Homology Theory Research Articles

(Co)homology of racks and multiple group racks for compact oriented surfaces in the 3-sphere

We introduce (co)homology theory for multiple group racks and construct cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles, where a multiple group rack is a rack consisting of a disjoint union of groups. We provide a method to find new rack cocycles and multiple group rack cocycles from existing rack cocycles and quandle cocycles. Evaluating cocycle invariants, we determine several symmetry types, caused by chirality and invertibility, of compact oriented surfaces in the 3-sphere.

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Mayer-Homology Learning Prediction of Protein-Ligand Binding Affinities

Artificial intelligence-assisted drug design is revolutionizing the pharmaceutical industry. Effective molecular features are crucial for accurate machine learning predictions, and advanced mathematics plays a key role in designing these features. Persistent homology theory, which equips topological invariants with persistence, provides valuable insights into molecular structures. The standard homology theory is based on a differential rule for the boundary operator that satisfies [Formula: see text] = 0. Our recent work has extended this rule by employing Mayer homology with generalized differentials that satisfy [Formula: see text] = 0 for [Formula: see text] 2, leading to the development of persistent Mayer homology (PMH) theory and richer topological information across various scales. In this study, we utilize PMH to create a novel multiscale topological vectorization for molecular representation, offering valuable tools for descriptive and predictive analyses in molecular data and machine learning prediction. Specifically, benchmark tests on established protein-ligand datasets, including PDBbind-v2007, PDBbind-v2013, and PDBbind-v2016, demonstrate the superior performance of our Mayer homology models in predicting protein-ligand binding affinities.

Equivariant oriented homology of the affine Grassmannian

We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the Goresky-Kottwitz-MacPherson (GKM) condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the formal affine Demazure algebra.

Braided Scalar Quantum Field Theory

AbstractWe formulate scalar field theories in a curved braided ‐algebra formalism and analyse their correlation functions using Batalin–Vilkovisky quantization. We perform detailed calculations in cubic braided scalar field theory up to two‐loop order and three‐point multiplicity. The divergent tadpole contributions are eliminated by a suitable choice of central curvature for the ‐structure, and we confirm the absence of UV/IR mixing. The calculations of higher loop and higher multiplicity correlators in homological perturbation theory are facilitated by the introduction of a novel diagrammatic calculus. We derive an algebraic version of the Schwinger–Dyson equations based on the homological perturbation lemma, and use them to prove the braided Wick theorem.

Hypernetwork modeling and topology of high-order interactions for complex systems

Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. Here, we integrate evolutionary game theory and behavioral ecology into a unified statistical mechanics framework, allowing all agents (modeled as nodes) and their bidirectional, signed, and weighted interactions at various orders (modeled as links or hyperlinks) to be coded into hypernetworks. Such hypernetworks can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurrence of active and passive HOI can drive complex systems to evolve at multiple time and space scales. We apply the model to reconstruct a hypernetwork of hexa-species microbial communities, and by dissecting the topological architecture of the hypernetwork using GLMY homology theory, we find distinct roles of pairwise interactions and HOI in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species.

Male genitalia, hierarchical homology, and the anatomy of the bullet ant (Paraponera clavata; Hymenoptera, Formicidae).

The male genitalia of insects are among the most variable, complex, and informative character systems for evolutionary analysis and taxonomic purposes. Because of these general properties, many generations of systematists have struggled to develop a theory of homology and alignment of parts. This struggle continues to the present day, where fundamentally different models and nomenclatures for the male genitalia of Hymenoptera, for example, are applied. Here, we take a multimodal approach to digitalize and comprehensively document the genital skeletomuscular anatomy of the bullet ant (Paraponera clavata; Hymenoptera: Formicidae), including hand dissection, synchrotron radiation microcomputed tomography, microphotography, scanning electron microscopy,confocal laser scanning microscopy,and 3D-printing. Through this work, we generate several new concepts for the structure and form of the male genitalia of Hymenoptera, such as for the endophallic sclerite (=fibula ducti), which we were able to evaluate in detail for the first time for any species. Based on this phenomic anatomical study and comparison with other Holometabola and Hexapoda, we reconsider the homologies of insect genitalia more broadly, and propose a series of clarifications in support of the penis-gonopod theory of male genital identity. Specifically, we use the male genitalia of Paraponera and insects more broadly as an empirical case for hierarchical homology by applying and refining the 5-category classification of serial homologs from DiFriscoet al.(2023) (DLW23) to all of our formalized concepts. Through this, we find that: (1) geometry is a critical attribute to account for in ontology, especially as all individually identifiable attributes are positionally indexed hence can be recognized as homomorphic; (2) the definition of "structure" proposed by DLW23 is difficult to apply, and likely heterogeneous; and (3) formative elements, or spatially defined foldings or in- or evaginations of the epidermis and cuticle, are an important yet overlooked class of homomorphs. We propose a morphogenetic model for male and female insect genitalia, and a model analogous to gene-tree species-tree mappings for the hierarchical homology of male genitalia specifically. For all of the structures evaluated in the present study, we provide 3D-printable models - with and without musculature, and in various states of digital dissection - to facilitate the development of a tactile understanding. Our treatment of the male genitalia of P. clavata serves as a basic template for future phenomic studies of male insect genitalia, which will be substantially improved with the development of automation and collections-based data processing pipelines, that is,collectomics. The Hymenoptera Anatomy Ontology will be a critical resource to include in this effort, and in best practice concepts should be linked.

Link splitting deformation of colored Khovanov–Rozansky homology

AbstractWe introduce a multiparameter deformation of the triply‐graded Khovanov–Rozansky homology of links colored by one‐column Young diagrams, generalizing the “‐ified” link homology of Gorsky–Hogancamp and work of Cautis–Lauda–Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2‐category of curved complexes of type A singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from our recent joint work and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply‐graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov–Rozansky homology and Hilbert schemes.

Antitumour Effects of Salvia Miltiorrhiza based on the Theory of Homology of Food and Medicine

Antitumour Effects of Salvia Miltiorrhiza based on the Theory of Homology of Food and Medicine

The homology digraph of a preordered space

This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology theory for the homology digraph. In particular, we prove a Künneth formula, which enables one to compute the homology digraph of a product of preordered spaces from the homology digraphs of the components.

Mini-Workshop: Homological aspects of TDLC-groups

This mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific topics include finiteness conditions for locally compact groups, Mackey functors and Bredon cohomology for topological groups, connections to condensed mathematics, connections to \ell^{2} -invariants and \Sigma -invariants.

The triviality of dihedral cohomology for operator algebras

This article delves into algebraic topology, specifically (co)homology theory, which is essential in various mathematical fields. It explores different types of (co)homology groups such as Hochschild, cyclic, reflexive, and dihedral, focusing on dihedral cohomology generated by the dihedral group operating on simplicial complexes. The text discusses when dihedral and reflexive cohomology vanishes and provides examples, along with proving a homomorphism between dihedral cohomology groups. Mathematical concepts like Banach spaces, C*-algebras, and tensor products are elucidated to understand the algebraic structures involved. Theorems and proofs establish the relationships and properties of these cohomology groups, culminating in a canonical isomorphism theorem for the product of two algebras. Overall, the article aims to provide a comprehensive exploration of dihedral cohomology and its interplay with other algebraic structures, offering insights into its applications and theoretical foundations. Additionally, it studies the (co)homology theory of C*-algebras, focusing on the triviality of cohomology groups of operator algebras and discovering the canonical isomorphism between any two unital K -algebras A and A′: HDn(A×A′)≅HDn(A)⊕HDn(A′).

The surjection property and computable type

We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the ϵ-surjection property, that were recently introduced to characterize the notion of computable type arising from computability theory. For a class of spaces including the finite simplicial complexes, we develop techniques to prove or disprove these properties using homotopy and homology theories, and give applications of these results. In particular, we answer an open question on the computable type property, showing that it is not preserved by taking products. We also observe that computable type is decidable for finite simplicial complexes.

Morse–Bott cohomology from homological perturbation theory

Morse–Bott cohomology from homological perturbation theory

On the cyclic homology theory of algebras

In this paper, the (Co)homology theory of pure algebra will be studied. We aim to study the homological theory of cyclic homology and its properties. The triviality property of cohomology groups of algebras will be obtained. We will study the Mayer-Vietortis of cyclic homology. The long exact sequence of cyclic homology and the relation between cyclic homology and dihedral homology will be proved. We will introduce the proof of isomorphism HCn(A×A′)≅HCn(A)⊕HCn(A′) and the definition of Morita invariance for cyclic homology.

Left Frobenius Pairs, Cotorsion Pairs and Weak Auslander-Buchweitz Contexts in Triangulated Categories

Let [Formula: see text] be a triangulated category with a proper class [Formula: see text] of triangles. We introduce the notions of left Frobenius pairs, left ([Formula: see text]-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to [Formula: see text] in [Formula: see text]. We show how to construct left cotorsion pais from left [Formula: see text]-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. Some applications are given in the Gorenstein homological theory of triangulated categories.

Localization for coarse homology theories

Localization for coarse homology theories

#1957 Microbial and metabolomic remodeling by Sichuan dark tea-based formula improves obesity-induced renal lipid metabolism disorder in mice

Abstract Background and Aims The theory of medicine-food homology is long-standing and has been widely applied in traditional Chinese medicine. The Sichuan dark tea-based medicated dietary formula (alternatively referred to as Qing, or clarity in Chinese) is previously used for its lipid-lowering properties in apoE mice. This study further investigated the renoprotective effects of Qing on diet-induced obesity (DIO) mice and explored the specific mechanisms involved. Method Male C57BL/6 mice were randomly assigned to three groups, a control group, a DIO group, and a Qing treatment group, or the Qing group, with 8 mice in each group. The mice in the control group were given normal maintenance feed and purified water, and the other two groups were fed a high-fat diet for 12 weeks to establish the DIO model. After that, high-fat diet continued in the DIO group, while the Qing group was given Qing at the same time for 12 weeks, during which period the weight of the mice was monitored and recorded every week. The mice were sacrificed after 12 weeks. Serum samples were collected and the levels of triglyceride (TG), total cholesterol (TC), alanine aminotransferase (ALT), aspartate aminotransferase (AST), and albumin were measured to evaluate liver function. In addition, renal lipids were extracted to determine the levels of TG and TC in the kidney and periodic acid-Schiff (PAS) and oil red O stainings were performed to evaluate kidney pathological injury. Western blot was performed to determine the phosphorylated AMPK (pAMPK)/AMPK ratio in the kidney tissue. RT-qPCR and Western blot were used to determine the expression of proteins related to fatty acid oxidation and key proteins related to lipid synthesis. 16SrRNA and metabolomics were applied to analyze the gut microbiota in the intestinal contents and its metabolites. Results Compared with those of the control group, the levels of liver mass (P=0.0003), serum ALT(P < 0.0001), AST (P=0.0001), TC (P=0.0191) and TG (P=0.0101) in the model group were significantly increased. Qing treatment can effectively reduce liver mass (P=0.0316), and improve AST (P=0.0012), ALT (P=0.0027), TC (P=0.0200), TG (P=0.0499) levels of mice compared to those of the DIO group. Qing treatment inhibited the expression of lipid synthesis related genes (SREBP-1, FASN and SCD1) and promoted the expression of fatty acid oxidation related genes (CPT1A, PGC1α and PPARγ) in kidneys (Fig. 1). Obesity-triggered dysfunction of structure and composition of gut microbiota were improved by Qing (Fig. 2), where the content of short-chain fatty acids, especially isovaleric acid and propionic acid, were also restored (Fig. 3). Conclusion Our data suggested that Sichuan dark tea-based medicated dietary formula may improve renal lipid metabolism by regulating gut microbiota and the levels of intestinal short-chain fatty acids, thereby protecting obesity-related kidney injury. Isovaleric acid and propionic acid may be the metabolites key to its regulation of gut microbiota.

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

Abstract We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math.521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

On the path homology of Cayley digraphs and covering digraphs

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a “bridge” between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.