scholarly journals Homotopy types of random cubical complexes

2021 ◽  
Author(s):  
K. Alex Dowling ◽  
Erik Lundberg
Keyword(s):  
Cubical Complexes ◽  
Homotopy Types

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals Extreme nonuniqueness of end-sum

2020 ◽  
pp. 1-43
Author(s):  
Jack S. Calcut ◽  
Craig R. Guilbault ◽  
Patrick V. Haggerty
Keyword(s):  
Abelian Groups ◽  
Cohomology Algebra ◽  
Homotopy Types ◽  
Proper Homotopy ◽  
Open Formula

We give explicit examples of pairs of one-ended, open [Formula: see text]-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

scholarly journals Strong Homotopy Types, Nerves and Collapses

2011 ◽  
Vol 47 (2) ◽  
pp. 301-328 ◽  
Author(s):  
Jonathan Ariel Barmak ◽  
Elias Gabriel Minian
Keyword(s):  
Homotopy Types

scholarly journals COMBINATORIAL AND TOPOLOGICAL PHASE STRUCTURE OF NON-PERTURBATIVE n-DIMENSIONAL QUANTUM GRAVITY

1992 ◽  
Vol 06 (11n12) ◽  
pp. 2109-2121
Author(s):  
M. CARFORA ◽  
M. MARTELLINI ◽  
A. MARZUOLI
Keyword(s):  
Quantum Gravity ◽  
Phase Structure ◽  
Gravity Models ◽  
Topological Phase ◽  
Geometrical Characterization ◽  
Homotopy Types ◽  
Geometrical Constraints ◽  
Riemannian Geometries

We provide a non-perturbative geometrical characterization of the partition function of ndimensional quantum gravity based on a rough classification of Riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.

scholarly journals Group-theoretic algebraic models for homotopy types

1991 ◽  
Vol 75 (3) ◽  
pp. 195-235 ◽  
Author(s):  
P. Carrasco ◽  
A.M. Cegarra
Keyword(s):  
Algebraic Models ◽  
Homotopy Types
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