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.Comment: 19 pages. Improved the codimension condition and the exposition.
Published in AG
