## Algebraic & Geometric Topology

### The rational homology of spaces of long links

Paul Arnaud Songhafouo Tsopméné

#### Abstract

We provide a complete understanding of the rational homology of the space of long links of $m$ strands in $ℝd$ when $d ≥ 4$. First, we construct explicitly a cosimplicial chain complex, $L∗∙$, 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 $L∗∙$ collapses at the $E2$ page, that the homology Bousfield–Kan spectral sequence associated to the Munson–Volić cosimplicial model for the space of long links collapses at the $E2$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 757-782.

Dates
Revised: 1 July 2015
Accepted: 11 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841151

Digital Object Identifier
doi:10.2140/agt.2016.16.757

Mathematical Reviews number (MathSciNet)
MR3493406

Zentralblatt MATH identifier
1354.57033

#### Citation

Songhafouo Tsopméné, Paul Arnaud. The rational homology of spaces of long links. Algebr. Geom. Topol. 16 (2016), no. 2, 757--782. doi:10.2140/agt.2016.16.757. https://projecteuclid.org/euclid.agt/1510841151

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