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Gregory Lupton

Publications and source records attributed to Gregory Lupton.

At least 19 recordsLinked to original sources
arXiv2025

The Simplicial Loop Space of a Simplicial Complex

Given a simplicial complex $X$, we construct a simplicial complex $\Omega X$ that may be regarded as a combinatorial version of the based loop space of a topological space. Our construction explicitly describes the simplices of $\Omega X$ directly in terms of the simplices of $X$. Working at a purely combinatorial level, we show two main results that confirm the (combinatorial) algebraic topology of our $\Omega X$ behaves like that of the topological based loop space. Whereas our $\Omega X$ is generally a disconnected simplical complex, each component of $\Omega X$ has the same edge group, up to isomorphism. We show an isomorphism between the edge group of $\Omega X$ and the combinatorial second homotopy group of $X$ as it has been defined in separate work (arxiv:2503.23651). Finally, we enter the topological setting and, relying on prior work of Stone, show a homotopy equivalence between the spatial realization of our $\Omega X$ and the based loop space of the spatial realization of $X$.

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arXiv2025

The Face Group of a Simplicial Complex

The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the ordinary (topological) fundamental group of the spatial realization. We define a counterpart to the edge group that likewise gives a combinatorial version of the second (higher) homotopy group. Working entirely combinatorially, we show our group is an abelian group and also respects products. We show that our combinatorially defined group is isomorphic to the ordinary (topological) second homotopy group of the spatial realization.

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arXiv2023

A Second Homotopy Group for Digital Images

We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.

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arXiv2022

The structuring effect of a Gottlieb element on the Sullivan minimal model of a space

We show a Gottlieb element in the rational homotopy of a simply connected space $X$ implies a structural result for the Sullivan minimal model, with different results depending on parity. In the even-degree case, we prove a rational Gottlieb element is a terminal homotopy element. This fact allows us to complete an argument of Dupont to prove an even-degree Gottlieb element gives a free factor in the rational cohomology of a formal space of finite type. We apply the odd-degree result to affirm a special case of the $2N$-conjecture on Gottlieb elements of a finite complex. We combine our results to make a contribution to the realization problem for the classifying space $B\mathrm{aut}_1(X)$. We prove a simply connected space $X$ satisfying $B\mathrm{aut}_1(X_{\mathbb{Q}}) \simeq S_{\mathbb{Q}}^{2n}$ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree $2n-1$ for $n= 1, 2, 3.$

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arXiv2020

The Digital Hopf Construction

Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and far between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration which mimics the smooth setting.

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arXiv2020

The universal fibration with fibre $X$ in rational homotopy theory

Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_\infty \colon UE \to \mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ ω\colon \mathrm{aut}_1(\mathrm{Baut}_1(X_{\mathbb Q})) \to \mathrm{Baut}_1(X_{\mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_\infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $\mathrm{Baut}_1(X_{\mathbb Q})$ as a consequence. We also prove that ${\mathbb C} P^n_{\mathbb Q}$ cannot be realized as $\mathrm{Baut}_1(X_{\mathbb Q})$ for $n \leq 4$ and $X$ with finite-dimensional rational homotopy groups.

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arXiv2019

Digital Fundamental Groups and Edge Groups of Clique Complexes

In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a graph. The clique complex is a simplicial complex and its edge group is well-known to be isomorphic to the ordinary (topological) fundamental group of its geometric realization. This identification of our intrinsic digital fundamental group with a topological fundamental group---extrinsic to the digital setting---means that many familiar facts about the ordinary fundamental group may be translated into their counterparts for the digital fundamental group: The digital fundamental group of any digital circle is $\mathbb{Z}$; a version of the Seifert-van Kampen Theorem holds for our digital fundamental group; every finitely presented group occurs as the (digital) fundamental group of some digital image. We also show that the (digital) fundamental group of every 2D digital image is a free group.

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arXiv2019

A Fundamental Group for Digital Images

We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously established versions of a fundamental group in the digital setting. Our development gives a prominent role to subdivision of digital images. We show that our fundamental group is preserved by subdivision.

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arXiv2019

Subdivision of Maps of Digital Images

With a view towards providing tools for analyzing and understanding digitized images, various notions from algebraic topology have been introduced into the setting of digital topology. In the ordinary topological setting, invariants such as the fundamental group are invariants of homotopy type. In the digital setting, however, the usual notion of homotopy leads to a very rigid invariance that does not correspond well with the topological notion of homotopy invariance. In this paper, we establish fundamental results about subdivision of maps of digital images with $1$- or $2$-dimensional domains. Our results lay the groundwork for showing that the digital fundamental group is an invariant of a much less rigid equivalence relation on digital images, that is more akin to the topological notion of homotopy invariance. Our results also lay the groundwork for defining other invariants of digital images in a way that makes them invariants of this less rigid equivalence.

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arXiv2019

Homotopy Theory in Digital Topology

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 homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

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arXiv2018

Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $π=π_1(X)$ and we denote it ${\sf TC}(π)$. We prove that ${\sf TC}(π)$ can be characterised as the smallest integer $k$ such that the canonical $π\timesπ$-equivariant map of classifying spaces $$E(π\timesπ) \to E_{\mathcal D}(π\timesπ)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(π\timesπ)$. The symbol $E(π\timesπ)$ denotes the classifying space for free actions and $E_{\mathcal D}(π\timesπ)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $π\timesπ$. Using this result we show how one can estimate ${\sf TC}(π)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(π) \le \max\{3, {\rm cd}_{\mathcal D}(π\timesπ)\},$ where ${\rm cd}_{\mathcal D}(π\timesπ)$ denotes the cohomological dimension of $π\timesπ$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.

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arXiv2018

An upper bound for topological complexity

In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\mathsf{TC}}^{\mathcal{D}}$ and the connections between ${\mathsf{TC}}^{\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\mathsf{TC}$-type invariant $\widetilde{\mathsf{TC}}$ that can be used to give an upper bound for $\mathsf{TC}$, $$\mathsf{TC}(X)\le {\mathsf{TC}}^{\mathcal{D}}(X) + \left\lceil \frac{2\dim X -k}{k+1}\right\rceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $\tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.

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arXiv2016

Hereditary properties of co-Kähler manifolds

We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: $\dim(H^*(M;\mathbb{Q})) \geq 2^r$, for any $r$-torus $T^r$ which acts almost freely on $M$.

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arXiv2016

Parallel forms, co-Kähler Manifolds and their Models

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context.

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arXiv2015

Realizing spaces as classifying spaces

Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.

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arXiv2013

A Mapping Theorem for Topological Complexity

We give new lower bounds for the (higher) topological complexity of a space, in terms of the Lusternik-Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and more generally for the rational sectional category of a map, in terms of the rational category of a certain auxiliary space. We use our results to deduce consequences for the global (rational) homotopy structure of simply connected, hyperbolic finite complexes.

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