The integral cohomology algebras of ordered configuration spaces of spheres

Abstract

We compute the cohomology algebras of spaces of ordered point configurations on spheres, $F(S^k,n)$, with integer coefficients. For $k=2$ we describe a product structure that splits $F(S^2,n)$ into well-studied spaces. For $k>2$ we analyze the spectral sequence associated to a classical fiber map on the configuration space. In both cases we obtain a complete and explicit description of the integer cohomology algebra of $F(S^k,n)$ in terms of generators, relations and linear bases. There is 2-torsion occuring if and only if $k$ is even. We explain this phenomenon by relating it to the Euler classes of spheres. Our rather classical methods uncover combinatorial structures at the core of the problem.