A bigroupoid's topology (or, Topologising the homotopy bigroupoid of a space)
David Michael Roberts
备注：
V1 26 pages, 8 figures. Preliminary version, comments welcome; V2 25 pages, 5 figures. HHA style, submitted for publication. License is CC-BY, V3 18 pages, final version to appear in JHRS. License is CC0
发表信息：
Journal of Homotopy and Related Structures Volume 11, Issue 4 (2016) pp 923-942
The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid.
Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the …
This book is an account of certain topics in general and algebraic topology. Instead of laying out a synopsis of each chapter, here is a sample of some of what is taken up: 1) Nilpotency and its role in homotopy theory. 2) Bousfield's theory of the localization of spaces and spectra. 3) Homotopy limits and colimits and their applications. 4) The James construction, symmetric products, and the Dold$-$Thom theorem. 5) Brown and Adams representability in the setting of triangulated categories 6) Operads and the May$-$Thomason theorem on the uniqueness of infinite loop space machines. 7) The plus …
D. K. Biss（Topology and its Applications 124（2002）355-371）介绍了拓扑基本群并给出了这个概念的一些有趣的基本性质。本文将上述概念推广到同伦群，并试图证明拓扑同伦群的一些类似的基本性质。我们还进一步研究了拓扑同伦群的拓扑结构，得到了拓扑结构离散的充要条件。此外，我们还证明了研究拓扑同伦群可能比拓扑基本群更有用。
We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp variables". This yields type theories that we call "spatial" and "cohesive", in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the "fundamental $\infty$-groupoid" or "shape"), disentangling the "iden…
In this paper, we present two types of Lefschetz numbers in the topology of digital images. Namely, the simplicial Lefschetz number $L(f)$ and the cubical Lefschetz number $\bar L(f)$. We show that $L(f)$ is a strong homotopy invariant and has an approximate fixed point theorem. On the other hand, we establish that $\bar L(f)$ is a homotopy invariant and has an $n$-approximate fixed point result. In essence, this means that the fixed point result for $L(f)$ is better than that for $\bar L(f)$ while the homotopy invariance of $\bar L(f)$ is better than that of $L(f)$. Unlike in classical topolo…
The space of orientation-compatible almost complex structures on the six-dimensional sphere naturally contains a copy of seven-dimensional real projective space. We show that the inclusion induces an isomorphism on fundamental groups and rational homotopy groups. We also compute the homotopy fiber of the inclusion and the homotopy groups of the space of almost complex structures in terms of the homotopy groups of the seven-dimensional sphere. Our approach lends itself to generalization to components of almost complex structures with vanishing first Chern class on six-manifolds.
For a discrete poset $\mathcal X$, McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$, from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, uZivaljević defined the notion of order complex for a topological poset. For a large class of such topological posets we prove the analog of McCord's theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. An example is the Grassmann poset of proper non-zero linear subspaces of $\R^{n+1}$. Here, Vassiliev had c…
Let X be a simply connected Riemannian manifold. Until now, quantitative topology has used Sullivan's rational homotopy theory as the bridge between geometric information on X and torsion-free homotopy theoretic information on X. In this paper we introduce Chen's iterated integrals on the based loop space of X as a new bridge between these two areas. We give two applications: finding upper bounds for Gromov's distortion of higher homotopy groups on X and also proving the non-existence of homologically non-trivial small-volume cycles in the space of loops on X of length at most L.
We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the c…
Uchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms $Sh^*(X,Y)$, for arbitrary topological spaces $X$ and $Y$. In particular, we can consider a topology on the coarse shape homotopy group of a topological space $(X,x)$, $Sh^*((S^k,*),(X,x))=\checkπ_k^{*}(X,x)$, which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopoy groups such as second countability, movabil…
We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function…
We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the homotopy type of their space of submanifolds
We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, ins…
An injective word over a finite alphabet $V$ is a sequence $w=v_1v_2\cdots v_t$ of distinct elements of $V$. The set $\mathrm{inj}(V)$ of injective words on $V$ is partially ordered by inclusion. A complex of injective words is the order complex $Δ(W)$ of a subposet $W \subset \mathrm{inj}(V)$. Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and are of independent interest in topology and combinatorics. In this article we mainly study Permutation Complexes, i.e. complexes of injective words $Δ(W)$, where $W$ is the downward closed subposet of …
The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial simplification of simplicial complexes and allows to see a certain duality on a class of simplicial complexes. This duality generalizes Poincare duality on cell subdivisions of manifolds. On the other hand, with the notion of a topological formal context, we review the classical proofs of two basic theorems of homotopy topology: Alexandrov Nerve theorem and Quillen-…
The Dupont homotopy, a classical construction in the algebraic topology of triangulated smooth manifolds, has been revived in the last decade in the construction of an effective field theory where it appears as a propagator. In this paper, we ask a question of relevance to the renormalization group of this theory: is Dupont's construction compatible with stellar subdivision?
In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a~standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a $Δ$-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an od…
By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $π_n^{wh}(X, x_0)$, we show that $π_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $π_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $π_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $π_n^{wh}(X, …
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.
