## Algebraic & Geometric Topology

### Homological stability for diffeomorphism groups of high-dimensional handlebodies

Nathan Perlmutter

#### 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

$colim g → ∞ H ∗ ( BDiff ( ( D n + 1 × S n ) ♮ g , D 2 n ) ; ℤ ) ≅ H ∗ ( Q 0 B O ( 2 n + 1 ) 〈 n 〉 + ; ℤ )$

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

$H k ( BDiff ( ( D n + 1 × S n ) ♮ g , D 2 n ) ; ℤ ) ≅ H k ( Q 0 B O ( 2 n + 1 ) 〈 n 〉 + ; ℤ )$

in the case that $k ≤ 1 2 ( g − 4 )$.

#### Article information

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

Dates
Revised: 11 March 2018
Accepted: 22 March 2018
First available in Project Euclid: 30 August 2018

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

#### 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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