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Geometric Realization of Simplicial Complexes
[chapter]
1999
Lecture Notes in Computer Science
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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]
2007
arXiv
pre-print
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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
1957
Annals of Mathematics
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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
2014
Advances in Mathematics
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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
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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
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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
2019
Notices of the American Mathematical Society
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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]
2020
arXiv
pre-print
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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]
2011
arXiv
pre-print
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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
2014
Applied Mathematical Sciences
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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]
2015
arXiv
pre-print
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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
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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
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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
2009
Commentarii Mathematici Helvetici
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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]
2015
arXiv
pre-print
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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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