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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Betti numbers and stability for configuration spaces via factorization homology

Ben Knudsen

Algebraic & Geometric Topology 17 (2017) 3137–3187

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.

configuration spaces, factorization homology, Lie algebras, homological stability
Mathematical Subject Classification 2010
Primary: 57R19
Secondary: 17B56, 55R80
Received: 8 December 2016
Revised: 9 December 2016
Accepted: 25 January 2017
Published: 19 September 2017
Ben Knudsen
Department of Mathematics
Harvard University
Cambridge, MA
United States