Jonas Kiessling
In this paper we prove certain properties of cellular and acyclic classes of chain complexes of modules over a commutative Noetherian ring. In particular we show that if X is finite and belongs to some cellular class C then Σ^n H_X also belongs to C, for every n.
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Let (R, m) be a local commutative ring. Suppose that m is principal and that m^2 = 0. We give a complete description of the cellular lattice of perfect chain complexes of modules over this ring.
Cellular complexes and embeddings into Euclidean spaces: Möbius strip, torus, and projective plane
本文研究莫比乌斯带、环面和实射影平面的胞腔复形表示与欧几里得空间嵌入，构造了从胞腔复形到3维和4维欧氏空间曲面的显式同胚及其逆映射。
Universal Assembly and Cellular Loop Spaces on Regular CW Complexes
本文为正则CW复形建立了代数手术理论中链复形的经典装配形式的类比，通过构造组合路径和环路对象，并引入通用装配函子，将局部化同伦范畴等同于导出范畴的Verdier商，证明了局部化理论可通过环路DG代数的DG模进行Morita描述。
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.
Let $M$ denote a two-dimensional Moore space (so $H_2(M; \Z) = 0$), with fundamental group $G$. The $M$-cellular spaces are those one can build from $M$ by using wedges, push-outs, and telescopes (and hence all pointed homotopy colimits). The question we address here is to characterize the class of $M$-cellular spaces by means of algebraic properties derived from the group $G$. We show that the cellular type of the fundamental group and homological information does not suffice, and one is forced to study a certain universal extension.
We construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_\infty$-operad. We use one of these to construct a cellular $E_\infty$-bialgebra structure on the interval and derive from it a natural cellular $E_\infty$-coalgebra structure on the geometric realization of a simplicial set which, passing to cellular chains, recovers up to signs the Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse and McClure-Smith. We use another prop, a quotient of the first…
We describe a "cellular" approach to the computation of the cohomology of a poset with coefficients in a presheaf. A cellular cochain complex is constructed, described explicitly and shown to compute the cohomology under certain circumstances. The descriptions are refined further for certain classes of posets including the cell posets of regular CW-complexes and geometric lattices.
Cellular Cohomology in Homotopy Type Theory
本文在同伦类型论中发展胞腔上同调，证明有限胞腔复形上的一类广泛上同调理论（含艾伦伯格-麦克莱恩空间定义的）可通过胞腔上同调计算，并在Agda中形式化。
Attending to Topological Spaces: The Cellular Transformer
本文提出细胞Transformer（CT），将图Transformer推广至细胞复形，利用关联关系与拓扑位置编码，在三个数据集上无需复杂增强即达最先进性能。
This paper exhibits a multiplicative and minimal cellular complex which allows explicit and complete (co)homological calculations for the symmetric products of a finite two dimensional CW complex. By considering cohomology, we observe that a classical theorem of Clifford on the dimension of various linear series on a projective curve has a purely topological statement. We give a "real" analog of this theorem for unoriented topological surfaces.
The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonz{á}lez, and Rudyak with the aim of constructing a cellular model of the configuration space of a sphere. Although the original aim was not achieved in the project, the notion of cellular stratified spaces turns out to be useful, at least, in the study of configuration spaces of graphs. In particular, the notion of totally normal cellular stratified spaces was used successfully in a joint work with the former students of the author arXiv:1312.7368 to study the homotopy type of configuration sp…
A distributive lattice of model structures relating to spectral sequences
本文证明过滤链复形与双复形上的$S$-模型范畴具有左固有、胞腔和稳定性质，利用胞腔化原则建立Quillen等价关系，并构建了由这些模型范畴组成的分配格。
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways these can be used to obtain lower bounds for the chromatic numbers of graphs, presented in a recent series of papers \cite{BK03a,BK03b,BK03c,CK1,CK2,K4,K5}. The structural theory is developed and put in the historical context, culminating in the proof of the Lovász Conjecture, which can be stated as follows: For a graph G, such that the complex Hom(C_{2r+1}…
We study the geometric properties of Cantor subshifts in the Besicovitch space, proving that sofic shifts occupy exactly the homotopy classes of simplicial complexes. In addition, we study canonical projections into subshifts, characterize the cellular automata that are contracting or isometric in the Besicovitch or Weyl spaces, study continuous functions that locally look like cellular automata, and present a new proof for the nonexistence of transitive cellular automata in the Besicovitch space.
It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathc…
We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including its genus and Euler characteristic. Our main result is that up to surgery at nullhomotopic curves minimizers are homotopic to cellwise coverings to the 2-skeleton. From this we conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2-dimensional CAT(-1)-complexes.
Given any topological group $G$, the topological classification of principal $G$-bundles over a finite CW-complex $X$ is long-known to be given by the set of free homotopy classes of maps from $X$ to the corresponding classifying space $BG$. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when $X$ has dimension $2$, it seems there is a case in which such explicit classification has not been explicitly considered. This is the case where $G$ is a Lie group, whose group of components acts non-trivially on its funda…
We show that coloring properties of a simplicial complex K are reflected by splitting properties of a bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley-Reisner algebra of K.
The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and incorporate the results into properties of the real moduli space of Riemann spheres. Connections are discussed with relation to classic combinatorial problems as well as to other sciences.
At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic …
Representation Stability for Marked Graph Complexes
本文证明带标记图复形的锐利表示稳定性，并推广Hersh-Reiner构型空间稳定性界至高亏格情形。
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