T-homotopy and refinement of observation (III) : Invariance of the branching and merging homologies

计算有向图与箭图的奇异单纯同调

Computing singular simplicial homologies of digraphs and quivers

本文为有向图与箭图发展同伦理论，构造函子实现奇异单纯同调与空间同调的同构，并提供计算持久同调的高效算法，适用于大规模网络分析。

The homotopy invariance of cyclic homology of $A_\infty$-algebras over rings

In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of $A_\infty$-algebras into isomorphisms of graded modules. As a corollary, it is obtained that the cyclic homology of an $A_\infty$-algebra over any field is isomorphic to the cyclic homology of the $A_\infty$-algebra of homologies for the source $A_\infty$-algebra.

The homotopy branching space of a flow

In this talk, I will explain the importance of the homotopy branching space functor (and of the homotopy merging space functor) in dihomotopy theory. The paper is a detailed abstract of math.AT/0304112 and math.AT/0305169.

(Co)Homology Self-closeness Numbers of Simply-connected Spaces

The (co)homology self-closeness number of a simply-connected based CW-complexes $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions$\leq k$ is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology self-closeness numbers of spaces in certain cofibrations, define the mod $p$ (co)homology self-closeness number of simply-connected $p$-local spaces with finitely generated homologies and stud…

前立方集的分支空间

Branching space of precubical set

本文提出前立方集的分支空间概念，并证明其在同伦等价下的不变性，同时推导出分支同调的不变性。

同伦分层空间的同伦截断

Homotopy truncations of homotopically stratified spaces

本研究在Quinn同伦分层空间框架下，采用Eckmann-Hilton意义下的对偶方法，用Postnikov逼近替换原链环构造分层空间，证明该构造还原了Gajer用于建立相交Dold-Thom定理的空间。

关于有向同伦等价与一种有向拓扑复杂度的观念

On directed homotopy equivalences and a notion of directed topological complexity

本文提出有向同伦等价与有向拓扑复杂度的概念，探讨其联合性质，指出有向同伦等价与双相似自然同调的关系，并讨论有向拓扑复杂度作为有向同伦等价不变量的特性。

Filtered simplicial homology, graph dissimilarity and überhomology

We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit expression for the associated graded object of these homologies when X is the matching complex of the Tait graph of a plane graph $G$, in terms of subgraphs determined by certain matchings on the dual of G. We then use one of these homologies, in the case where X is a graph, to define a conjecturally optimal dissimilarity pseudometric for graphs; we prove various re…

对称性破缺与链环同调 III

Symmetry Breaking and Link Homologies III

本文在前期构造的严格破缺对称性谱基础上，应用基于环路群 level n 表示的 Dominant K-theory，推导出一个谱序列，其 E_2 项形变自 sl(n)-链环同调，揭示了潜在的 level-rank 对偶性。

对称性破缺与链环同调 II

Symmetry Breaking and Link Homologies II

本文构造过滤的U(r)-等变稳定同伦谱，证明其谱序列E_1项同构于Soergel双模的Hochschild同调，从而计算三度链环同调，并通过扭曲的Borel等变上同调将普适幂级数特化得到sl(n)-链环同调。

Homotopy invariants of higher dimensional categories and concurrency in computer science

The strict globular $ω$-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) $ω$-category $\C$ three homology theories. The first one is called the globular homology. It contains the oriented loops of $\C$. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of $\C$. Two natural linear maps called the negative (resp. the positive) Hurewic…

球形细分是双同伦等价

Globular subdivisions are dihomotopy equivalences

本文证明了球形细分在 $d$-空间中产生双同伦等价，并展示了相关流的同调理论和同伦类型的同构性。