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

TL;DR

This paper establishes a homological stability theorem for diffeomorphism groups of certain high-dimensional manifolds, using linking forms and intersection theory of embedded manifolds, with additional disjunction results for embeddings.

Contribution

It introduces a new homological stability result for (4n+1)-dimensional manifolds and develops a geometric model for linking forms via intersections of $ ext{Z}/k$-manifolds.

Findings

Proved homological stability for diffeomorphism groups of (4n+1)-manifolds.

Constructed a geometric model for linking forms using intersections.

Established disjunction results for embeddings and immersions of $ ext{Z}/k$-manifolds.

Abstract

Let $n \geq 2$. 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 particular, we construct a geometric model for the linking form using the intersections of embedded and immersed $\mathbb{Z}/k$-manifolds. In addition to our main homological stability theorem, we prove several disjunction results for the embeddings and immersions of $\mathbb{Z}/k$-manifolds that could be of independent interest.