Homological stability for diffeomorphism groups of high-dimensional handlebodies

Perlmutter, Nathan

doi:10.2140/agt.2018.18.2769

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Homological stability for diffeomorphism groups of high-dimensional handlebodies

Nathan Perlmutter

Algebraic & Geometric Topology 18 (2018) 2769–2820

DOI: 10.2140/agt.2018.18.2769

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 $D^{p + 1} \times S^{q}$ for $| q - p | < min {p - 2, q - 3}$ . In a recent joint paper with B Botvinnik, we prove that there is an isomorphism

{colim}_{g \to \infty} H_{*} (BDiff ({(D^{n + 1} \times S^{n})}^{♮ g}, D^{2 n}); ℤ) ≅ H_{*} (Q_{0} B O (2 n + 1) {〈 n 〉}_{+}; ℤ)

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

H_{k} (BDiff ({(D^{n + 1} \times S^{n})}^{♮ g}, D^{2 n}); ℤ) ≅ H_{k} (Q_{0} B O (2 n + 1) {〈 n 〉}_{+}; ℤ)

in the case that $k \leq \frac{1}{2} (g - 4)$ .

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

Volume 18, issue 5 (2018)