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Geometric Realization of Simplicial Complexes
[chapter]

1999
*
Lecture Notes in Computer Science
*

We show that an abstract

doi:10.1007/3-540-46648-7_33
fatcat:eewmmrfn4bf27p4vnd3jl5otxi
*simplicial**complex*∆ may be*realized*on a grid*of*IR d−1 , where d = dim P (∆) is the order dimension (Dushnik-Miller dimension)*of*the face poset*of*∆. ... We shall say that an abstract*simplicial**complex*∆ is*realizable*in IR n if there exists a*geometric**realization**of*∆ in IR n . ... If V (∆) is a set*of*points in IR n , ∆ is thus a*geometric**simplicial**complex*if and only if the identity is a*geometric**realization**of*∆. ...##
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Necessary Conditions for Geometric Realizability of Simplicial Complexes
[article]

2007
*
arXiv
*
pre-print

We associate with any

arXiv:0705.1912v2
fatcat:fqaizppyobfehbec2o4v62kofy
*simplicial**complex*and any integer m a system*of*linear equations and inequalities. If has a*simplicial*embedding in ^m then the system has an integer solution. ... This result extends the work*of*I. Novik (2000). ... Thus, infeasibility*of*a certain integer program might prove that a*complex*K has no*geometric**realization*. ...##
###
The Geometric Realization of a Semi-Simplicial Complex

1957
*
Annals of Mathematics
*

*of*the

*geometric*

*realization*, let C be an ordered simplicialcomplex with space ( C / . ... Since the remaining conditions for a CW-

*complex*are easily verified, this proves Theorem 1. The space I C / is homeomorphic to the

*geometric*

*realization*I K j . ...

##
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Helly numbers of acyclic families

2014
*
Advances in Mathematics
*

As an application, we obtain several explicit bounds on Helly numbers in

doi:10.1016/j.aim.2013.11.004
fatcat:3yns3opur5ghzkqspopjix32t4
*geometric*transversal theory for which only ad hoc*geometric*proofs were previously known; in certain cases, the bound we obtain ... We call a family F*of*open subsets*of*Γ acyclic if for any non-empty sub-family G ⊆ F, each connected component*of*the intersection*of*the elements*of*G is a Q-homology cell. ... The authors would like to thank Jürgen Eckhoff for helpful comments on a preliminary version*of*this paper. ...##
###
Page 458 of American Journal of Mathematics Vol. 79, Issue 3
[page]

1957
*
American Journal of Mathematics
*

The

*simplicial*singular*complex**of*the*geometrical**realization*. ... We shall now use the results*of*Section 5 in order to compare the*simplicial*singular*complex**of*the*geometrical**realization**of*a c.s.s.*complex*AK with Ex® K. ...##
###
Quantitative Homotopy Theory in Topological Data Analysis

2013
*
Foundations of Computational Mathematics
*

We introduce the "contiguity

doi:10.1007/s10208-013-9177-5
fatcat:b364a6ozxjefxbc3tqzh2ljgzm
*complex*", a*simplicial**complex**of*maps between*simplicial**complexes*defined in terms*of*the combinatorial notion*of*contiguity. ... We generalize the*Simplicial*Approximation Theorem to show that the contiguity*complex*approximates the homotopy type*of*the mapping space as we subdivide the domain. ... Review*of**Simplicial**Complexes*This section reviews the basic definitions*of**simplicial**complex*and*geometric**realization*. ...##
###
Fall Sectional Sampler: Face Numbers: Centrally Symmetric Spheres versus Centrally Symmetric Polytopes

2019
*
Notices of the American Mathematical Society
*

Each

doi:10.1090/noti1966
fatcat:zi6gqz36vneu3bsgop2aux5mly
*simplicial**complex*Δ admits a*geometric**realization*|Δ| that contains a*geometric*-simplex for eachface*of*Δ. ... This allows us to talk about*simplicial*spheressimplicial*complexes*whose*geometric**realizations*are homeomorphic to a sphere. ...##
###
Contiguity Distance between Simplicial Maps
[article]

2020
*
arXiv
*
pre-print

In particular, we show that

arXiv:2012.10627v1
fatcat:t7vqszmkivdeni5boujjn5fqoy
*simplicial*versions*of*LS-category and topological*complexity*are particular cases*of*this more general notion. ... We study properties*of*contiguity distance between*simplicial*maps. ... The second author was partially supported by the Scientific and Technological Research Council*of*Turkey (TÜBİTAK) [grant number 11F015]. ...##
###
Free loop space and homology
[article]

2011
*
arXiv
*
pre-print

We introduce all the relevant technical tools, namely

arXiv:1110.0405v1
fatcat:epo2ssqarzg77mwvzqwmwalesu
*simplicial*and cyclic objects, and we provide the various steps*of*the proofs, which are scattered around in the literature. ... The aim*of*this paper is to explain the relationship between the (co)homology*of*the free loop space and the Hochschild homology*of*its singular cochain algebra. ... Let K · be a*simplicial*set whose*geometric**realization*is a finite cell*complex**of*dimension dim(K · ). ...##
###
Simplicial complexes: from continuous to discrete

2014
*
Applied Mathematical Sciences
*

We expose some basic concepts

doi:10.12988/ams.2014.49691
fatcat:zbvzvyoyrjf53oxthyn5qjqfay
*of*combinatorial topology (*simplicial**complex*, polyhedron,*simplicial*map,*simplicial*approximation*of*a continuous map) together with a brief description*of*the popular examples ...*of*Voronoi cells and*of*Delaunay triangulation. ... In the section 3 we expose the notion*of*abstract*simplicial**complex*and that*of*its*geometric**realization*. ...##
###
Lusternik-Schnirelmann category of simplicial complexes and finite spaces
[article]

2015
*
arXiv
*
pre-print

In a similar way to the classical case, we also develop a notion

arXiv:1501.07540v2
fatcat:s7pkysr5wvay5av4nniwsnlj5a
*of**geometric*category for*simplicial**complexes*. ... This category has the property*of*being homotopy invariant under strong equivalences, and only depends on the*simplicial*structure rather than its*geometric**realization*. ... Unlike other topological notions established for the*geometric**realization**of*the*complex*, our approach is directly based on the*simplicial*structure. ...##
###
Page 334 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 87, Issue 2
[page]

1958
*
American Mathematical Society. Transactions of the American Mathematical Society
*

For every c.s.s.

*complex*K K@z is its*geometrical**realization*(by a CW-*complex**of*which the n-cells are in one- to-one correspondence with the nondegenerate simplices*of*K; (see [6]). ... For every topological space X uX: AY(2, X) @ TX is the (natural) map*of*the*geometrical**realization**of*the*simplicial*singular com- plex*of*X onto X (see [6]}). ...##
###
An elementary illustrated introduction to simplicial sets
[article]

2021
*
arXiv
*
pre-print

This is an expository introduction to

arXiv:0809.4221v7
fatcat:v27cyapxy5fwvilyntbsctoocy
*simplicial*sets and*simplicial*homotopy theory with particular focus on relating the combinatorial aspects*of*the theory to their*geometric*/topological origins. ... It is intended to be accessible to students familiar with just the fundamentals*of*algebraic topology. ... Theorem 4.9 that the*realization**of*a*simplicial*set is always a CW*complex*. ...##
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EG for systolic groups

2009
*
Commentarii Mathematici Helvetici
*

Systolic group,

doi:10.4171/cmh/156
fatcat:aus777a47nhk7gjkaa3xsiuhra
*simplicial*nonpositive curvature, classifying space for the family*of*finite subgroups. ... ., if it acts properly and cocompactly on a systolic*complex*X, then an appropriate Rips*complex*constructed from X is a finite model for EG. Mathematics Subject Classification (2000). 20F67, 20F65. ... The*geometric**realization**of*a poset C is the*simplicial**complex*, whose set*of*vertices is the set*of*objects*of*C and a simplex is spanned on each subset which forms a chain. ...##
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Monoids, Segal's condition and bisimplicial spaces
[article]

2015
*
arXiv
*
pre-print

The condition that characterizes such

arXiv:1505.05010v1
fatcat:ldhymqij6nevhao5i5yi34pkwy
*simplicial*objects is a strictification*of*Segal's condition guaranteeing that the loop space*of*the*geometric**realization**of*a*simplicial*space X and the space X_ ... This generalization gives conditions guaranteing that the double loop space*of*the*geometric**realization**of*a bisimplicial space X and the space X_11 are*of*the same homotopy type. ... It turns out that this condition is a strictification*of*Segal's condition guaranteeing that the loop space*of*the*geometric**realization**of*a*simplicial*space X and the space X 1 are*of*the same homotopy ...
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