The cohomology ring of polygon spaces

Hausmann, Jean-Claude; Knutson, Allen

Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 281-321.

On calcule l’anneau de cohomologie entière des espaces de configurations de polygones dans ${\mathbf{R}}^3$ tels qu’ils sont introduits dans [F. Kirwan, Cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 853-906] et [M. Kapovich & J. Millson, the symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513]. Pour cela, on plonge ces espaces dans certaines variétés toriques; on montre que le plongement obtenu induit un épimorphisme en cohomologie dont on calcule le noyau à l’aide des technique des bases de Gröbner. En tensorisant par $\mathbf{Z}/2\mathbf{Z}$ et en divisant les degrés par 2, on obtient l’anneau de cohomologie $\mathrm{mod}2$ des espaces de polygones planaires. Dans le cas d’un polygone équilatéral, où le groupe symétrique agit en permutant les arêtes, on montre que l’action induite sur le second groupe de cohomologie n’est pas standard, bien qu’elle le soit pour la cohomologie rationnelle, cf.F. Kirwan, op. loc. On obtient aussi des expressions pour les polynômes de Poincaré qui sont calculatoirement plus efficaces que celles connues jusqu’ici.

We compute the integer cohomology rings of the “polygon spaces”introduced in [F. Kirwan, Cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 853-906] and [M. Kapovich & J. Millson, the symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Gröbner bases. Since we do not invert the prime 2, we can tensor with ${\mathbf{Z}}_2$; halving all degrees we show this produces the ${\mathbf{Z}}_2$ cohomology rings of the planar polygon spaces. In the equilateral case, where there is an action of the symmetric group permuting the edges, we show that the induced action on the integer cohomology is not the standard one, despite it being so on the rational cohomology, cf.F. Kirwan, op. loc. Finally, our formulae for the Poincaré polynomials are more computationally effective than those known, cf.F. Kirwan op. loc.

[Br] M. Brion, Cohomologie équivariante des points semi-stables, J. Reine Angew. Math., 421 (1991), 125-140. | MR | Zbl

[DM] P. Deligne & G. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publication de l'IHES, 43 (1986), 5-90. | Numdam | MR | Zbl

[DJ] M. Davis & T. Januszkiewicz Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62 (1991), 417-451. | MR | Zbl

[Ei] D. Eisenbud, Commutative algebra, Springer-Verlag, 1995. | MR | Zbl

[Fu] W. Fulton, Introduction to toric varieties, Princeton Univ. Press, 1993. | MR | Zbl

[GGMS] I. Gel'Fand & M.Goresky & R. Macpherson & V. Serganova, Combinatorial geometries, convex polyedra and Schubert cells, Adv. Math., 63 (1987), 301-316. | MR | Zbl

[GM] I. Gel'Fand & R. Macpherson Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., 44 (1982), 279-312. | MR | Zbl

[GH] P. Griffiths & J. Harris, Principles of algebraic geometry, Wiley Classics Library, 1978. | MR | Zbl

[GS] V. Guillemin, & S. Sternberg, The coefficients of the Duistermaat-Heckman polynomial and the cohomology ring of reduced spaces. Geometry, topology, and physics, Conf. Proc. Lecture Notes Geom. Topology, VI International Press, 1995. | MR | Zbl

[Gu] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian Tn-spaces, Birkhäuser, 1994. | MR | Zbl

[Ha] J-C. Hausmann, Sur la topologie des bras articulés. In “Algebraic Topology, Poznan”, Springer Lectures Notes, 1474 (1989), 146-159. | MR | Zbl

[HK] J-C. Hausmann & A. Knutson, Polygon spaces and Grassmannians, L'Enseignement Mathématique, 43 (1997), 173-198. | MR | Zbl

[Hu] D. Husemoller, Fibre bundles (2nd ed.), Springer Verlag, 1975. | MR | Zbl

[KM] M. Kapovich, & J.Millson, The symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513. | MR | Zbl

[K1] F. Kirwan, The cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 902 and 904. | MR | Zbl

[K2] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984. | MR | Zbl

[K1] A. Klyachko, Spatial polygons and stable configurations of points in the projective line. in : Algebraic geometry and its applications (Yaroslavl, 1992), Aspects Math., Vieweg, Braunschweig (1994) 67-84. | MR | Zbl

[Le] E. Lerman, Symplectic cuts, Math. Res. Lett., 2 (1995), 247-258. | MR | Zbl

[Ma] S. Martin, Doctoral thesis (in preparation).

[MS] J. Milnor & J. Stasheff, Characteristic Classes, Princeton Univ. Press, 1974. | MR | Zbl

[Popp] H. Popp, Moduli theory and classification of algebraic varieties, Springer-Verlag, 1977. | MR | Zbl

[Sp] E. Spanier, Algebraic topology, McGraw-Hill, 1966. | MR | Zbl