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On the topology of graph picture spaces | | Jeremy L. Martin
; | | Date: |
31 Jul 2003 | | Journal: | Adv. Math. 191, no. 2 (2005), 312--338 | | Subject: | Combinatorics; Algebraic Geometry; Algebraic Topology MSC-class: 05C10 (Primary) 05B35, 14N20, 52C35 (Secondary) | math.CO math.AG math.AT | | 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. | | Source: | arXiv, math.CO/0307405 | | Services: | Forum | Review | PDF | Favorites | |
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