## Finiteness Properties for Handlebody Mapping Class GroupsWe also derive substantial partial results for establishing homological stability for the 1-patched handlebody mapping class group. We note that in the rather different contexts of DAC and finiteness properties on the one hand, and the arena of homological stability on the other, the notion of "patches" is a fruitful analog allowing (modifications of) many familiar constructions to be applied, and familiar results to be established, as if the handlebody had one or many localized boundary components. |

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action acts analogous apply arc representatives arc segments arc system assume automorphism boundary bounded Chapter choose circles closed collection compact complex component condition connected consider construction contains continuous contractible corresponding defined definition deformation denote diffeomorphism dimension discs disjoint edges embedded endpoints equivalent essential discs establishing example exist fact Figure Finally finite fixes follows genus g given gives handlebody mapping class Hence holds homological stability homotopy identity implies induction intersection isomorphism isotopy classes isotopy extension Lemma mapping class group maximal minimal modified Note obtained orientation pair parallel particular patches pointwise preserves PROOF Proposition prove punctures quotient representatives respectively restriction result satisfying separating sequence similarly simple system simplex simplicial map simply-connected space sphere splitting step subcomplex subgroup subset Suppose surface surgery system of discs taking Theorem vertex vertices