Ayse Borat, Mehmetcik Pamuk, Tane Vergili
We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion.
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Higher Contiguity Distance
本文提出了邻接距离的高阶类似物，探讨了其与单纯Lusternik-Schnirelmann范畴和离散拓扑复杂度的关系，并研究了单纯映射的几何实现与重心重分对其的影响。
Sequential $m$-contiguity distance
本文引入序列$m$-邻接距离概念，研究其基本性质，定义单纯复形的序列$m$-离散拓扑复杂度，将拓扑复杂度经典结果扩展到单纯和$m$维情形。
m-Contiguity Distance
本文系统发展了单纯映射间的m-连续性距离，作为simplicial复形中同调复杂度的离散近似框架，构建了逼近连续性距离的递增不变量序列，并建立了其在巴列中心细分、组合和积不等式下的基本性质，定义了m-单纯 Lusternik-Schnirelmann范畴和m-离散拓扑复杂度。
A Computational (Co)homological Approach to Contiguity Distance
本文提出了一种基于同调的计算方法，用于计算连续映射之间的距离，并通过具体计算验证了该方法的有效性。
Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexity of the realization $\|K\|$. Our approach is well suited for designing and implementing algorithms that search for optimal motion planners for autonomous systems in real-life applications.
We define the notion of valuation on simplicial maps between geometric realizations of simplicial complexes in $\mathbb{R}^n$. Valuations on simplicial maps are analogous to valuations on sets. In particular, we define the Lefschetz volumes, which are analogous to the intrinsic volumes of subsets of $\mathbb{R}^n$. Our definition not only provides a generalization of the Lefschetz number, but also yields a Hadwiger-style classification theorem for all such valuations.
Discrete Topological Complexities of Simplical Maps
该研究聚焦满单纯纤维化的离散拓扑复杂度，分析其与离散几何结构、计算效率的关联及高维性质，还探讨高邻接距离性质，提出可简化离散TC的计算方式。
Discrete version of topological complexity of maps
研究映射拓扑复杂性的离散版本，引入并研究其离散类似物，证明是邻接不变量且等价，还建立了单纯映射离散拓扑复杂性的理论性质与计算方面内容。
This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain explicit combinatorial descriptions of Steenrod k-th powers exclusively in terms of face operators.
The Leray spectral sequence of a map $f$ computes the homology of the domain of $f$ from the fibers of $f$. In this expository paper, we relate in full detail the Leray spectral sequence associated to a simplicial map $f$ to the Leray cosheaves of $f$. We then give applications to level set persistent homology and Reeb spaces.
Given a map of simplicial topological spaces, mild conditions on degeneracies and the levelwise maps imply that the geometric realization of the simplicial map is a cofibration. These conditions are not formal consequences of model category theory, but depend on properties of spaces, and similar results have not been available for any model for the stable homotopy category. In this paper we prove such results for symmetric spectra. Consequently, we get a set of conditions which ensure that the geometric realization of a map of simplicial symmetric spectra is a cofibration. These conditions are…
We show that the space $\tilde{A}_{d}(m,n)$ consisting of all real projective classes of $(n+1)$-tuples of real coefficients homogeneous polynomials of degree $d$ in $(m+1)$ variables, without common real roots except zero, has the same homology as the space $ \Map(\RP^m,\Bbb \RP^n)$ of continuous maps from the $m$-dimensional real projective space $\RP^m$ into the $n$ real dimensional projective space $\RP^n$ up to dimension %in dimensions smaller than $(n-m)(d+1)-1$. This considerably improves the main result of \cite{AKY1}.
On essential simplicial maps $S^3 \rightarrow S^2$
本文证明了本质单纯映射 $S^3 \rightarrow S^2$ 的复杂性上界，并展示了 Madahar 和 Sarkaria 构造的 Hopf 映射三角化在同伦类中是最小的。
We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant-$\infty$-operads in equivariant dendroidal sets.
As a higher dimensional version of the theory of Morse functions, there have been various studies of smooth manifolds using generic smooth maps. As fundamental results, in these studies, they have found that inverse images of such maps often restrict the types of the source manifolds. For example, if a generic map such that the inverse image of a regular value is not null-cobordant, then the homology group of its {\it Reeb} space, which is defined as the space of all the connected components of inverse images of the map and a fundamental tool in the theory of generic smooth maps, is known to b…
Let $L$ be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of $L$. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some maps on the space of probability functions on sub-hypergraphs of $L$. We study the compositions of these maps as well as their actions on the space of probability functions.
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $\mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provide…
2-Segal maps associated to a category with cofibrations
该文研究带余纤维化范畴的Waldhausen $S_\bullet$-构造的2-Segal性质，确定了几类单纯对象中恒为等价的2-Segal映射，约化了相关判定问题并给出充分条件，还引入新范畴概念与反例。
Extendability of simplicial maps is undecidable
该研究证明了单纯映射的可扩展性问题在余维数>1时是不可判定的，并指出了相关证明中的漏洞。
In this work we propose a realization of Lurie's prediction that inner fibrations $p: X \rightarrow A$ are classified by $A$-indexed diagrams in a ``higher category" whose objects are $\infty$-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and $\infty$-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categor…
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$. In this work, we further develop our previous work for classification of toric spaces. For a star-sh…
A spectral sequence for spaces of maps between operads
针对满足温和条件的单纯增广operad，研究者构建了逼近其导出映射空间的纤维化塔，给出第n阶的构造方式，并通过主定理刻画了塔的各层结构。
We introduce a finitely presented prop $\mathcal{S} = \{\mathcal{S}(n,m)\}$ in the category of differential graded modules whose associated operad $U(\mathcal{S})=\{\mathcal{S}(1,m)\}$ is a model for the $E_\infty$-operad. This finite presentation allows us to describe a natural $E_\infty$-coalgebra structure on the chains of any simplicial set in terms of only three maps: the Alexander-Whitney diagonal, the augmentation map, and an algebraic version of the join of simplices. The first appendix connects our construction to the Surjection operad of McClure-Smith and Berger-Fresse. The second es…
This paper develops links between the Burnside ring of a finite group $G$ and the slice Burnside ring}. The goal is to gain a better understanding of ghost maps, idempotents, prime spectrum of these Burnside rings and connections between them.
Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \textsf{Mapper}, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points. The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scal…
We introduce the notion of nonevasive reduction, and show that for any monotone poset map $ϕ:P\to P$, the simplicial complex $Δ(P)$ {\tt NE}-reduces to $Δ(Q)$, for any $Q\supseteq{\text{\rm Fix}}ϕ$. As a corollary, we prove that for any order-preserving map $ϕ:P\to P$ satisfying $ϕ(x)\geq x$, for any $x\in P$, the simplicial complex $Δ(P)$ collapses to $Δ(ϕ(P))$. We also obtain a generalization of Crapo's closure theorem.
The simplicial coalgebra of chains under three different notions of weak equivalence
本文针对三种弱等价概念，在约化单纯集和连通单纯余交换余代数上构造模型结构，并证明链的单纯余代数在代数闭域上定义同伦完全忠实的左 Quillen 函子，在完美域上通过同伦不动点比较各模型结构。
We employ projective Fraïssé theory to define the "generic combinatorial $n$-simplex" as the pro-finite, simplicial complex that is canonically associated with a family of simply defined selection maps between finite triangulations of the simplex. The generic combinatorial $n$-simplex is a combinatorial object that can be used to define the geometric realization of a simplicial complex without any reference to the Euclidean space. It also reflects dynamical properties of its homeomorphism group down to finite combinatorics. As part of our study of the generic combinatorial simplex, we define a…
We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if $T$ is the reduced monad associated wi…
On the toric lifting properties for simplicial $3$-spheres
研究单纯 3 - 球面模 2 特征映射提升问题，借助通用复形\(X(\mathbb{Z}_2^4)\)的坏块划分证明避免准则，得出顶点至多 20 的单纯 3 - 球面有环面提升性质等结果。
Bures--Kuratowski metrics and simplicial complexes for completely bounded maps
研究Bures-Kuratowski度量在完全有界映射上的应用，结合Kuratowski嵌入与正则表示Hausdorff坐标，提出度量的l^p楔分解方法，并探讨其在单纯复形中的几何计算与同调特征。
Given topological spaces X and Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X -> Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve this problem in the stable range, where for some d >= 2, we have dim X <= 2d - 2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere S^d. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplici…
In this paper we obtain results indicating that fine shape is tractable and "not too strong" even in the non-locally compact case, and can be used to better understand infinite-dimensional metrizable spaces and their homology theories. We show that every Polish space $X$ is fine shape equivalent to the limit of an inverse sequence of simplicial maps between metric simplicial complexes. A deeper result is that if $X$ is locally finite dimensional, then the simplicial maps can be chosen to be non-degenerate. They cannot be chosen to be non-degenerate if $X$ is the Taylor compactum.
It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a large-cardinal axiom called Vopenka's principle.In this article we extend the validity of this result to any left proper, combinatorial, simplicial model category $\cat M$ and show that, under additional assumptions on $\cat M$, every homotopy idempotent functor is in fact a localization with respect to some set of maps. These results are valid for the homotopy category of spectra, among other applications.
Relations among higher Whitehead maps
本文定义了聚合同态积之间的广义高阶Whitehead映射，通过研究聚合同态积的同伦性质与单纯复形的组合性质之间的关系，描述了新的映射关系家族，并恢复和推广了已知的Whitehead积恒等式。
A Cartesian Promonoidal Kernel on $Δ$ and a Hadamard Contraction of $Δ^n$
本研究构造了单纯形范畴Δ上带笛卡尔单位的对称前幺半核，导出阿达马自然变换，取q=1得到经典的Δⁿ到0顶点的单纯同伦，其前幺半表述及诱导映射为新成果。
We consider two basic problems of algebraic topology, the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity. The extension problem is the following: Given topological spaces X and Y, a subspace A\subseteq X, and a (continuous) map f:A->Y, decide whether f can be extended to a continuous map \bar{f}:X->Y. All spaces are given as finite simplicial complexes and the map f is simplicial. Recent positive algorithmic results, proved in a series of companion papers, show that for (k-1)-connected Y, k>=2, the extension…
We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. This contrasts with the ordinary, unstructured notion of a Kan fibration. We show that fundamental properties of Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, we …
We introduce an axiomatic definition for the Kodaira dimension and classify Thurston geometries in dimensions $\leq 5$ in terms of this Kodaira dimension. We show that the Kodaira dimension is monotone with respect to the partial order defined by maps of non-zero degree between 5-manifolds. We study the compatibility of our definition with traditional notions of Kodaira dimension, especially the highest possible Kodaira dimension. To this end, we establish a connection between the simplicial volume and the holomorphic Kodaira dimension, which in particular implies that any smooth Kähler 3-fold…
The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends with open questions, such as the existence of Sperner's lemma for a polytopal complex or the existence of a cubical approximation theorem.
We give a particular choice of the higher Eilenberg-MacLane maps by a recursive formula.This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras.
In a previous paper we lifted Charles Rezk's complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of "relative categories," and our aim in this successor paper is to obtain a more explicit description of the weak equivalences in this model structure by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.
If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…
Weak cartesian properties of simplicial sets
本文定义单纯集的弱拉回型完备性条件，刻画其提升性质并给出基于 $Δ$ 中推出方形分解的检验判据，例子涵盖拟范畴、compositories、gleaves 及单子代数 bar 构造。
The Stable Adjunction in $\mathbb{A}^1$-Homotopy Theory
研究动机稳定同伦理论中悬架谱与零阶空间函子伴随的同伦单子性定理，通过验证满足一般单子性定理假设来证明，过程中展示六个单纯动机同伦理论结果及相关单子框架，或为动机无限环空间猜想提供工具。
Connecting Discrete Morse Functions via Birth-Death Transitions
本文通过生灭转换研究有限单纯复形上离散Morse函数间的转换，证明了任意两函数均可由有限转换序列连接，并以此给出Forman基本结果的替代证明及探讨函数空间的拓扑结构。
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (in our case, a simplicial complex). Modern packages for persistent homology often construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing these complexes and the effects of making associated choices during simplicial-complex construction on the output of persistent-homology algorithms. We present two new methods for constructing simplicial complexe…
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