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Geometric Realization of Simplicial Complexes [chapter]

Patrice Ossona de Mendez
1999 Lecture Notes in Computer Science  
We show that an abstract 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 ∆.  ... 
doi:10.1007/3-540-46648-7_33 fatcat:eewmmrfn4bf27p4vnd3jl5otxi

Necessary Conditions for Geometric Realizability of Simplicial Complexes [article]

Dagmar Timmreck
2007 arXiv   pre-print
We associate with any 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.  ... 
arXiv:0705.1912v2 fatcat:fqaizppyobfehbec2o4v62kofy

The Geometric Realization of a Semi-Simplicial Complex

John Milnor
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 .  ... 
doi:10.2307/1969967 fatcat:3hjj3hoqongytjq47vzeuxd6ey

Helly numbers of acyclic families

Éric Colin de Verdière, Grégory Ginot, Xavier Goaoc
2014 Advances in Mathematics  
As an application, we obtain several explicit bounds on Helly numbers in 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.  ... 
doi:10.1016/j.aim.2013.11.004 fatcat:3yns3opur5ghzkqspopjix32t4

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

Andrew J. Blumberg, Michael A. Mandell
2013 Foundations of Computational Mathematics  
We introduce the "contiguity 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.  ... 
doi:10.1007/s10208-013-9177-5 fatcat:b364a6ozxjefxbc3tqzh2ljgzm

Fall Sectional Sampler: Face Numbers: Centrally Symmetric Spheres versus Centrally Symmetric Polytopes

Isabella Novik
2019 Notices of the American Mathematical Society  
Each 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.  ... 
doi:10.1090/noti1966 fatcat:zi6gqz36vneu3bsgop2aux5mly

Contiguity Distance between Simplicial Maps [article]

Ayse Borat, Mehmetcik Pamuk, Tane Vergili
2020 arXiv   pre-print
In particular, we show that 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].  ... 
arXiv:2012.10627v1 fatcat:t7vqszmkivdeni5boujjn5fqoy

Free loop space and homology [article]

Jean-Louis Loday
2011 arXiv   pre-print
We introduce all the relevant technical tools, namely 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 · ).  ... 
arXiv:1110.0405v1 fatcat:epo2ssqarzg77mwvzqwmwalesu

Simplicial complexes: from continuous to discrete

Sara Dragotti
2014 Applied Mathematical Sciences  
We expose some basic concepts 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.  ... 
doi:10.12988/ams.2014.49691 fatcat:zbvzvyoyrjf53oxthyn5qjqfay

Lusternik-Schnirelmann category of simplicial complexes and finite spaces [article]

D. Fernández-Ternero and E. Macías-Virgós and J.A. Vilches
2015 arXiv   pre-print
In a similar way to the classical case, we also develop a notion 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.  ... 
arXiv:1501.07540v2 fatcat:s7pkysr5wvay5av4nniwsnlj5a

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]

Greg Friedman
2021 arXiv   pre-print
This is an expository introduction to 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.  ... 
arXiv:0809.4221v7 fatcat:v27cyapxy5fwvilyntbsctoocy

EG for systolic groups

Piotr Przytycki
2009 Commentarii Mathematici Helvetici  
Systolic group, 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.  ... 
doi:10.4171/cmh/156 fatcat:aus777a47nhk7gjkaa3xsiuhra

Monoids, Segal's condition and bisimplicial spaces [article]

Zoran Petric
2015 arXiv   pre-print
The condition that characterizes such 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  ... 
arXiv:1505.05010v1 fatcat:ldhymqij6nevhao5i5yi34pkwy
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