Volume 18, issue 5 (2018)

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Homological stability for diffeomorphism groups of high-dimensional handlebodies

Nathan Perlmutter

Algebraic & Geometric Topology 18 (2018) 2769–2820
Abstract

We prove a homological stability theorem for the diffeomorphism groups of high-dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product Dp+1 × Sq for |q p| < min{p 2,q 3}. In a recent joint paper with B Botvinnik, we prove that there is an isomorphism

colimgH(BDiff((Dn+1 × Sn)g,D2n); )H (Q0BO(2n + 1)n+; )

in the case that n 4. By combining this “stable homology” calculation with the homological stability theorem of this paper, we obtain the isomorphism

Hk(BDiff((Dn+1 × Sn)g,D2n); )H k(Q0BO(2n + 1)n+; )

in the case that k 1 2(g 4).

Keywords
manifolds, homological stability, moduli spaces
Mathematical Subject Classification 2010
Primary: 57R15, 57R50, 57R65, 57S05
References
Publication
Received: 23 April 2017
Revised: 11 March 2018
Accepted: 22 March 2018
Published: 22 August 2018
Authors
Nathan Perlmutter
Department of Mathematics
Stanford University
Stanford, CA
United States