# The cohomology ring of polygon spaces

Jean-Claude Hausmann; Allen Knutson

Annales de l'institut Fourier (1998)

- Volume: 48, Issue: 1, page 281-321
- ISSN: 0373-0956

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topHausmann, Jean-Claude, and Knutson, Allen. "The cohomology ring of polygon spaces." Annales de l'institut Fourier 48.1 (1998): 281-321. <http://eudml.org/doc/75280>.

@article{Hausmann1998,

abstract = {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 $\{\bf Z\}_2$; halving all degrees we show this produces the $\{\bf Z\}_2$ 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.},

author = {Hausmann, Jean-Claude, Knutson, Allen},

journal = {Annales de l'institut Fourier},

keywords = {polygon spaces; toric varieties; Gröbner bases; integer cohomology rings},

language = {eng},

number = {1},

pages = {281-321},

publisher = {Association des Annales de l'Institut Fourier},

title = {The cohomology ring of polygon spaces},

url = {http://eudml.org/doc/75280},

volume = {48},

year = {1998},

}

TY - JOUR

AU - Hausmann, Jean-Claude

AU - Knutson, Allen

TI - The cohomology ring of polygon spaces

JO - Annales de l'institut Fourier

PY - 1998

PB - Association des Annales de l'Institut Fourier

VL - 48

IS - 1

SP - 281

EP - 321

AB - 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 ${\bf Z}_2$; halving all degrees we show this produces the ${\bf Z}_2$ 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.

LA - eng

KW - polygon spaces; toric varieties; Gröbner bases; integer cohomology rings

UR - http://eudml.org/doc/75280

ER -

## References

top- [Br] M. BRION, Cohomologie équivariante des points semi-stables, J. Reine Angew. Math., 421 (1991), 125-140. Zbl0729.14015MR92i:14010
- [DM] P. DELIGNE & G. MOSTOW, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publication de l'IHES, 43 (1986), 5-90. Zbl0615.22008MR88a:22023a
- [DJ] M. DAVIS & T. JANUSZKIEWICZConvex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62 (1991), 417-451. Zbl0733.52006MR92i:52012
- [Ei] D. EISENBUD, Commutative algebra, Springer-Verlag, 1995. Zbl0819.13001MR97a:13001
- [Fu] W. FULTON, Introduction to toric varieties, Princeton Univ. Press, 1993. Zbl0813.14039MR94g:14028
- [GGMS] I. GEL'FAND & M.GORESKY & R. MACPHERSON & V. SERGANOVA, Combinatorial geometries, convex polyedra and Schubert cells, Adv. Math., 63 (1987), 301-316. Zbl0626.00007MR88f:14045
- [GM] I. GEL'FAND & R. MACPHERSONGeometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., 44 (1982), 279-312. Zbl0504.57021MR84b:57014
- [GH] P. GRIFFITHS & J. HARRIS, Principles of algebraic geometry, Wiley Classics Library, 1978. Zbl0408.14001MR80b: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. Zbl0869.57029MR96k:58082
- [Gu] V. GUILLEMIN, Moment maps and combinatorial invariants of Hamiltonian Tn-spaces, Birkhäuser, 1994. Zbl0828.58001MR96e:58064
- [Ha] J-C. HAUSMANN, Sur la topologie des bras articulés. In “Algebraic Topology, Poznan”, Springer Lectures Notes, 1474 (1989), 146-159. Zbl0736.57014MR93a:57035
- [HK] J-C. HAUSMANN & A. KNUTSON, Polygon spaces and Grassmannians, L'Enseignement Mathématique, 43 (1997), 173-198. Zbl0888.58007MR98e:58035
- [Hu] D. HUSEMOLLER, Fibre bundles (2nd ed.), Springer Verlag, 1975. Zbl0307.55015MR51 #6805
- [KM] M. KAPOVICH, & J.MILLSON, The symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513. Zbl0889.58017MR98a:58027
- [K1] F. KIRWAN, The cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 902 and 904. Zbl0804.14010MR93g:14016
- [K2] F. KIRWAN, Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984. Zbl0553.14020MR86i:58050
- [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. Zbl0820.51016MR95k:14015
- [Le] E. LERMAN, Symplectic cuts, Math. Res. Lett., 2 (1995), 247-258. Zbl0835.53034MR96f:58062
- [Ma] S. MARTIN, Doctoral thesis (in preparation).
- [MS] J. MILNOR & J. STASHEFF, Characteristic Classes, Princeton Univ. Press, 1974. Zbl0298.57008MR55 #13428
- [Popp] H. POPP, Moduli theory and classification of algebraic varieties, Springer-Verlag, 1977. Zbl0359.14005MR57 #6024
- [Sp] E. SPANIER, Algebraic topology, McGraw-Hill, 1966. Zbl0145.43303MR35 #1007

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