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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. [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]

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. [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]

The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on $ \mathbb{S}^n $ to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes. [ABSTRACT FROM AUTHOR]

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework. [ABSTRACT FROM AUTHOR]

Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K -algebra A , we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A # H , involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial "Hopf" algebra H p a r is involved in the definition and we construct a projective resolution of B. [ABSTRACT FROM AUTHOR]

Let x _ denote a system of elements of a commutative ring R. For an R -module M we investigate when x _ is M -pro-regular resp. M -weakly pro-regular as generalizations of M -regular sequences. This is done in terms of Čech co-homology resp. homology, defined by H i ( C ˇ x _ ⊗ R ⋅) resp. by H i (R Hom R ( C ˇ x _ , ⋅)) ≅ H i (Hom R (L x _ , ⋅)) , where C ˇ x _ denotes the Čech complex and L x _ is a bounded free resolution of it as constructed in [17] resp. [16]. The property of x _ being M -pro-regular resp. M -weakly pro-regular follows by the vanishing of certain Čech co-homology resp. homology modules, which is related to completions. This extends previously work by Greenlees and May (see [5]) and Lipman et al. (see [1]). This contributes to a further understanding of Čech (co-)homology in the non-Noetherian case. As a technical tool we use one of Emmanouil's results (see [4]) about the inverse limits and its derived functor. As an application we prove a global variant of the results with an application to prisms in the sense of Bhatt and Scholze (see [3]). [ABSTRACT FROM AUTHOR]

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 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]

Given a finite group G acting on a surface S , the centralizer of G in the mapping class group Mod (S) has a natural representation given by its action on the homology H 1 (S ; Q). We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3. [ABSTRACT FROM AUTHOR]

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying τG$\tau _{\mathrm{G}}$, defined by the second author via the minus flavors KHI̲−$\underline{\operatorname{KHI}}^-$ and KHM̲−$\underline{\operatorname{KHM}}^-$ of the knot homologies, with τG♯$\tau ^\sharp _{\mathrm{G}}$, defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute KHI̲−$\underline{\operatorname{KHI}}^-$ and KHM̲−$\underline{\operatorname{KHM}}^-$ for twist knots. [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]

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]

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]

Putman and Wieland conjectured that if Σ∼→Σ$\widetilde{\Sigma }\rightarrow \Sigma$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of H1(Σ∼;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ under the action of lifts to Σ∼$\widetilde{\Sigma }$ of mapping classes on Σ$\Sigma$ are infinite. We prove that this holds if H1(Σ∼;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ is generated by the homology classes of lifts of simple closed curves on Σ$\Sigma$. We also prove that the subspace of H1(Σ∼;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2‐holed spheres, and we prove that H1(Σ∼;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ is generated by the homology classes of lifts of loops on Σ$\Sigma$ lying on subsurfaces homeomorphic to 3‐holed spheres. [ABSTRACT FROM AUTHOR]

Summary: Previous research on the motivational factors of safety performance has predominantly focused on one's willingness to directly enact safety behaviors or safety‐specific motivation. The current study extends beyond this view and examines an additional motivational force, altruistic motivation, as a main predictor of employees' safety performance at both individual and team levels. Further, we provide that servant leadership serves as a critical precursor of employee altruistic motivation. A sample of 416 nurses in 42 workgroups and their respective supervisors from a hospital in China completed a two‐wave survey. Hierarchical linear modeling was used to analyze the data. The results indicated that altruistic motivation was positively related to safety performance at both individual and team levels, with a stronger effect at the team level, supporting a proportional theory of homology. Multilevel mediation results showed that servant leadership was positively related to altruistic motivation, which in turn positively led to safety performance at both individual and team levels. These findings highlight both the theoretical and practical importance of encouraging altruistic motivation to improve workplace safety. [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]

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]

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]

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]

Let |$T$| be a torus, |$X$| a smooth separated scheme of finite type equipped with a |$T$| -action, and |$[X/T]$| the associated quotient stack. Given any localizing |${\mathbb {A}}^{1}$| -homotopy invariant of dg categories |$E$| (homotopy |$K$| -theory, algebraic |$K$| -theory with coefficients, étale |$K$| -theory with coefficients, |$l$| -adic algebraic |$K$| -theory, |$l$| -adic étale |$K$| -theory, semi-topological |$K$| -theory, topological |$K$| -theory, periodic cyclic homology, etc), we prove that the derived completion of |$E([X/T])$| at the augmentation ideal |$I$| of the representation ring |$R(T)$| of |$T$| agrees with the classical Borel construction associated to the |$T$| -action on |$X$|⁠. Moreover, for certain localizing |${\mathbb {A}}^{1}$| -homotopy invariants, we extend this result to the case of a linearly reductive group scheme |$G$|⁠. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic |$K$| -theory, of Thomason's completion theorem in étale |$K$| -theory with coefficients, and also of Atiyah-Segal's completion theorem in topological |$K$| -theory (for those topological |$M$| -spaces |$X^{\textrm {an}}$| arising from analytification; |$M$| is a(ny) maximal compact Lie subgroup of |$G^{\textrm {an}}$|⁠). These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field |$k$| no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological |$M$| -space |$X^{\textrm {an}}$| no longer needs to be compact and the |$M$| -equivariant topological |$K$| -theory groups of |$X^{\textrm {an}}$| no longer need to be finitely generated over the representation ring |$R(M)$|⁠. As a second application, we obtain new completion theorems in |$l$| -adic étale |$K$| -theory, in semi-topological |$K$| -theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, |$l$| -adic, morphic, Betti, de Rham, etc) in terms of derived completion. Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy |$K$| -theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of semi-topological |$K$| -theory. [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 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]

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]

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]

We study the space of slice torus invariants. In particular we characterise the set of values that slice torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local Thom conjecture implies the existence of slice torus invariants without having to appeal to any explicit construction from a knot homology theory. [ABSTRACT FROM AUTHOR]

Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half‐integral packing. We prove a far‐reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half‐integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example, cycles of length at least ℓ$\ell$, cycles of length p$p$ modulo q$q$, cycles intersecting a prescribed set of vertices at least t$t$ times and cycles contained in given Z2$\mathbb {Z}_2$‐homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties. [ABSTRACT FROM AUTHOR]

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY theory) in a general simplicial setting. The new theory includes as particular cases the GLMY theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square‐commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau. [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 compute $ku^*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ and $ku_*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ , the connective $KU$ -cohomology and connective $KU$ -homology groups of the mod- $p$ Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,2\right)$ , using the Adams spectral sequence. We obtain a striking interaction between $h_0$ -extensions and exotic extensions. The mod- $p$ connective $KU$ -cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups. [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]

The well-rounded retract for SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space X_n of metric graphs of rank n and the Teichmüller space \mathcal {T}_g of closed hyperbolic surfaces of genus g, we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for \mathcal {T}_g by Thurston) are spines for X_n but that their dimension is larger than the virtual cohomological dimension of Out(F_n) in general. [ABSTRACT FROM AUTHOR]

Hypergraphs are useful mathematical models for describing complex relationships among members of a structured graph, while hyperdigraphs serve as a generalization that can encode asymmetric relationships in the data. However, obtaining topological information directly from hyperdigraphs remains a challenge. To address this issue, we introduce hyperdigraph homology in this work. We also propose topological hyperdigraph Laplacians, which can extract both harmonic spectra and non-harmonic spectra from directed and internally organized data. Moreover, we introduce persistent hyperdigraph homology and persistent hyperdigraph Laplacians through filtration, enabling the capture of topological persistence and homotopic shape evolution of directed and structured data across multiple scales. The proposed methods offer new multiscale algebraic topology tools for topological data analysis. [ABSTRACT FROM AUTHOR]

Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$ , where $n_{1}. Let $\mathcal {G}_k$ be the gauge group of the principal $G$ -bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod- $p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}. [ABSTRACT FROM AUTHOR]

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible σ -compact space in which the corona sits as a Z -set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space B G , then our constructions yield a Z -structure for the group. [ABSTRACT FROM AUTHOR]

We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology , that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to $+\infty$ on the open subset but remain close to $0$ and positive on the rest of the boundary. [ABSTRACT FROM AUTHOR]

Human diseases involve metabolic alterations. Metabolomic profiles have served as a vital biomarker for the early identification of high-risk individuals and disease prevention. However, current approaches can only characterize individual key metabolites, without taking into account the reality that complex diseases are multifactorial, dynamic, heterogeneous, and interdependent. Here, we leverage a statistical physics model to combine all metabolites into bidirectional, signed, and weighted interaction networks and trace how the flow of information from one metabolite to the next causes changes in health state. Viewing a disease outcome as the consequence of complex interactions among its interconnected components (metabolites), we integrate concepts from ecosystem theory and evolutionary game theory to model how the health state-dependent alteration of a metabolite is shaped by its intrinsic properties and through extrinsic influences from its conspecifics. We code intrinsic contributions as nodes and extrinsic contributions as edges into quantitative networks and implement GLMY homology theory to analyze and interpret the topological change of health state from symbiosis to dysbiosis and vice versa. The application of this model to real data allows us to identify several hub metabolites and their interaction webs, which play a part in the formation of inflammatory bowel diseases. The findings by our model could provide important information on drug design to treat these diseases and beyond. [ABSTRACT FROM AUTHOR]

Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Topology is considered an advanced field in mathematics, and it might seem off-putting to people with no previous experience in mathematics. The classification theorem, which lies within the field of algebraic topology, is fascinating, but understanding it requires extensive mathematical knowledge. In this manuscript, we present a modular object that provides a glimpse into the construction of surfaces as defined by the classification theorem. This allows any user to explore different options and discover possible surfaces with three holes or fewer. This tactile experience of exploration grants the user intuition about the meaning of the theorem, without needing the rigorous proof; thus, it simplifies the depth of the classification theorem and its respective proof for non-mathematical audiences. We strive to accurately illustrate these esoteric concepts by giving them a form and to describe the modeling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding. We believe that this playful object will be exciting for a variety of communities: university students, graduate students, teachers, professional mathematicians, and especially designers, who can use it to produce new tools. [ABSTRACT FROM AUTHOR]

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 Spinc 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 π1(Sr³​ (K))→SU(2) unless r ≥ 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. [ABSTRACT FROM AUTHOR]

Let A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Torelli group of a genus g$g$ oriented surface Σg$\Sigma _g$ is the subgroup Ig$\mathcal {I}_g$ of the mapping class group Mod(Σg)${\rm Mod}(\Sigma _g)$ consisting of all mapping classes that act trivially on H1(Σg,Z)${\rm H}_1(\Sigma _g, \mathbb {Z})$. The quotient group Mod(Σg)/Ig${\rm Mod}(\Sigma _g) / \mathcal {I}_g$ is isomorphic to the symplectic group Sp(2g,Z)${\rm Sp}(2g, \mathbb {Z})$. The cohomological dimension of the group Ig$\mathcal {I}_g$ equals to 3g−5$3g-5$. The main goal of the present paper is to compute the top homology group of the Torelli group in the case g=3$g = 3$ as Sp(6,Z)${\rm Sp}(6, \mathbb {Z})$‐module. We prove an isomorphism H4(I3,Z)≅IndS3⋉SL(2,Z)×3Sp(6,Z)Z,$$\begin{equation*} \hspace*{4pc}{\rm H}_4(\mathcal {I}_3, \mathbb {Z}) \cong {\rm Ind}^{{\rm Sp}(6, \mathbb {Z})}_{S_3 \ltimes {\rm SL}(2, \mathbb {Z})^{\times 3}} \mathcal {Z}, \end{equation*}$$where Z$\mathcal {Z}$ is the quotient of Z3$\mathbb {Z}^3$ by its diagonal subgroup Z$\mathbb {Z}$ with the natural action of the permutation group S3$S_3$ (the action of SL(2,Z)×3${\rm SL}(2, \mathbb {Z})^{\times 3}$ is trivial). We also construct an explicit set of generators and relations for the group H4(I3,Z)${\rm H}_4(\mathcal {I}_3, \mathbb {Z})$. [ABSTRACT FROM AUTHOR]

Advances in developmental genetics and evo-devo in the last several decades have enabled the growth of novel developmental approaches to the classic theme of homology. These approaches depart from the more standard phylogenetic view by contending that homology between morphological characters depends on developmental-genetic individuation and explanation. This article provides a systematic re-examination of the relationship between developmental and phylogenetic homology in light of current evidence from developmental and evolutionary genetics and genomics. I present a qualitative model of the processes that cause de-coupling of morphological and molecular evolution by developmental system drift (DSD), and hypothesize that DSD is a widespread and regular stochastic process that is predictable from parameters such as population size, pleiotropy, and regulatory redundancy. These theoretical findings support an integrative approach in which models of DSD aid in determining when developmental-genetic explanations of homology apply and when other explanations such as stabilizing selection are needed. I argue that a phylogenetic monism about the definition and criteria of homology supports the integration of different explanations into a theory of homology better than pluralism does. [ABSTRACT FROM AUTHOR]

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]

A complex algebraic surface S$S$ is a Q$\mathbb {Q}$‐homology plane if Hi(S,Q)=0$H_{i}(S,\mathbb {Q})=0$ for i>0$i>0$. The Negativity Conjecture of Palka asserts that κ(KX+12D)=−∞$\kappa (K_{X}+\tfrac{1}{2}D)=-\infty$, where (X,D)$(X,D)$ is a log smooth completion of S$S$. We give a complete description of smooth Q$\mathbb {Q}$‐homology planes satisfying the Negativity Conjecture. We restrict our attention to those of log general type, as otherwise their geometry is well‐understood. We show that, as conjectured by tom Dieck and Petrie, they can be arranged in finitely many discrete series, each obtained in a uniform way from an arrangement of lines and conics on P2$\mathbb {P}^{2}$. We infer that these surfaces satisfy the Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which asserts that #Aut(S)⩽6$\#\operatorname{Aut}(S)\leqslant 6$. [ABSTRACT FROM AUTHOR]

In this paper, we build up a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that lc-homology satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its corresponding lc-cohomology satisfies exactness axiom for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first lc-homology group. [ABSTRACT FROM AUTHOR]