scholarly journals The Implementation of the Colored Abstract Simplicial Complex and Its Application to Mesh Generation

2019 ◽  
Vol 45 (3) ◽  
pp. 1-20 ◽  
Author(s):  
Christopher T. Lee ◽  
John B. Moody ◽  
Rommie E. Amaro ◽  
J. Andrew Mccammon ◽  
Michael J. Holst
Keyword(s):  
Simplicial Complex ◽  
Mesh Generation ◽  
Abstract Simplicial Complex

Cup-products for the polyhedral product functor

2012 ◽  
Vol 153 (3) ◽  
pp. 457-469 ◽  
Author(s):  
A. BAHRI ◽  
M. BENDERSKY ◽  
F. R. COHEN ◽  
S. GITLER
Keyword(s):  
Simplicial Complex ◽  
Cohomology Ring ◽  
Decomposition Theorem ◽  
Ring Structure ◽  
Polyhedral Product ◽  
Cup Product ◽  
Rational Cohomology ◽  
Geometric Decomposition ◽  
Abstract Simplicial Complex ◽  
Cup Products

AbstractDavis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

scholarly journals Cosheaf representations of relations and Dowker complexes

2021 ◽  
Author(s):  
Michael Robinson
Keyword(s):  
Weight Function ◽  
Simplicial Complex ◽  
Binary Relation ◽  
Homotopy Equivalent ◽  
Covariant Functor ◽  
The Matrix ◽  
Abstract Simplicial Complex ◽  
Integer Weight

AbstractThe Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

scholarly journals The Canonical Join Complex

10.37236/7866 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Emily Barnard
Keyword(s):  
Number Theory ◽  
Simplicial Complex ◽  
Finite Lattice ◽  
Noncrossing Partitions ◽  
Prime Factorization ◽  
Finite Lattices ◽  
Abstract Simplicial Complex ◽  
Order Ideals ◽  
Minimal Factorization

A canonical join representation is a certain minimal "factorization" of an element in a finite lattice $L$ analogous to the prime factorization of an integer from number theory. The expression $\bigvee A =w$ is the canonical join representation of $w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where "lowest" is made precise by comparing order ideals under containment). Canonical join representations appear in many familiar guises, with connections to comparability graphs and noncrossing partitions. When each element in $L$ has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets $A$ such that $\bigvee A$ is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of $L$.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

Mesh generation for computational analysis. Part I: Electromagnetic and technical considerations for mesh generation

1986 ◽  
Vol 3 (5) ◽  
pp. 190 ◽  
Author(s):  
T.I. Boubez ◽  
W.R.J. Funnell ◽  
D.A. Lowther ◽  
A.R. Pinchuk ◽  
P.P. Silvester
Keyword(s):  
Mesh Generation ◽  
Computational Analysis ◽  
Technical Considerations

Multiscale mesh generation on the sphere

2008 ◽  
Vol 58 (5-6) ◽  
pp. 461-473 ◽  
Author(s):  
Jonathan Lambrechts ◽  
Richard Comblen ◽  
Vincent Legat ◽  
Christophe Geuzaine ◽  
Jean-François Remacle
Keyword(s):  
Mesh Generation
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