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

Published electronically:
April 21, 2009

MathSciNet review:
2506433

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the intersection ring of the space of polygons in . We find homology cycles dual to generators of this ring and prove a recursion relation in (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 with 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 '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.

**[1]**Michel Brion,*Cohomologie équivariante des points semi-stables*, J. Reine Angew. Math.**421**(1991), 125–140 (French). MR**1129578**, 10.1515/crll.1991.421.125**[2]**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**, 10.1007/BF01399506**[3]**Victor Guillemin,*Moment maps and combinatorial invariants of Hamiltonian 𝑇ⁿ-spaces*, Progress in Mathematics, vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994. MR**1301331****[4]**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****[5]**V. Guillemin and S. Sternberg,*Geometric quantization and multiplicities of group representations*, Invent. Math.**67**(1982), no. 3, 515–538. MR**664118**, 10.1007/BF01398934**[6]**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**, 10.1007/BFb0084743**[7]**Jean-Claude Hausmann and Allen Knutson,*Polygon spaces and Grassmannians*, Enseign. Math. (2)**43**(1997), no. 1-2, 173–198. MR**1460127****[8]**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****[9]**Lisa C. Jeffrey,*Extended moduli spaces of flat connections on Riemann surfaces*, Math. Ann.**298**(1994), no. 4, 667–692. MR**1268599**, 10.1007/BF01459756**[10]**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**, 10.1007/s002080050024**[11]**Yael Karshon,*Periodic Hamiltonian flows on four-dimensional manifolds*, Mem. Amer. Math. Soc.**141**(1999), no. 672, viii+71. MR**1612833**, 10.1090/memo/0672**[12]**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**, 10.1090/S0894-0347-1992-1145826-8**[13]**Alexander A. Klyachko,*Spatial polygons and stable configurations of points in the projective line*, Algebraic geometry and its applications (Yaroslavl′, 1992) Aspects Math., E25, Vieweg, Braunschweig, 1994, pp. 67–84. MR**1282021****[14]**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****[15]**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****[16]**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****[17]**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****[18]**Eugene Lerman,*Symplectic cuts*, Math. Res. Lett.**2**(1995), no. 3, 247–258. MR**1338784**, 10.4310/MRL.1995.v2.n3.a2**[19]**A. Mandini,*The geometry of the moduli space of polygons in the Euclidean space*, Ph.D. Thesis, Università di Bologna, 2007.**[20]**Tatsuru Takakura,*Intersection theory on symplectic quotients of products of spheres*, Internat. J. Math.**12**(2001), no. 1, 97–111. MR**1812066**, 10.1142/S0129167X01000678**[21]**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**, 10.2996/kmj/1111588046**[22]**K. Walker,*Configuration spaces of linkages*, Undergraduate Thesis, Princeton (1985).**[23]**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**, 10.1016/S0040-9383(96)00036-5**[24]**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****[25]**Edward Witten,*On quantum gauge theories in two dimensions*, Comm. Math. Phys.**141**(1991), no. 1, 153–209. MR**1133264****[26]**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****[27]**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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
53D20,
58D99,
53D35

Retrieve articles in all journals with MSC (2000): 53D20, 58D99, 53D35

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:
http://dx.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.