We show that J_n, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron A_n. This resolution is not minimal for n > 5; in this case the Betti numbers of J_n are strictly smaller th
an the f-vector of A_n. We show that in fact the Betti numbers of J_n are in bijection with the number of standard Young tableaux of shape (d+1, 2, 1^{n-d-3}). This complements the fact that the number of (d-1)-dimensional faces of A_n are given by the number of standard Young tableaux of (super)shape (d+1, d+1, 1^{n-d-3}); a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of J_n that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.
One can iteratively obtain a free resolution of any monomial ideal $I$ by considering the mapping cone of the map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal resolutio
n if $I$ has linear quotients, in which case the mapping cone in each step cones a Koszul complex onto the previously constructed resolution. Here we consider cellular realizations of these resolutions. Extending a construction of Mermin we describe a regular CW-complex that supports the resolutions of Herzog and Takayama in the case that $I$ has a `regular decomposition function. By varying the choice of chain map we recover other known cellular resolutions, including the `box of complexes resolutions of Corso, Nagel, and Reiner and the related `homomorphism complex resolutions of Dochtermann and Engstrom. Other choices yield combinatorially distinct complexes with interesting structure, and suggests a notion of a `space of cellular resolutions.
The Laplacian matrix of a graph G describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The lattice ideal associated to the lattice generated by the columns of the Laplacian provides an alge
braic perspective on this recently (re)emerging field. This ideal I_G has a distinguished monomial initial ideal M_G, characterized by the property that the standard monomials are in bijection with the G-parking functions of the graph G. The ideal M_G was also introduced by Postnikov and Shapiro (2004) in the context of monotone monomial ideals. We study resolutions of M_G and show that a minimal free cellular resolution is supported on the bounded subcomplex of a section of the graphical arrangement of G. This generalizes constructions from Postnikov and Shapiro (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and of Perkinson et al. on the commutative algebra of Sandpiles. As a corollary we verify a conjecture of Perkinson et al. regarding the Betti numbers of M_G, and in the process provide a combinatorial characterization in terms of acyclic orientations.
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.
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.
