The rational homology of spaces of long links

Abstract

We provide a complete understanding of the rational homology of the space of long links of $m$ strands in ${ℝ}^d$ when $d\ge 4$. First, we construct explicitly a cosimplicial chain complex, $L_{\ast }^{\bullet }$, 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 $L_{\ast }^{\bullet }$ collapses at the $E^2$ page, that the homology Bousfield–Kan spectral sequence associated to the Munson–Volić cosimplicial model for the space of long links collapses at the $E^2$ 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 $m$ goes to infinity.