Abelian quotients of mapping class groups of highly connected manifolds (Q289874)

Let \(W^{2n}_{g} = g(S^{n} \;\times \;S^{n})\) where the \(g\) denotes the \(g\)-fold connected sum. Let \(\mathrm{Diff}^{+}(W^{2n}_{g})\) be the group of orientation preserving diffeomorphisms of \(W^{2n}_{g}\) and let \(\mathrm{Diff}(W^{2n}_{g}, D^{2n})\) be the subgroup of those diffeomorphisms that fix an open neighbourdood of the disk. The mapping class groups are \[ \Gamma^{n}_{g,1} = \pi_{0}D(\mathrm{Diff}(W^{2n}_{g}, D^{2n})\text{ and } \Gamma^{n}_{g} = \pi_{0}(\mathrm{Diff}^{+}(W^{2n}_{g})). \] It is a result of \textit{M. Kreck} that these two groups are isomorphic for \(n \geq 3\) [Lect. Notes Math. 763, 643--663 (1979; Zbl 0421.57009)]. This paper computes the abelianizations of these two groups for \(n \geq 3\) and \(g \geq 5\). The elements of the abelianizations are determined by a pair consisting of an element in the first homology group of the automorphisms of the quadratic forms on \(H^{n}(W^{2n}_{g})\) defined by \textit{C. T. C. Wall} [Ann. Math. (2) 75, 163--189 (1962; Zbl 0218.57022)] and an element in a cobordism group \(\Omega ^{\langle n\rangle}_{2n+1}\). This element is obtained by taking the mapping torus of an element of \(\mathrm{Diff}(W^{2n}_{g}, D^{2n})\) and performing surgery on the embedded \(D^{2n} \;\times S^{1}\).