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Eric Babson

Publications and source records attributed to Eric Babson.

27 records · Page 2Linked to original sources
arXiv2005

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 Lovász 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 $χ(G)\geq k+m$; 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.

math.CO↗
arXiv2003

Group Actions on Posets

In this paper we study quotients of posets by group actions. In order to define the quotient correctly we enlarge the considered class of categories from posets to loopfree categories: categories without nontrivial automorphisms and inverses. We view group actions as certain functors and define the quotients as colimits of these functors. The advantage of this definition over studying the quotient poset (which in our language is the colimit in the poset category) is that the realization of the quotient loopfree category is more often homeomorphic to the quotient of the realization of the original poset. We give conditions under which the quotient commutes with the nerve functor, as well as conditions which guarantee that the quotient is again a poset.

math.CT↗
arXiv2003

Topological obstructions to graph colorings

For any two graphs $G$ and $H$ Lovász has defined a cell complex $Hom(G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with $G$ a cycle of odd length. More specifically, we show that: if $Hom(C_{2r+1},G)$ is $k$-connected, then $χ(G)\geq k+4$. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney classes.

math.CO↗
arXiv2002

Geometry of the tetrahedron space

Let X^{circ} be the space of all labeled tetrahedra in P^{3}. In [BGS] we constructed a smooth symmetric compactification X-tilde of X^{circ}. In this article we show that the complement X-tilde smallsetminus X^{circ} is a divisor with normal crossings, and we compute the cohomology ring H^*(X-tilde;Q).

math.AG↗
arXiv2002

The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases

We provide a polynomial time algorithm for computing the universal Gröbner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the state polyhedron of any member of the Hilbert scheme Hilb^d_n of n-long d-variate ideals, enabled by introducing the Hilbert zonotope H^d_n and showing that it simultaneously refines all state polyhedra of ideals on Hilb^d_n.

math.CO↗
arXiv2002

Symmetric iterated Betti numbers

We define a set of invariants of a homogeneous ideal $I$ in a polynomial ring called the symmetric iterated Betti numbers of $I$. For $I_Γ$, the Stanley-Reisner ideal of a simplicial complex $Γ$, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of $Γ$ introduced by Duval and Rose. We show that the symmetric iterated Betti numbers of an ideal $I$ coincide with those of a particular reverse lexicographic generic initial ideal $\Gin(I)$ of $I$, and interpret these invariants in terms of the associated primes and standard pairs of $\Gin(I)$. We verify that for an ideal $I=I_Γ$ the extremal Betti numbers of $I_Γ$ are precisely the extremal (symmetric or exterior) iterated Betti numbers of $Γ$. We close with some results and conjectures about the relationship between symmetric and exterior iterated Betti numbers of a simplicial complex.

math.CO↗
arXiv1997

Complexes of not $i$-connected graphs

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not $(n-2)$-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not $(n-3)$-connected graphs we provide a formula for the generating function of the Euler characteristic.

math.CO↗
arXiv1997

Diagrams of classifying spaces and $k$-fold Boolean algebras

In this paper we study the problem of determining the homology groups of a quotient of a topological space by an action of a group. The method is to represent the original topological space as a homotopy limit of a diagram, and then act with the group on that diagram. Once it is possible to understand what the action of the group on every space in the diagram is, and what it does to the morphisms, we can compute the homology groups of the homotopy limit of this quotient diagram. Our motivating example is the symmetric deleted join of a simplicial complex. It can be represented as a diagram of symmetric deleted products. In the case where the simplicial complex in question is a simplex, we perform the complete computation of the homology groups with $\mathbb Z_p$ coefficients. For the infinite simplex the spaces in the quotient diagram are classifying spaces of various direct products of symmetric groups and diagram morphisms are induced by group homomorphisms. Combining Nakaoka's description of the $\mathbb Z_p$-homology of the symmetric group with a spectral sequence, we reduce the computation to an essentially combinatorial problem, which we then solve using the braid stratification of a sphere. Finally, we give another description of the problem in terms of posets and complete the computation for the case of a finite simplex.

math.CO↗