The cohomology ring of polygon spaces

TL;DR

This paper computes the integer cohomology rings of polygon spaces by embedding them into toric varieties, using Gr"obner bases, and explores their properties over Z/2, providing more efficient formulas for Poincaré polynomials.

Contribution

It introduces a method to compute cohomology rings of polygon spaces via embeddings into toric varieties and analyzes their properties over Z/2, including group actions.

Findings

Computed integer cohomology rings of polygon spaces.

Established surjectivity of the restriction map on cohomology.

Provided more computationally effective formulas for Poincaré polynomials.

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 Gr\"obner 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 Poincar\'e polynomials are more computationally effective than those known [Kl].