On the topology of graph picture spaces

Details On the topology of graph picture spaces On the topology of graph picture spaces

2003-07-31

arxiv.org/abs/math/0307405v2

Adv. Math. 191, no. 2 (2005), 312--338

Mathematics

Combinatorics

LaTeX (uses xypic package), 22 pages. Final version, to appear in Advances in Mathematics. The title has been changed slightly

Scientific paper

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.

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