Homological stability for diffeomorphism groups of high-dimensional handlebodies

Perlmutter, Nathan

doi:10.2140/agt.2018.18.2769

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

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)