Algebraic & Geometric Topology

Homological stability for diffeomorphism groups of high-dimensional handlebodies

Nathan Perlmutter

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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 × 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 H ( BDiff ( ( D n + 1 × S n ) g , D 2 n ) ; ) H ( Q 0 B O ( 2 n + 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

H k ( BDiff ( ( D n + 1 × S n ) g , D 2 n ) ; ) H k ( Q 0 B O ( 2 n + 1 ) n + ; )

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

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2769-2820.

Dates
Received: 23 April 2017
Revised: 11 March 2018
Accepted: 22 March 2018
First available in Project Euclid: 30 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1535594422

Digital Object Identifier
doi:10.2140/agt.2018.18.2769

Mathematical Reviews number (MathSciNet)
MR3848399

Zentralblatt MATH identifier
06935820

Subjects
Primary: 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 57R50: Diffeomorphisms 57R65: Surgery and handlebodies 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms

Keywords
manifolds homological stability moduli spaces

Citation

Perlmutter, Nathan. Homological stability for diffeomorphism groups of high-dimensional handlebodies. Algebr. Geom. Topol. 18 (2018), no. 5, 2769--2820. doi:10.2140/agt.2018.18.2769. https://projecteuclid.org/euclid.agt/1535594422


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