Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology

Abstract: We show that the Torelli group of a simply connected closed 5-manifold $M$, which is spin and has no 2-torsion elements in homology, is isomorphic to the bordism group $Ω^{{\mathrm{Spin}}_6(K(H_2(M),} 2))$. For $H_2(M)$ with no 2- and 3-torsion we determine this bordism group, and give explicit constructions for the generators of the Torelli group. Furthermore, for the $g$-fold connected sum ${#}^g(S² \times S<sup>3)$ we completely determine its mapping class group. We apply our results to compute the stabilization and abelianization of the mapping class group of ${#}^g(S² \times S<sup>3)$, determine the group of isotopy classes of diffeomorphisms of $M$ that are homotopic to the identity, and study the embeddings of $S<sup>3$ in $S² \times S<sup>3$.