The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres

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.

1991 Mathematics Subject Classification: Primary 55M99; Secondary: 57N65, 55R20, 52C35

Keywords and Phrases: spheres, ordered configuration spaces, subspace arrangements, integral cohomology algebra, fibration, Serre spectral sequence

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