Stable moduli spaces of high‐dimensional handlebodies

Abstract

Abstract. We study the moduli space of handlebodies diffeomorphic to () of the group of diffeomorphisms that restrict to the identity near an embedded diskWe prove that there is a natural mapwhich induces an isomorphism in integral homology when n ≥ 4 . Above, BO(2n + 1) n denotes the n -connective cover of BO(2n + 1) .

Infinite loop spaces from operads with homological stability

“…However this is the homological stability that is needed for condition 2 of our definition. The same remark holds for diffeomorphism groups for handlebodies which also have been shown to have homological stability by Perlmutter [Per], and the stable homology been identified by Botvinnik and Permutter [BP17]. Similarly, Nariman [Nara] proves homological stability results for diffeomorphism groups of manifolds made discrete.…”

Homological stability for diffeomorphism groups of high-dimensional handlebodies

“…To prove Theorem 1.1, we will need to recast the direct system (1.1) as arising from concatenation with a relative cobordism between two compact manifold pairs. Doing this will make our constructions consistent with the cobordism categories considered in [2] and [1]. Below we introduce some definitions and terminology that we will use throughout the paper.…”

“…For certain choices of p and q we can also identify the homology of the colimit. In joint work with B. Botvinnik [1] we construct a map, colim g→∞ BDiff((D n+1 × S n ) ♮g , D 2n ) −→ Q 0 BO(2n + 1) n + , which we prove induces an isomorphism in H * (−, Z) in the case that n ≥ 4. Combining this homological equivalence with Theorem 1.1 we obtain the following corollary which lets us compute the homology of the classifying space BDiff((D n+1 × S n ) ♮g , D 2n ) in low degrees relative to g .…”

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.…”