomology vanishes in its lowest degree, while for an homology manifold it coincides with the fundamental class in its top degree. We compute the überhomology on several classes of examples and infinite families and prove some of its properties; namely that, in its extremal degrees, it is well-behaved under coning and taking suspension. We then focus on the case where X is a simple graph and prove a detection result. Finally, we define some singly graded homologies for graphs obtained by specialising the überhomology in certain bi-degrees, provide some computations and use computer-aided calculations to make some conjectures.

Pre-crossed modules and rack homology

We define a homology theory for pre-crossed modules that specifies to rack homology in the case when the pre-crossed module is freely generated by a rack.

Path Topology in Molecular and Materials Sciences.

The structures of molecules and materials determine their functions. Understanding the structure and function relationship is the holy grail of molecular and materials sciences. However, the rational design of molecules and materials with desirable functions remains a grand challenge despite decades of efforts. A major obstacle is the lack of an intrinsic mathematical characteristic that attributes to a specific function. This work introduces persistent path topology (PPT) to effectively characterize directed networks extracted from functional units, such as constitutional isomers, cis-trans isomers, chiral molecules, Jahn-Teller isomerism, and high-entropy alloy catalysts. Path homology (PH) theory is utilized to decipher the role of mirror-symmetric sublattices that hinder the formation of periodic unit cells in amorphous solids. Topological perturbation analysis (TPA) is proposed to reveal the critical target in the blood coagulation system. The proposed topological tools can be directly applied to systems biology, omics sciences, topological materials, and machine learning study of molecular and materials sciences.

An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fibre bundle

We present an elementary approach to local combinatorial formulae for the Euler class of a fibre-oriented triangulated spherical fibre bundle. This approach is based on sections averaging technique and very basic knowledge of simplicial (co)homology theory. Our formulae are close relatives of those due to Mnëv. Bibliography: 9 titles.

A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science

Abstract Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely, persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the sugar, fish, flower, and gravel dataset produced for the study of mesoscale organization of clouds by Rasp et al. We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and we explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that readers of this paper will leave with a better understanding of TDA and persistent homology, will be able to identify problems and datasets of their own for which persistent homology could be helpful, and will gain an understanding of the results they obtain from applying the included GitHub example code. Significance Statement Information such as the geometric structure and texture of image data can greatly support the inference of the physical state of an observed Earth system, for example, in remote sensing to determine whether wildfires are active or to identify local climate zones. Persistent homology is a branch of topological data analysis that allows one to extract such information in an interpretable way—unlike black-box methods like deep neural networks. The purpose of this paper is to explain in an intuitive manner what persistent homology is and how researchers in environmental science can use it to create interpretable models. We demonstrate the approach to identify certain cloud patterns from satellite imagery and find that the resulting model is indeed interpretable.

E-infinity structure in hyperoctahedral homology

Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.

Synthetic spectra and the cellular motivic category

To an Adams-type homology theory we associate the notion of a synthetic spectrum; this is a product-preserving sheaf on the site of finite spectra with projective E-homology. We show that the $$\infty $$ -category of synthetic spectra based on E is in a precise sense a deformation of the $$\infty $$ -category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $$E_{*}$$ -based Adams spectral sequence. We describe a symmetric monoidal functor from the $$\infty $$ -category of cellular motivic spectra over $$\text {Spec}({\mathbb {C}})$$ into an even variant of synthetic spectra based on $$\textrm{MU}$$ and show that it induces an equivalence between the $$\infty $$ -categories of p-complete objects for all primes p. In particular, it follows that the p-complete cellular motivic category can be described purely in terms of chromatic homotopy theory.

A bivariant Yoneda lemma and (∞,2)–categories of correspondences

Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula. Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for bivariant functors. Moreover, conditioned on the existence of a 2-dimensional Grothendieck construction, I provide a proof of the aforementioned universal property. The methods, morally speaking, employ the `internal logic' of higher category theory: they make no explicit use of any particular model.

Rational torus-equivariant stable homotopy V: The torsion Adams spectral sequence

We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, At(G) of finite injective dimension, a homology theory π⁎At(⋅) taking values in At(G) based on the homology of the Borel construction, and a finite Adams spectral sequenceExtAt(G)⁎,⁎(π⁎At(X),π⁎At(Y))⇒[X,Y]⁎G for rational G-spectra X and Y.

Deformation quantization of the simplest Poisson orbifold

Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by Kontsevich Formality. We consider the simplest example of this situation: R2 with the reflection symmetry Z2. The usual quantization leads to the Weyl algebra. While Weyl algebra is rigid, the algebra of even or twisted by Z2 functions has one more deformation, which was identified by Wigner and is related to Feigin's glλ and to fuzzy sphere. With the help of homological perturbation theory we obtain explicit formula for the deformed product, the first order of which can be extracted from Shoikhet–Tsygan–Kontsevich formality.

A Hilbert reciprocity law on 3-manifolds

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.

Extensions, crossed modules and pseudo quadratic Lie type superalgebras

Extensions and crossed modules of Lie type superalgebras are introduced and studied. We construct homology and cohomology theories of Lie-type superalgebras. The notion of left super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.

Instantons and L-space surgeries

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the \operatorname{Spin}^c decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for r a positive rational number and K a nontrivial knot in the 3 -sphere, there exists an irreducible homomorphism \pi_1(S^3_r(K)) \to SU(2) unless r \geq 2g(K)-1 and K is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to SU(2) . In another application, we show that a slight enhancement of the A -polynomial detects infinitely many torus knots, including the trefoil.

Method for persistent topological features extraction of schizophrenia patients' electroencephalography signal based on persistent homology.

With the development of network science and graph theory, brain network research has unique advantages in explaining those mental diseases, the neural mechanism of which is unclear. Additionally, it can provide a new perspective in revealing the pathophysiological mechanism of brain diseases from the system level. The selection of threshold plays an important role in brain networks construction. There are no generally accepted criteria for determining the proper threshold. Therefore, based on the topological data analysis of persistent homology theory, this study developed a multi-scale brain network modeling analysis method, which enables us to quantify various persistent topological features at different scales in a coherent manner. In this method, the Vietoris–Rips filtering algorithm is used to extract dynamic persistent topological features by gradually increasing the threshold in the range of full-scale distances. Subsequently, the persistent topological features are visualized using barcodes and persistence diagrams. Finally, the stability of persistent topological features is analyzed by calculating the Bottleneck distances and Wasserstein distances between the persistence diagrams. Experimental results show that compared with the existing methods, this method can extract the topological features of brain networks more accurately and improves the accuracy of diagnostic and classification. This work not only lays a foundation for exploring the higher-order topology of brain functional networks in schizophrenia patients, but also enhances the modeling ability of complex brain systems to better understand, analyze, and predict their dynamic behaviors.

Hamiltonian perturbations in contact Floer homology

We study the contact Floer homology \(\mathrm{HF}_*(W, h)\) introduced by Merry–Uljarević in [21], which associates a Floer-type homology theory with a Liouville domain W and a contact Hamiltonian h on its boundary. The main results investigate the behavior of \(\mathrm{HF}_*(W, h)\) under the perturbations of the input contact Hamiltonian h. In particular, we provide sufficient conditions that guarantee \(\mathrm{HF}_*(W, h)\) to be invariant under the perturbations. This can be regarded as a contact geometry analog of the continuation and bifurcation maps along the Hamiltonian perturbations of Hamiltonian Floer homology in symplectic geometry. As an application, we give an algebraic proof of a rigidity result concerning the positive loops of contactomorphisms for a wide class of contact manifolds.

The effects and underlying mechanisms of medicine and food homologous flowers on the prevention and treatment of related diseases.

The theory of medicine and food homology has a long history in China. Numerous traditional Chinese medicinal could be used as both medicine and food. Many flower medicinal materials also belong to the homology of medicine and food, such as Chrysanthemum morifolium, Lonicera japonica, Crocus sativus, and Lonicera macranthoides. They mainly contain flavonoids, organic acids, terpenoids, and other active ingredients, which have a variety of medicinal values, including anti-inflammatory, anti-tumor, and antioxidant. There are many formulations and functional foods containing these plants in Chinese medicine, which have a variety of nutritional and health effects on the human body. In this review, 10 widely used flowers were selected to review their pharmacological activities, prevention and treatment of related diseases and underlying mechanisms, and discussed the current limitations and future development prospects, hoping to provide references for the research on the development and utilization of natural medical flowers. PRACTICAL APPLICATIONS: The "homology of medicine and food" flowers have a wide range of uses and are of great research value. In this paper, we introduce 10 "homology of medicine and food" flowers. Their active ingredients, pharmacological activities, and treatments for related diseases are reviewed, and the limitations and development prospects of the "homology of medicine and food" flowers are discussed. It is hoped that this will contribute to the development of the food and pharmacological fields.

H0 Tensions in Cosmology and Axion Pseudocycles in the Stringy Universe

The tension between early and late H0 is revised in the context of axion dark matter arising naturally from string theoretical integrations of antisymmetric tensor fields over non-trivial cycles. Certain early universe cycles may appear non-trivial from the perspective of a homology analysis focused on the early universe, while they may become trivial when analysed from the perspective of a homology theory reaching out to lower energies and later times. Such phenomena can introduce variations in the axion potential that would explain the observed H0 tension. The decay of such pseudo-axions when the pseudo-cycles dissipate trigger axion-two-photon (otherwise having an extremely long lifetime) and axion-gravitational processes mediated by Chern–Simons couplings with observable electromagnetic or gravitational wave signals originating in the early universe.

A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods

It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier's homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds, and experimented with 64 complete TRSs.

A Shadow Perspective on Equivariant Hochschild Homologies

Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.

Henning Krause: “Homological Theory of Representations”

Henning Krause: “Homological Theory of Representations”