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In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.
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The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex.
Let $mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $mathcal{F}subsetmathcal{H}$ with covering number at most $p$ is $left(binom{r+p}{r}-1right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $mathcal{F}subsetmathcal{H}$ is $frac{1}{2}binom{2r}{r}$-collapsible.
The family of contractible graphs, introduced by A. Ivashchenko, consists of the collection $mathfrak{I}$ of graphs constructed recursively from $K_1$ by contractible transformations. In this paper we show that every graph in a subfamily of $mathfrak{I}$ (the strongly contractible ones) is a collapsible graph (in the simplicial sense), by providing a sequence of elementary collapses induced by removing contractible vertices or edges. In addition, we introduce an algorithm to identify the contractible vertices in any graph and show that there is a natural homomorphism, induced by the inclusion map of graphs, between the homology groups of the clique complex of graphs with the contractible vertices removed. Finally, we show an application of this result to the computation of the persistent homology for the Vietoris-Rips filtration.
We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) rightarrow THH(E(2))rightarrow overline{THH}(E(2))$ and describe $overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 leq i leq n$. In particular, we compute $K(i)_*THH(E(n))$.
Manifolds and fiber bundles, while superficially different, have strong parallels; in particular, they are both defined in terms of equivalence classes of atlases or in terms of maximal atlases, with the atlases treated as mere adjuncts. This paper presents a unified view of atlases for manifolds and fiber bundles as mathematical entities in their own right. It defines some convenient notation, defines categories of atlases and defines functors among them. The paper Local Coordinate Spaces: a proposed unification of manifolds with fiber bundles, and associated machinery (Arxiv:1801.05775) introduced some of the ideas presented here, but many of the details are not needed there. This paper fleshes out the concepts in more detail than would be relevant there.
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