Mapping class groups of highly connected \((4k+2)\)-manifolds
Let \(n\ge 3\) an \textit{odd} integer and \(g\ge 1\) an integer. Let us denote by \(W_g^n\) the connected sum of \(g\) copies of \(\mathbb{S}^n\times\mathbb{S}^n\) and by \(\Gamma_g^n\) the mapping class group of this manifold, that is the group of isotopy classes of orientation-preserving diffeomorphisms of \(W_g^n\). The action of such diffeomorphisms on the homology \(H_n(W_g^n;\mathbb{Z})\simeq\mathbb{Z}^{2g}\) induces a group morphism \(W_g^n\to\operatorname{GL}_{2g}(\mathbb{Z})\) whose kernel is the \textit{Torelli group} \(T_g^n\) and whose image \(G_g^n\) is known -- it is an explicit subgroup of finite index of the symplectic group \(\operatorname{Sp}_g(\mathbb{Z})\subset\operatorname{GL}_{2g}(\mathbb{Z})\). This was proved in [\textit{M. Kreck}, Lect. Notes Math. 763, 643--663 (1979; Zbl 0421.57009)], in which Keck exhibits also the group \(\Theta_{2n+1}\) of homotopy \((2n+1)\)-spheres introduced in [\textit{M. A. Kervaire} and \textit{J. W. Milnor}, Ann. Math. (2) 77, 504--537 (1963; Zbl 0115.40505)] as a central subgroup of \(\Gamma_g^n\) lying into \(T_g^n\); moreover, the quotient group \(T_g^n/\Theta_{2n+1}\) is known. But the structure of the corresponding group extensions was not well-known before the article under review, except for special values of \(n\) or \(g\). The author proves that the surjective group morphism \(\Gamma_g^n/\Theta_{2n+1}\twoheadrightarrow G_g^n\) splits if and only if \(n\notin\{3,7\}\) or \(g=1\) (\textit{Theorem A}). Moreover, he describes the class of the central extension \(0\to\Theta_{2n+1}\to\Gamma_g^n\to\Gamma_g^n/\Theta_{2n+1}\to 1\) in \(H^2(\Gamma_g^n/\Theta_{2n+1};\Theta_{2n+1})\), proving especially that it is trivial if and only if \(n\equiv 1\quad (\mathrm{mod}\;4)\) and \(g=1\) (\textit{Theorem B}). This has implications for \(T_g^n\) (see \textit{Corollary C}) and allows to describe the abelianisations of \(\Gamma_g^n\) and \(T_g^n\) (see \textit{Corollary D}). The article proves also further more technical consequences and gives a few partial results for an even \(n\).
scientific article; zbMATH DE number 7296293
scientific article