Exotic diffeomorphisms of 4-manifolds with b_+ = 2

TL;DR

This paper demonstrates the existence of exotic smooth structures on certain 4-manifolds with $b_+ = 2$, showing that their Torelli groups are infinitely generated and their mapping class groups are not finitely generated, using Seiberg-Witten invariants.

Contribution

It proves that for specific 4-manifolds with $b_+ = 2$, the Torelli and mapping class groups have complex, infinitely generated structures, advancing understanding of smooth structures in 4-manifold topology.

Findings

Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$ for $n \ge 10

Mapping class group of $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated

Seiberg-Witten invariants and gluing formulas are key tools in the proofs

Abstract

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The mapping class group of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The Torelli group of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg-Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg-Witten invariants plays an important role.