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

TL;DR

This paper computes the mapping class group of certain 6-manifolds resembling complete intersections, revealing algebraic properties, residual finiteness, and parallels with Riemann surface mapping class groups.

Contribution

It provides the first detailed algebraic and structural analysis of the mapping class group for these specific 6-manifolds, including generators, relations, and homological properties.

Findings

Mapping class group is residually finite modulo its center

Computed abelianization and center of the group

Identified generators and relations for the group and its subgroups

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 $\pi_{3}(M)$.