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.

Locations

• Documenta Mathematica - View - PDF