On the topology of graph picture spaces

Abstract

We study the space ${\mathcal X}^{d}(G)$ of pictures of a graph $G$ in complex projective $d$-space. The main result is that the homology groups (with integer coefficients) of ${\mathcal X}^{d}(G)$ are completely determined by the Tutte polynomial of $G$. One application is a criterion in terms of the Tutte polynomial for independence in the {\it $d$-parallel matroids} studied in combinatorial rigidity theory. For certain special graphs called \defterm{orchards}, the picture space is smooth and has the structure of an iterated projective bundle. We give a Borel presentation of the cohomology ring of the picture space of an orchard, and use this presentation to develop an analogue of the classical Schubert calculus.