Homological stability for diffeomorphism groups of high-dimensional handlebodies

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 $|q-p|<min\left\{p-2,q-3\right\}$. In a recent joint paper with B Botvinnik, we prove that there is an isomorphism

$${colim}_{g\to \infty }{H}_{\ast }\left(BDiff\left({\left(D^{n+1}\times S^n\right)}^{♮g},D^{2n}\right);\mathbb{Z}\right)\cong H_{\ast }\left(Q_{0}BO\left(2n+1\right){\langle n\rangle }_{+};\mathbb{Z}\right)$$

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

$$H_k\left(BDiff\left({\left(D^{n+1}\times S^n\right)}^{♮g},D^{2n}\right);\mathbb{Z}\right)\cong H_k\left(Q_{0}BO\left(2n+1\right){\langle n\rangle }_{+};\mathbb{Z}\right)$$

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