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Let    n  ⩾  2   . We prove a homological stability theorem for the diffeomorphism groups of    (  4  n    +    1  )   ‐dimensional manifolds, with respect to forming the connected sum with certain    (  2  n  ‐  1  )   ‐connected,    (  4  n  +  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 particular, 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 disjunction results for the embeddings and immersions of    Z  /  k   ‐manifolds that could be of independent interest.

Linking forms and stabilization of diffeomorphism groups of manifolds of dimension 4 n + 1