# The cohomology ring of polygon spaces

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

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## Abstract

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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.

## How to cite

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Hausmann, 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 -

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