Algebraic Topology

The equivariant cohomology ring of the representation variety $\Hom(\Z^2,\mathrm{GL}_n(\C))$

We give a presentation of the $\mathrm{GL}_n(\C)$-equivariant cohomology ring with $\Z$-coefficients of the variety $\Hom(\Z^2,\mathrm{GL}_n(\C))\subseteq \mathrm{GL}_n(\C)^2$ for any $n$. It is torsion free and minimally generated as a $H^\ast B\mathrm{GL}_n(\C)$-algebra by $3n$ elements. The ideal of relations is the saturation of an $n$-generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide $n!$, we also give a presentation of the non-equivariant cohomology ring of $\Hom(\Z^2,\mathrm{GL}_n(\C))$.