Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe universality of Hom complexes
88
0
0.0
(
0
)
Authors
Anton Dochtermann
Ask ChatGPT about the research
No Arabic abstract
It is shown that if T is a connected nontrivial graph and X is an arbitrary finite simplicial complex, then there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials, and subdivision are established that may be of independent interest.
rate research
Read More
The notion of $times$-homotopy from cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space $Hom_*(G,H)$ with the homotopy groups of $Hom_*(G,H^I)$. Here $Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $pi_i big(Hom_*(G,H) big) cong [G,Omega^i H]_{times}$, where $Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{times}$ is the set of $times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.
We investigate a notion of $times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $times$-homotopy is characterized by the topological properties of the $Hom$ complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; $Hom$ complexes were introduced by Lov{a}sz and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of $Hom$ complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. Graph $times$-homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of $times$-homotopy equivalence to the class of dismantlable graphs to get a list of conditions that again characterize these. We end with a discussion of graph homotopies arising from other internal homs, including the construction of `$A$-theory associated to the cartesian product in the category of reflexive graphs.
We introduce new methods for understanding the topology of $Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $Hom(T,-)$ and $Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $Hom$ complexes in terms of spaces of equivariant poset maps and $Gamma$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $X$, this provides a useful way to control the topology of $Hom$ complexes. Our foremost application of these results is the construction of new families of `test graphs with arbitrarily large chromatic number - graphs $T$ with the property that the connectivity of $Hom(T,G)$ provides the best possible lower bound on the chromatic number of $G$. In particular we focus on two infinite families, which we view as higher dimensional analogues of odd cycles. The family of `spherical graphs have connections to the notion of homomorphism duality, whereas the family of `twisted toroidal graphs lead us to establish a weakened version of a conjecture (due to Lov{a}sz) relating topological lower bounds on chromatic number to maximum degree. Other structural results allow us to show that any finite simplicial complex $X$ with a free action by the symmetric group $S_n$ can be approximated up to $S_n$-homotopy equivalence as $Hom(K_n,G)$ for some graph $G$; this is a generalization of a result of Csorba. We conclude the paper with some discussion regarding the underlying categorical notions involved in our study.
We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.
The family of contractible graphs, introduced by A. Ivashchenko, consists of the collection $mathfrak{I}$ of graphs constructed recursively from $K_1$ by contractible transformations. In this paper we show that every graph in a subfamily of $mathfrak{I}$ (the strongly contractible ones) is a collapsible graph (in the simplicial sense), by providing a sequence of elementary collapses induced by removing contractible vertices or edges. In addition, we introduce an algorithm to identify the contractible vertices in any graph and show that there is a natural homomorphism, induced by the inclusion map of graphs, between the homology groups of the clique complex of graphs with the contractible vertices removed. Finally, we show an application of this result to the computation of the persistent homology for the Vietoris-Rips filtration.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
