#### Volume 17, issue 5 (2017)

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Betti numbers and stability for configuration spaces via factorization homology

### Ben Knudsen

Algebraic & Geometric Topology 17 (2017) 3137–3187
##### Abstract

Using factorization homology, we realize the rational homology of the unordered configuration spaces of an arbitrary manifold $M$, possibly with boundary, as the homology of a Lie algebra constructed from the compactly supported cohomology of $M$. By locating the homology of each configuration space within the Chevalley–Eilenberg complex of this Lie algebra, we extend theorems of Bödigheimer, Cohen and Taylor and of Félix and Thomas, and give a new, combinatorial proof of the homological stability results of Church and Randal-Williams. Our method lends itself to explicit calculations, examples of which we include.

##### Keywords
configuration spaces, factorization homology, Lie algebras, homological stability
##### Mathematical Subject Classification 2010
Primary: 57R19
Secondary: 17B56, 55R80