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In his 1957 paper, John Milnor introduced link invariants which measure the homotopy class of the longitudes of a link relative to the lower central series of the link group. Consequently, these invariants determine the lower central series quotients of the link group. This work has driven decades of research with profound influence. One of Milnor's original problems remained unsolved: to extract similar invariants from the transfinite lower central series of the link group. We reformulate and extend Milnor's invariants in the broader setting of 3-manifolds, with his original invariants as special cases. We present a solution to Milnor's problem for general 3-manifold groups, developing a theory of transfinite invariants and realizing nontrivial values. [ABSTRACT FROM AUTHOR]

In this paper, we explore the concept of topologically dense injectivity of monoid acts. It is shown that topologically dense injective acts constitute a class strictly larger than the class of ordinary injective ones. We determine a number of acts satisfying topologically dense injectivity. Specifically, any strongly divisible as well as strongly torsion free S -act over a monoid S is topologically dense injective if and only if S is a left reversible monoid. Furthermore, we establish a counterpart of the Skornjakov criterion and also identify a class of acts satisfying the Baer criterion for topologically dense injectivity. Lastly, some homological classifications for monoids by means of this type of injectivity of monoid acts are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants θ(c)$\theta ^{(c)}$ defined by the first author satisfy θ(c)(Ta,b)=(a−1)(b−1)/2$\theta ^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever c$c$ is a prime not dividing ab$ab$. Since θ(c)$\theta ^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere Y=Σ(a1,⋯,ar)$Y = \Sigma (a_1, \dots, a_r)$ does not extend smoothly to any homology 4‐ball bounding Y$Y$. In the case of a non‐free cyclic group action of prime order, we prove that if the rank of HFred+(Y)$HF_{red}^+(Y)$ is greater than p$p$ times the rank of HFred+(Y/Zp)$HF_{red}^+(Y/\mathbb {Z}_p)$, then the Zp$\mathbb {Z}_p$‐action on Y$Y$ does not extend smoothly to any homology 4‐ball bounding Y$Y$. Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres. [ABSTRACT FROM AUTHOR]

It is known that homology and inverse limit functors do not commute. In the paper, we consider this very problem and find its application for various homology theories. In particular, on the category of general topological spaces, there are some exact homology functors induced by different non-free cochain complexes. The relation between them and other classical homology theories is given. In addition, for the defined homology functors, the tautness and the continuous properties are obtained. [ABSTRACT FROM AUTHOR]

The data-informativity approach in data-driven control focuses on data and their matching model sets for system design and analysis. The approach offers a new mathematical formulation different from model-based control and is expected to progress. In model-based control, the introduction of equivalent transformations has made system analysis and design easier and facilitated theoretical development. In this study, we focus on data transformations and their transformation of matching model sets. We first introduce an algebraic sequence representing the relationship between the data and model set, and using this algebraic approach, we utilize propositions from homology theory, such as kernel universality, to analyze data and model transformations. This technique is significant not only mathematically but also in engineering. Further, we demonstrate how this technique can be applied to derive controllability judgments for data informativity-based analysis. Finally, we prove that design problems can be reduced to analysis problems involving controller inclusion. [ABSTRACT FROM AUTHOR]

As data used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of data analysis. This can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely predominantly on the manifold hypothesis which asserts that the underlying space is a smooth manifold. This assumption fails for many physical models where the underlying space contains singularities. In this paper, we develop a machine learning pipeline that captures finegrain geometric information without having to rely on any smoothness assumptions. Our approach involves working within the scope of algebraic geometry and algebraic varieties instead of differential geometry and smooth manifolds. In the setting of the variety hypothesis, the learning problem becomes to find the underlying variety using sample data. We cast this learning problem into a Maximum A Posteriori optimization problem which we solve in terms of an eigenvalue computation. Having found the underlying variety, we explore the use of Gröbner bases and numerical methods to reveal information about its geometry. In particular, we propose a heuristic for numerically detecting points lying near the singular locus of the underlying variety. [ABSTRACT FROM AUTHOR]

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality. [ABSTRACT FROM AUTHOR]

A previous result about the decategorified bordered (sutured) Heegaard Floer invariants of surfaces glued together along intervals, generalizing the decategorified content of Rouquier and the author's higher-tensor-product-based gluing theorem in cornered Heegaard Floer homology, was proved only over |${\mathbb{F}}_2$| and without gradings. In this paper we add signs and prove a graded version of the interval gluing theorem over |${\mathbb{Z}}$|⁠ , enabling a more detailed comparison of these aspects of decategorified Heegaard Floer theory with modern work on non-semisimple 3d TQFTs in mathematics and physics. [ABSTRACT FROM AUTHOR]

Given a hermitian line bundle L→M$L\rightarrow M$ on a closed Riemannian manifold (Mn,g)$(M^n,g)$, the self‐dual Yang–Mills–Higgs energies are a natural family of functionals Eε(u,∇):=∫M|∇u|2+ε2|F∇|2+(1−|u|2)24ε2$$\protect\begin{align*} E_{\epsilon} (u,\nabla):=\int _M{\left(|\nabla u|^2+\epsilon ^2|F_{\nabla} |^2+\frac{{(1-|u|^2)}^2}{4\epsilon ^2}\right)} \protect\end{align*}$$defined for couples (u,∇)$(u,\nabla)$ consisting of a section u∈Γ(L)$u\in \Gamma (L)$ and a hermitian connection ∇ with curvature F∇$F_\nabla$. While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as ε→0$\epsilon \rightarrow 0$ (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of Eε$E_\epsilon$ to (2π times) the codimension two area: more precisely, given a family of couples (uε,∇ε)$(u_\epsilon ,\nabla _\epsilon)$ with supεEε(uε,∇ε)<∞$\sup _\epsilon E_\epsilon (u_\epsilon ,\nabla _\epsilon)<\infty$, we prove that a suitable gauge invariant Jacobian J(uε,∇ε)$J(u_\epsilon ,\nabla _\epsilon)$ converges to an integral (n−2)$(n-2)$‐cycle Γ, in the homology class dual to the Euler class c1(L)$c_1(L)$, with mass 2πM(Γ)≤lim infε→0Eε(uε,∇ε)$2 \pi \mathbb {M}(\Gamma)\le \liminf _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon)$. We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the (n−2)$(n-2)$‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of Eε$E_{\epsilon }$. [ABSTRACT FROM AUTHOR]

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the 1-dimensional homology classes with Z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}_2$$\end{document} coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. (LATIN 2018: Theoretical Informatics, Springer International Publishing, Cham, 2018), runs in O(Nmω-1+nmg)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(N m^{\omega -1} + n m g)$$\end{document} time, where N denotes the total number of simplices in K, m denotes the number of edges in K, n denotes the number of vertices in K, g denotes the rank of the 1-homology group of K, and ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega $$\end{document} denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in O~(mω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{O}(m^\omega )$$\end{document} time, the second algorithm runs in O(Nmω-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(N m^{\omega -1})$$\end{document} time and the third algorithm runs in O~(N2g+Nmg2+mg3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{O}(N^2\,g + N m g{^2} + m g{^3})$$\end{document} time which is nearly quadratic time when g=O(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g=O(1)$$\end{document}. We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in O(mω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(m^\omega )$$\end{document} time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in O~(mω)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{O}(m^\omega )$$\end{document} time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop). [ABSTRACT FROM AUTHOR]

In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the "length" of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs. [ABSTRACT FROM AUTHOR]

A well-known conjecture of Rasmussen states that for any knot K in S^{3}, the rank of the reduced Khovanov homology of K is greater than or equal to the rank of the reduced knot Floer homology of K. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture. [ABSTRACT FROM AUTHOR]

In this brief note, we investigate the \mathbb {CP}^2-genus of knots, i.e., the least genus of a smooth, compact, orientable surface in \mathbb {CP}^2\smallsetminus \mathring {B^4} bounded by a knot in S^3. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the \mathbb {CP}^2-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in \mathbb {CP}^2\# \mathbb {CP} ^2. [ABSTRACT FROM AUTHOR]

Secondary homological stability is a recently discovered stability pattern for a sequence of spaces exhibiting homological stability and it holds outside the range where the homology stabilizes. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is closed because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider coalgebra measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology. We also consider comodule measurings between stable anti-Yetter Drinfeld (SAYD) modules over Hopf algebroids. These give an enrichment of the global category of SAYD modules over comodules. These measurings also induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients, which are compatible with Hopf-Galois maps. Finally, we consider non-symmetric operads with multiplication and modules over them which have both cyclic and Gerstenhaber type structures, known as cyclic unital comp modules. We obtain an enrichment of cyclic unital comp modules over comodules, as well as morphisms on cyclic homology induced by comodule measurings of comp modules over operads with multiplication. [ABSTRACT FROM AUTHOR]

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy ψ 1 of a symplectic manifold (M , ω) fixing a relatively exact Lagrangian L setwise must act trivially on R ∗ (L) , where R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over Z / 2 and over Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that ψ 1 | L acts trivially on R ∗ (L L) , where L L is the free loop space of L. From this we deduce that when L is a surface or a K (π , 1) , ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over Z / 2 . [ABSTRACT FROM AUTHOR]

Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and the main interest is in identifying the most persisting ones since they correspond to the Betti number values. Given the randomness inherent in the sampling process, and the complex structure of the space where persistence diagrams take values, estimation of Betti numbers is not straightforward. The approach followed in this work makes use of features' lifetimes and provides a full Bayesian clustering model, based on random partitions, in order to estimate Betti numbers. A simulation study is also presented. [ABSTRACT FROM AUTHOR]

In Crew and Spirkl (2020), the authors generalize Stanley's chromatic symmetric function (Stanley, 1995) to vertex-weighted graphs. In this paper we find a categorification of their new invariant extending the definition of chromatic symmetric homology to vertex-weighted graphs. We prove the existence of a deletion–contraction long exact sequence for chromatic symmetric homology which gives a useful computational tool and allow us to answer two questions left open in Chandler et al. (2019). In particular, we prove that, for a graph G with n vertices, the maximal index with nonzero homology is not greater that n − 1. Moreover, we show that the homology is non-trivial for all the indices between the minimum and the maximum with this property. [ABSTRACT FROM AUTHOR]

This manuscript explores the capability of topological data analysis (TDA) based on homology theory (HT: a subfield of algebraic topology) to extract relevant information for recognition of confusing Arabic machine-printed numerals. In fact, topological properties may significantly reduce the confusion between some numerals such as "1" and "4" in the context of small data sets. These two latter digits differ in the sense that digit 1 has no hole and digit 4 has one hole. Our contribution consists of evaluating the contribution of TDA with its invariant descriptors such as Betti numbers in machine-printed Arabic numerals recognition. Our investigation is driven by the following set of actions: (i) we extract Betti numbers invariant features of each numeral image and partition the ten numerals into three different clusters with respect to these features. (ii) We then perform a classification by assigning a test image to its corresponding cluster, and map this image to a numeral using dynamic-time warping as a metric defined in the Freemans' chaincode space. We compared our proposed approach with major state-of-the-art methods depicting various ways of using TDA in character recognition. The advantages and limitations of TDA (including its pros and cons) are discussed further based on numeral recognition results. [ABSTRACT FROM AUTHOR]

We study the relation between the set of oriented Z/d-homology 3-spheres and the level-d mapping class groups, the kernels of the canonical maps from the mapping class group of an oriented surface to the symplectic group with coefficients in Z/dZ. We formulate a criterion to decide whenever a Z/d-homology 3-sphere can be constructed from a Heegaard splitting with gluing map an element of the level-d mapping class group. Then, we give a tool to construct invariants of Z/d-homology 3-spheres from families of trivial 2-cocycles on the level-d mapping class groups. We apply this tool to find all the invariants of Z/p-homology 3-spheres constructed from families of 2-cocycles on the abelianization of the level-p mapping class group with p prime and to disprove the conjectured extension of the Casson invariant modulo a prime p to rational homology 3-spheres due to B. Perron. [ABSTRACT FROM AUTHOR]

We construct an operator on ... (N)/ link homology with coefficients in a ring whose characteristic divides N. If P is a prime number, we use this operator to exhibit structural features of ... (P) link homology that are special to characteristic P that generalize well-known features of Khovanov homology that are special to characteristic 2. [ABSTRACT FROM AUTHOR]