Moduli spaces of geometric graphs

Abstract: In this talk I will investigate the structure of the "moduli space" $W(G,d)$ of a geometric graph $G$, i.e. the set of all possible geometric realizations in $\mathbb R^d$ of a given graph $G$ on $n$ vertices. Such moduli space is Spanier–Whitehead dual to a real algebraic discriminant.

For example, in the case of geometric realizations of $G$ on the real line, the moduli space $W(G, 1)$ is a component of the complement of a hyperplane arrangement in $\mathbb R^n$. (Another example: when $G$ is the empty graph on n vertices, $W(G, d)$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb R^d$.) Numerous questions about graph enumeration can be formulated in terms of the topology of this moduli space.

I will explain how to associate to a graph $G$ a new graph invariant which encodes the asymptotic structure of the moduli space when $d$ goes to infinity, for fixed $G$. Surprisingly, the sum of the Betti numbers of $W(G,d)$ stabilizes as $d$ goes to infinity, and gives the claimed graph invariant $B(G)$, even though the cohomology of $W(G,d)$ "shifts" its dimension. We call the invariant $B(G)$ the "Floer number" of the graph $G$, as its construction is reminiscent of Floer theory from symplectic geometry.

Joint work with M. Belotti and A. Newman.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

Topology and geometry: extremal and typical

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