Homological stability for diffeomorphism groups of high-dimensional handlebodies

Abstract

In this paper 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, q} − 2 . In a recent joint paper with B. Botvinnik, we prove that there is an isomorphismin the case that n ≥ 4 . By combining this "stable homology" calculation with the homological stability theorem of this current paper, we obtain the isomorphism H k (BDiff((D n+1 × S n ) ♮g , D 2n ); Z) ∼ = H k (Q0BO(2n + 1) n +; Z) in the case that k ≤ 1 2 (g − 4) .

Stable moduli spaces of high‐dimensional handlebodies

“…In the second author of this paper proves that the map on homology induced by , is an isomorphism when and . By combining Theorem A with this homological stability result, we obtain the immediate corollary:…”

“…In [13] it is shown that the direct system (1.5) satisfies homological stability. As a result of this homological stability theorem we obtain the following corollary:…”

“…We will also need to use a higher‐dimensional generalization of the previous proposition. The next proposition follows from Theorem in Appendix (see also [, Theorem A.14] for a generalization).…”

Some finiteness results for groups of automorphisms of manifolds

“…The input to our proofs is a number of deep theorems in manifold theory: the homological stability results of Galatius and Randal-Williams [GRW18,GRW14] and Botvinnik and Perlmutter [BP17,Per18], the embedding calculus of Weiss [Wei99,BdBW13], the excision estimates of Goodwillie and Klein [GK15], and the arithmeticity results of Sullivan [Sul77,Tri95].…”

“…This discussion is not complete and does not cover the results used in our argument, e.g. [GRW14, GRW18, Wei15, BP17,Per18]. We start with diffeomorphisms of a disk.…”