Linking forms and stabilization of diffeomorphism groups of manifolds of dimension \(4n + 1\) (Q2813668)

For a smooth compact connected manifold \(M\) with boundary \(\partial M\not=\emptyset\), the symbol \(\mathrm{BD}^\partial(M)\) denotes the classifying space of the group of self-diffeomorphisms of \(M\) which are identical near the boundary \(\partial M\). The author studies the natural stabilisation maps NEWLINE\[NEWLINE\mathrm{BD}^\partial(M)\to \mathrm{BD}^\partial(M \sharp W)\to \dots \to \mathrm{BD}^\partial(M \sharp W^{\sharp g})\to \dotsNEWLINE\]NEWLINE where \(W\) is a closed connected manifold of dimension \(\dim W =m=\dim M\). The main result of the paper states that for \(m=4n+1\), if \(M\) is 2-connected and \(W\) is \((2n-1)\)-connected, then the stabilisation map induces an isomorphism NEWLINE\[NEWLINEH_\ell({\mathrm{BD}^\partial}(M\sharp W^{\sharp g});\mathbb Z) \to H_\ell(\mathrm{BD}^\partial(M\sharp W^{\sharp {g+1}});\mathbb Z)NEWLINE\]NEWLINE for \(g\geq 2\ell +3\), provided \(W\) is stably parallelizable with \(H_{2n}(W; \mathbb Z)\) finite without 2-tosion. This result is an odd-dimensional analogue of theorems of \textit{S. Galatius} and \textit{O. Randal-Williams} [``Homological stability for moduli spaces of high dimensional manifolds'', Preprint, \url{arXiv:1203.6830}; ``Homological stability for moduli spaces of high dimensional manifolds. I'', Preprint, \url{arXiv:1403.2334}]. As one of the main tools, the author uses the action of the diffeomorphism group of a manifold \(M\) on \textit{the linking form} associated to the homology groups of \(M\). The author also proves several disjunction results for embeddings and immersions of \(\mathbb Z/k\)-manifolds which could be of independent interest.