Mapping class group of manifolds which look like $3$-dimensional complete intersections

Abstract

In this paper we compute the mapping class group of closed simply-connected 6-manifolds $M$ which look like complete intersections, i.~e.~ $H_2(M;\mathbb Z) \cong \mathbb Z $ and $x^3 \ne 0$ where $x \in H^2(M; \mathbb Z)$ is a generator. We determine some algebraic properties of the mapping class group; for example we compute its abelianization and its center. We show that modulo the center the mapping class group is residually finite and virtually torsion-free. We also study low dimensional homology groups. The results are very similar to the computation of the mapping class group of Riemann surfaces. We give generators of the mapping class group, and generators and relations for the subgroup acting trivially on $\pi_{3}(M)$.