Intersection numbers of polygon spaces
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- by José Agapito and Leonor Godinho PDF
- Trans. Amer. Math. Soc. 361 (2009), 4969-4997 Request permission
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.References
- Michel Brion, Cohomologie équivariante des points semi-stables, J. Reine Angew. Math. 421 (1991), 125–140 (French). MR 1129578, DOI 10.1515/crll.1991.421.125
- J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259–268. MR 674406, DOI 10.1007/BF01399506
- Victor Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Mathematics, vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301331, DOI 10.1007/978-1-4612-0269-1
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
- Jean-Claude Hausmann, Sur la topologie des bras articulés, Algebraic topology Poznań 1989, Lecture Notes in Math., vol. 1474, Springer, Berlin, 1991, pp. 146–159 (French). MR 1133898, DOI 10.1007/BFb0084743
- Jean-Claude Hausmann and Allen Knutson, Polygon spaces and Grassmannians, Enseign. Math. (2) 43 (1997), no. 1-2, 173–198. MR 1460127
- J.-C. Hausmann and A. Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 281–321 (English, with English and French summaries). MR 1614965, DOI 10.5802/aif.1619
- Lisa C. Jeffrey, Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann. 298 (1994), no. 4, 667–692. MR 1268599, DOI 10.1007/BF01459756
- Lisa Jeffrey and Jonathan Weitsman, Toric structures on the moduli space of flat connections on a Riemann surface. II. Inductive decomposition of the moduli space, Math. Ann. 307 (1997), no. 1, 93–108. MR 1427677, DOI 10.1007/s002080050024
- Yael Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672, viii+71. MR 1612833, DOI 10.1090/memo/0672
- Frances Kirwan, The cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), no. 4, 853–906. MR 1145826, DOI 10.1090/S0894-0347-1992-1145826-8
- Alexander A. Klyachko, Spatial polygons and stable configurations of points in the projective line, Algebraic geometry and its applications (Yaroslavl′, 1992) Aspects Math., E25, Friedr. Vieweg, Braunschweig, 1994, pp. 67–84. MR 1282021
- Hiroshi Konno, The intersection pairings on the configuration spaces of points in the projective line, J. Math. Kyoto Univ. 41 (2001), no. 2, 277–284. MR 1852984, DOI 10.1215/kjm/1250517633
- Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23. MR 1171758, DOI 10.1007/BF02099526
- Michael Kapovich and John J. Millson, The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), no. 3, 479–513. MR 1431002
- Yasuhiko Kamiyama and Michishige Tezuka, Symplectic volume of the moduli space of spatial polygons, J. Math. Kyoto Univ. 39 (1999), no. 3, 557–575. MR 1718781, DOI 10.1215/kjm/1250517868
- Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
- A. Mandini, The geometry of the moduli space of polygons in the Euclidean space, Ph.D. Thesis, Università di Bologna, 2007.
- Tatsuru Takakura, Intersection theory on symplectic quotients of products of spheres, Internat. J. Math. 12 (2001), no. 1, 97–111. MR 1812066, DOI 10.1142/S0129167X01000678
- Vu The Khoi, On the symplectic volume of the moduli space of spherical and Euclidean polygons, Kodai Math. J. 28 (2005), no. 1, 199–208. MR 2122200, DOI 10.2996/kmj/1111588046
- K. Walker, Configuration spaces of linkages, Undergraduate Thesis, Princeton (1985).
- Jonathan Weitsman, Geometry of the intersection ring of the moduli space of flat connections and the conjectures of Newstead and Witten, Topology 37 (1998), no. 1, 115–132. MR 1480881, DOI 10.1016/S0040-9383(96)00036-5
- Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR 1144529
- Edward Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209. MR 1133264, DOI 10.1007/BF02100009
- Edward Witten, On the Kontsevich model and other models of two-dimensional gravity, Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 176–216. MR 1225112
- Takahiko Yoshida, The generating function for certain cohomology intersection pairings of the moduli space of flat connections, J. Math. Sci. Univ. Tokyo 8 (2001), no. 3, 541–558. MR 1855458
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
- MR Author ID: 684216
- ORCID: 0000-0002-6329-3002
- Email: lgodin@math.ist.utl.pt
- 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. - © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2506433