We provide a complete understanding of the rational homology of the space of long links of
strands
in
when
.
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
collapses
at the
page, that the homology Bousfield–Kan spectral sequence associated to the
Munson–Volić cosimplicial model for the space of long links collapses at the
page
rationally, solving a conjecture of B Munson and I Volić. Our method enables us
also to determine the rational homology of high-dimensional analogues of spaces
of long links. Our last result states that the radius of convergence of the
Poincaré series for the space of long links (modulo immersions) tends to zero as
goes
to infinity.
Keywords
long links, embeddings calculus, module over operads,
spectral sequences
