Linking Forms, Singularities, and Homological Stability for Diffeomorphism Groups of Odd Dimensional Manifolds
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Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)-dimensional manifolds, with respect to forming the connected sum with (2n-1)-connected, (4n+1)-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest.