The rational homology of spaces of long links

We provide a complete understanding of the rational homology of the space of long links of m strands in the euclidean space of dimension d > 3. First, we construct explicitly a cosimplicial chain complex whose totalization is quasi-isomorphic to the singular chain complex of the space of long links. Next we show (using the fact that the Bousfield-Kan spectral sequence associated to this cosimplcial chain complex collapses at the page 2) that the homology Bousfield-Kan spectral sequence associated to the Munson-Volic cosimplicial model for the space of long links collapses at the page 2 rationally, and this solves a conjecture of Munson-Volic. Our method enables us also to determine the rational homology of the high dimensional analogues of spaces of long links. The last result of this paper states that the radius of convergence of the Poincar\'e series for the space of long links (modulo immersions) tends to zero as m goes to the infinity.