Gregory Lupton
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View article: The Simplicial Loop Space of a Simplicial Complex
The Simplicial Loop Space of a Simplicial Complex Open
Given a simplicial complex $X$, we construct a simplicial complex $ΩX$ that may be regarded as a combinatorial version of the based loop space of a topological space. Our construction explicitly describes the simplices of $ΩX$ directly in …
View article: The Face Group of a Simplicial Complex
The Face Group of a Simplicial Complex Open
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 ordin…
View article: A second homotopy group for digital images
A second homotopy group for digital images Open
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…
View article: A Second Homotopy Group for Digital Images
A Second Homotopy Group for Digital Images Open
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…
View article: The structuring effect of a Gottlieb element on the Sullivan minimal model of a space
The structuring effect of a Gottlieb element on the Sullivan minimal model of a space Open
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 Gottli…
View article: The Digital Hopf Construction
The Digital Hopf Construction Open
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 ar…
View article: Digital Fundamental Groups and Edge Groups of Clique Complexes
Digital Fundamental Groups and Edge Groups of Clique Complexes Open
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 consi…
View article: Bredon cohomology and robot motion planning
Bredon cohomology and robot motion planning Open
We study the topological invariant [math] reflecting the complexity of algorithms for autonomous robot motion. Here, [math] stands for the configuration space of a system and [math] is, roughly, the minimal number of continuous rules which…
View article: Parallel Forms, Co-Kähler Manifolds and their Models
Parallel Forms, Co-Kähler Manifolds and their Models Open
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 o…
View article: Parallel forms, co-Kahler manifolds and their models
Parallel forms, co-Kahler manifolds and their models Open
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 o…
View article: Topological Complexity, Robotics and Social Choice
Topological Complexity, Robotics and Social Choice Open
Topological complexity is a number that measures how hard it is to plan motions (for robots, say) in terms of a particular space associated to the kind of motion to be planned. This is a burgeoning subject within the wider area of Applied …