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Intersection numbers of polygon spaces


Authors: José Agapito and Leonor Godinho
Journal: Trans. Amer. Math. Soc. 361 (2009), 4969-4997
MSC (2000): Primary 53D20, 58D99; Secondary 53D35
DOI: https://doi.org/10.1090/S0002-9947-09-04796-5
Published electronically: April 21, 2009
MathSciNet review: 2506433
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Abstract: We study the intersection ring of the space $ \mathcal{M}(\alpha_1,\ldots,\alpha_m)$ of polygons in $ \mathbb{R}^3$. We find homology cycles dual to generators of this ring and prove a recursion relation in $ m$ (the number of edges) for their intersection numbers. This result is an analog of the recursion relation appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves and in the work of Weitsman on moduli spaces of flat connections on two-manifolds of genus $ g$ with $ m$ marked points. Based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples. Among others, we study the special case of equilateral polygon spaces (where all $ \alpha_i$'s are the same) and compare our results with the expressions for these particular spaces that have been determined by Kamiyama and Tezuka. Finally, we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of Yoshida, as well as with equivalent expressions for polygon spaces obtained by Takakura and Konno through different techniques.


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Additional Information

José Agapito
Affiliation: Departamento De Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
Email: agapito@math.ist.utl.pt

Leonor Godinho
Affiliation: Departamento De Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
Email: lgodin@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9947-09-04796-5
Received by editor(s): November 2, 2007
Published electronically: April 21, 2009
Additional Notes: The first author was partially supported by FCT (Portugal) through program POCTI/FEDER and grant POCTI/SFRH/BPD/20002/2004
The second author was partially supported by FCT through program POCTI/FEDER and grant POCTI/MAT/57888/2004, and by Fundação Calouste Gulbenkian.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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