Betti numbers and stability for configuration spaces via factorization homology

Knudsen, Ben

doi:10.2140/agt.2017.17.3137

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Betti numbers and stability for configuration spaces via factorization homology

Ben Knudsen

Algebraic & Geometric Topology 17 (2017) 3137–3187

DOI: 10.2140/agt.2017.17.3137

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

References

Publication

Received: 8 December 2016

Revised: 9 December 2016

Accepted: 25 January 2017

Published: 19 September 2017

Authors

Ben Knudsen
Department of Mathematics Harvard University Cambridge, MA United States
http://scholar.harvard.edu/knudsen/

Volume 17, issue 5 (2017)