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The cohomology ring of polygon spaces. (English) Zbl 0903.14019

Summary: We compute the integer cohomology rings of the “polygon spaces” introduced by F. Kirwan [J. Am. Math. Soc. 5, No. 4, 853-906 (1992; Zbl 0804.14010)] and M. Kapovich and J. J. Millson [J. Differ. Geom. 44, No. 3, 479-513 (1996; Zbl 0889.58017)]. 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 \(\mathbb{Z}_2\); halving all degrees we show this produces the \(\mathbb{Z}_2\) 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 is so on the rational cohomology [cf. F. Kirwan, op. loc.]. Finally, our formulae for the Poincaré polynomials are computationally more effective than those known [cf. F. Kirwan, op. loc.].

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14F25 Classical real and complex (co)homology in algebraic geometry
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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