Homotopy types of Hom complexes of graphs

The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the chromatic number of $G$. For a fixed finite graph $T$, we show that there is no homotopy invariant of ${\rm Hom}(T,G)$ which gives an upper bound for the chromatic number of $G$. More precisely, for a non-bipartite graph $G$, we construct a graph $H$ such that ${\rm Hom}(T,G)$ and ${\rm Hom}(T,H)$ are homotopy equivalent but $\chi(H)$ is much larger than $\chi(G)$. The equivariant homotopy type of ${\rm Hom}(T,G)$ is also considered.Comment: 13 pages, final version. In the present version, the proofs are simplifie

Complexes of graph homomorphisms

$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lov\'asz to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$ is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have \varpi_1^k(\thom(K_m,G))\neq 0, then $\chi(G)\geq k+m$; here $Z_2$-action is induced by the swapping of two vertices in $K_m$, and $\varpi_1$ is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of $Hom(G,H)$ induces a homotopy equivalence. It then follows that $Hom(F,K_n)$ is homotopy equivalent to a direct product of $(n-2)$-dimensional spheres, while $Hom(\bar{F},K_n)$ is homotopy equivalent to a wedge of spheres, where $F$ is an arbitrary forest and $\bar{F}$ is its complement.Comment: This is the first part of the series of papers containing the complete proofs of the results announced in "Topological obstructions to graph colorings". This is the final version which is to appear in Israel J. Math., it has an updated list of references and new remarks on latest development

Topology of Hom complexes and test graphs for bounding chromatic number

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.Comment: 42 pages, 3 figures; incorporated referee's comments and corrections, to appear in Israel J. Math

On inverse powers of graphs and topological implications of Hedetniemi's conjecture

We consider a natural graph operation $\Omega_k$ that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the $\mathbb{Z}_2$-homotopy type) of the box complex, a basic tool in topological combinatorics. Moreover, we prove that the box complex of a graph G admits a $\mathbb{Z}_2$-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph $\Omega_k(G)$ admits a homomorphism to H, for high enough k. This allows to show that if Hedetniemi's conjecture on the chromatic number of graph products were true for n-colorings, then the following analogous conjecture in topology would also also true: If X,Y are $\mathbb{Z}_2$-spaces (finite $\mathbb{Z}_2$-simplicial complexes) such that X x Y admits a $\mathbb{Z}_2$-map to the (n-2)-dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture

Derived equivalence classification of nonstandard selfinjective algebras of domestic type

We give a complete derived equivalence classification of all nonstandard representation-infinite domestic selfinjective algebras over an algebraically closed field. As a consequence, also a complete stable equivalence classification of these algebras is obtained.Comment: 13 pages, 7 figure

Paths of homomorphisms from stable Kneser graphs

We denote by SG_{n,k} the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k. For k congruent 3 (mod 4) and n\ge2 we show that there is a component of the \chi-colouring graph of SG_{n,k} which is invariant under the action of the automorphism group of SG_{n,k}. We derive that there is a graph G with \chi(G)=\chi(SG_{n,k}) such that the complex Hom(SG_{n,k}, G) is non-empty and connected. In particular, for k congruent 3 (mod 4) and n\ge2 the graph SG_{n,k} is not a test graph.Comment: 7 page

Homological Coordinatization

In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in the context of topological data analysis, can be viewed as an extension of conventional unsupervised learning methods to simplicial complexes

Perverse sheaves of categories and some applications

We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on $\mathbb{P}^1$ bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi--Yau varieties.Comment: Material on mirror symmetry and noncommutative projective planes removed. Many other changes made at referee's request. Comments still welcome! 40 page

Matching groups and gliding systems

With every matching in a graph we associate a group called the matching group. We study this group using the theory of non-positively curved cubed complexes. Our approach is formulated in terms of so-called gliding systems.Comment: 24 pages, 1 figure. This paper generalizes and replaces my paper on dimer spaces, arXiv:1211.397

Andre-Quillen homology of commutative algebras

These notes are an introduction to basic properties of Andre-Quillen homology for commutative algebras. They are an expanded version of my lectures at the summer school: Interactions between homotopy theory and algebra, University of Chicago, 26th July - 6th August, 2004. The aim is to give fairly complete proofs of characterizations of smooth homomorphisms and of locally complete intersection homomorphisms in terms of vanishing of Andre-Quillen homology. The choice of the material, and the point of view, are guided by these goals.Comment: 32 pages; to appear in: Interactions between homotopy theory and algebra (Chicago 2004), Contemp. Mat