The cohomology ring of polygon spaces
Annales de l'Institut Fourier, Tome 48 (1998) no. 1, p. 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.
@article{AIF_1998__48_1_281_0,
author = {Hausmann, Jean-Claude and Knutson, Allen},
title = {The cohomology ring of polygon spaces},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {48},
number = {1},
year = {1998},
pages = {281-321},
doi = {10.5802/aif.1619},
mrnumber = {1614965},
zbl = {0903.14019},
mrnumber = {99a:58027},
language = {en},
url = {http://www.numdam.org/item/AIF_1998__48_1_281_0}
}

Hausmann, Jean-Claude; Knutson, Allen. The cohomology ring of polygon spaces. Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 281-321. doi : 10.5802/aif.1619. http://www.numdam.org/item/AIF_1998__48_1_281_0/

[Br] M. Brion, Cohomologie équivariante des points semi-stables, J. Reine Angew. Math., 421 (1991), 125-140. | MR 92i:14010 | Zbl 0729.14015

[DM] P. Deligne & G. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publication de l'IHES, 43 (1986), 5-90. | Numdam | MR 88a:22023a | Zbl 0615.22008

[DJ] M. Davis & T. Januszkiewicz Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62 (1991), 417-451. | MR 92i:52012 | Zbl 0733.52006

[Ei] D. Eisenbud, Commutative algebra, Springer-Verlag, 1995. | MR 97a:13001 | Zbl 0819.13001

[Fu] W. Fulton, Introduction to toric varieties, Princeton Univ. Press, 1993. | MR 94g:14028 | Zbl 0813.14039

[GGMS] I. Gel'Fand & M.Goresky & R. Macpherson & V. Serganova, Combinatorial geometries, convex polyedra and Schubert cells, Adv. Math., 63 (1987), 301-316. | MR 88f:14045 | Zbl 0626.00007

[GM] I. Gel'Fand & R. Macpherson Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., 44 (1982), 279-312. | MR 84b:57014 | Zbl 0504.57021

[GH] P. Griffiths & J. Harris, Principles of algebraic geometry, Wiley Classics Library, 1978. | MR 80b:14001 | Zbl 0408.14001

[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 96k:58082 | Zbl 0869.57029

[Gu] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian Tn-spaces, Birkhäuser, 1994. | MR 96e:58064 | Zbl 0828.58001

[Ha] J-C. Hausmann, Sur la topologie des bras articulés. In “Algebraic Topology, Poznan”, Springer Lectures Notes, 1474 (1989), 146-159. | MR 93a:57035 | Zbl 0736.57014

[HK] J-C. Hausmann & A. Knutson, Polygon spaces and Grassmannians, L'Enseignement Mathématique, 43 (1997), 173-198. | MR 98e:58035 | Zbl 0888.58007

[Hu] D. Husemoller, Fibre bundles (2nd ed.), Springer Verlag, 1975. | MR 51 #6805 | Zbl 0307.55015

[KM] M. Kapovich, & J.Millson, The symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513. | MR 98a:58027 | Zbl 0889.58017

[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 93g:14016 | Zbl 0804.14010

[K2] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984. | MR 86i:58050 | Zbl 0553.14020

[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 95k:14015 | Zbl 0820.51016

[Le] E. Lerman, Symplectic cuts, Math. Res. Lett., 2 (1995), 247-258. | MR 96f:58062 | Zbl 0835.53034

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

[MS] J. Milnor & J. Stasheff, Characteristic Classes, Princeton Univ. Press, 1974. | MR 55 #13428 | Zbl 0298.57008

[Popp] H. Popp, Moduli theory and classification of algebraic varieties, Springer-Verlag, 1977. | MR 57 #6024 | Zbl 0359.14005

[Sp] E. Spanier, Algebraic topology, McGraw-Hill, 1966. | MR 35 #1007 | Zbl 0145.43303