The cohomology ring of polygon spaces
Jean-Claude Hausmann, Allen Knutson
TL;DR
The paper develops a toric-geometric framework to compute the integral cohomology rings of polygon spaces Pol(α). By embedding Pol(α) as a transverse intersection inside the ambient toric upper path space UP(α) and applying Danilov’s toric cohomology together with Gröbner-basis techniques to determine annihilators, the authors obtain explicit presentations for H^*(APol(α)) and H^*(Pol(α)) in terms of natural generators R,V_i, c_i, and p. They further analyze natural SO(2)-bundles over Pol(α) via Chern classes c_j, relate these to the symplectic form and Duistermaat–Heckmann theory, and treat special cases such as equilateral and planar polygons, including actions of the symmetric group and the resulting rational/cohomological structures. The work yields computable Poincaré polynomials, demonstrates how integral and mod-2 cohomology differ, and provides concrete examples that illustrate the nuanced dependence on combinatorial data and edge-length symmetries.
Abstract
We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. 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 Z/2; halving all degrees we show this produces the Z/2 cohomology rings of 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 [Kl]. Finally, our formulae for the Poincaré polynomials are more computationally effective than those known [Kl].