A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem

半粗空间：基本广群与范坎彭定理

Semi-coarse Spaces: Fundamental Groupoid and the van Kampen Theorem

本文构造了半粗空间上的基本广群不变量，证明了其对一般半粗空间良定义且对粗空间非平凡，并证明了其满足范坎彭定理。

Groupoids, the Phragmen-Brouwer Property and the Jordan Curve Theorem

We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points.

Models and van Kampen theorems for directed homotopy theory

We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and m…

The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action

The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (Ellis …

Fundamental groupoids and homotopy types of non-compact surfaces

The paper contains an application of van Kampen theorem for groupoids for computation of homotopy types of certain class of non-compact foliated surfaces obtained by gluing at most countably many strips $\mathbb{R}\times(0,1)$ with boundary intervals in $\mathbb{R}\times\{\pm1\}$ along some of those intervals.

Modelling and Computing Homotopy Types: I

The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces.