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

TL;DR

This paper analyzes the homological growth of the space of long links in Euclidean space using spectral sequences, providing explicit calculations and demonstrating exponential Betti number growth.

Contribution

It introduces explicit Euler-Poincaré series calculations for the spectral sequence of long links, revealing exponential Betti number growth.

Findings

Explicit Euler-Poincaré series for the spectral sequence

Demonstration of exponential Betti number growth

Insights into the homology of long link spaces

Abstract

We study the Bousfield-Kan spectral sequence associated to the Munson-Voli\'c cosimplicial model for the space of long links with $l$ strings in $\mathbb{R}^N$ . We compute explicitely some Euler-Poincar\'e series associated to the second page of that spectral sequence and deduce exponential growth of their Betti numbers.