Principal bundles on 2-dimensional CW-complexes with disconnected structure group

On the rational homotopical nilpotency index of principal bundles

Let $\rm{Aut}(p)$ denote the space of all self-fibre homotopy equivalences of a principal $G$-bundle $p: E\rightarrow X$ of simply connected CW complexes with $E$ finite. When $G$ is a compact connected topological group, we show that there exists an inequality $$n-{\rm N}(p)\leq {\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_0)\leq n$$ for any space $X$, where $n$ is the number of non-trivial rational homotopy groups of $G$ and ${\rm N}(p)$ is defined in Section 2. In particular, ${\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_{0})=n$ if $p$ is a fibre-homotopy trivial bundle and X is finite.

Spaces of maps into classifying spaces for equivariant crossed complexes, II: The general topological group case

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory from Tonks' thesis, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equ…

Topology of random clique complexes

In a seminal paper, Erdos and Renyi identified the threshold for connectivity of the random graph G(n,p). In particular, they showed that if p >> log(n)/n then G(n,p) is almost always connected, and if p << log(n)/n then G(n,p) is almost always disconnected, as n goes to infinity. The clique complex X(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There a…

On Complexes Equivalent to $\mathbb{S}^3$-bundles over $\mathbb{S}^4$

There has been renewed interest in $\mathbb{S}^3$-bundles over $\mathbb{S}^4$ since K. Grove and W. Ziller constructed metrics on nonnegative curvature on the total spaces of these bundles. In this paper we write down necessary and sufficient conditions for a CW complex to be homotopy equivalent to such a bundle. We also show that for a manifold homotopy equivalent to such a bundle, in certain cases, there is no obstruction to homeomorphism. We use this to show that the Berger manifold, $\text{Sp}(2)/\text{Sp}(1)$, is PL-homeomorphic to such a bundle. This question was raised in the paper by G…

Reconstructing the spectrum of F_1 from the stable homotopy category

The finite stable homotopy category S_0 has been suggested as a candidate for a category of perfect complexes over the monoid scheme Spec F_1. We apply a reconstruction theorem from algebraic geometry to S_0, and show that one recovers the one point topological space. We also classify filtering subsets of the set of principal thick subcategories of S_0, and of its p-local versions. This is motivated by a result saying that the analogous classification for the category of perfect complexes over an affine scheme provides topological information.

高连通8维流形上的$SU(2)$丛

$SU(2)$-bundles over highly connected $8$-manifolds

该研究分析3连通8维庞加莱对偶复形上主$SU(2)$丛全空间的同伦型，同时分类了由$S^4$与$S^7$楔和粘贴11维胞腔得到的3连通11维复形$E$。

Classifying spaces and Bredon (co)homology for transitive groupoids

We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.

通过椭球体进行持久同调

Persistent Homology via Ellipsoids

本文提出了一种基于椭球体的Rips型复形，用于改进拓扑数据分析中的持久同调计算，通过主成分分析估计切线空间并验证其在分类任务中的优越性。

A classification of fibre bundles over 2-dimensional spaces

The classification problem for principal fibre bundles over two-dimensional CW-complexes is considered. Using the Postnikov factorization for the base space of a universal bundle a Puppe sequence that gives an implicit solution for the classification problem is constructed. In cases, when the structure group $G$ is path-connected or $π_{1}(G) = 0$, the classification can be given in terms of cohomology groups.

The Higher-Dimensional Skeletonization Problem

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when given data points split into well-separated clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as hol…

局部紧右角Artin群的可推广表示与紧性性质

Generalisable presentations and compactness properties of locally compact right-angled Artin groups

本文提出可推广表示的概念，引入拓扑右角Artin群，研究其几何与紧性性质，并构造了具有特定有限性性质的离散群例子。

Equivariant Bundles and Isotropy Representations

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $Γ$-spaces. We show that equivariant principal $G$-bundles over split $Γ$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=Γ\backslash X$ is a graph, with all edge stabilizers toral subgroups of $Γ$, we obtain a purely combinatorial classification of bundles with structural group $G$ a compact connected Lie group. If $G$ is abelian, our approach gives combinatorial and geometric descriptions of some results …