Euler series, Stirling numbers and the growth of the homology of the space of long links (Q2636791)

This paper is concerned with the Bousfield-Kan spectral sequence associated to \(\mathcal L_l\), the space of long links with \(l\) strands modulo immersions. This is the space of embeddings of \(l\) copies of \(\mathbb R\) in \(\mathbb R^N\), \(N\geq 3\), that are linear outside some compact set, but then one also takes the homotopy fiber of the inclusion of this space into the space of immersions (defined similarly). When \(l=1\), one gets the space of long knots \(\mathcal K\). In [\textit{B. Munson} and \textit{I. Volić}, ``Cosimplicial models for spaces of links'', to appear in Journal of Homotopy and Related Structures; \url{arxiv:0906.2589}], the authors define a cosimplicial model for these spaces which consists of configuration spaces and whose totalization is, for \(N\geq 4\), equivalent to \(\mathcal L_l\). This is a generalization of the cosimplicial model for \(\mathcal K\) defined by Sinha. The cosimplicial model comes equipped with the cohomology spectral sequence that converges to its totalization for \(N\geq 4\) (and hence it conveges to \(\mathcal L_l\)). Using the fact that this spectral sequence has a lower and an upper vanishing line which makes all the non-trivial entries be concentrated in a ``steep wedge'', again for \(N\geq 4\), the authors derive a combinatorial formula for the \textit{Euler series} (which they define) of the \(E_1\) page of this spectral sequence. The upshot is that, if entries in the cosimplicial space are finite type, as configuration spaces that make up the cosimplicial model for \(\mathcal L_l\) are, the Euler series gives a lower bound for the Betti numbers of the \(E_2\) page provided that the spectral sequence collapses at that page. The authors go on to find the Euler series for the cosimplicial model for \(\mathcal L_l\) using the Poincaré series for configuration spaces. The latter are related to Stirling numbers and the authors find a combinatorial relationship between the two kinds of Stirling numbers which in turn allows them to find the desired Euler series. The authors also consider the \(l\)-fold product of long knots and show that this is a retract of the space of long links. Since this is a retract on the level of cosimplicial models, one gets a spectral sequence for the pair \((\mathcal L_l, \mathcal K^l)\). Since the spectral sequence for \(\mathcal K\) collapses at the \(E_2\) page [\textit{P. Lambrechts} et al., Geom. Topol. 14, No. 4, 2151--2187 (2010; Zbl 1222.57020)], it is natural to conjecture that the spectral sequences for \(\mathcal L_l\) and \((\mathcal L_l, \mathcal K^l)\) also collapse (this is likely). As an application of their results, the authors at the end show that, if the latter spectral sequence indeed collapses, then the Betti numbers of the totalization of the \(E_2\) page would have exponential growth.