Feichtner, Eva Maria; Ziegler, Günter M.
The integral cohomology algebras of ordered configuration spaces of spheres
Doc. Math., J. DMV 5, 115-139 (2000)
Summary: 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.
Mathematics Subject Classification
55M99, 57N65, 55R20, 52C35
spheres, ordered configuration spaces, subspace arrangements, integral cohomology algebra, fibration, Serre spectral sequence