Exotic diffeomorphisms on $4$-manifolds with $b_2^+ = 2$

TL;DR

This paper introduces a new method based on winding numbers to find exotic diffeomorphisms on certain 4-manifolds, specifically demonstrating that the smallest known example is the connected sum of two complex projective planes with ten reversed orientations.

Contribution

The paper develops a novel approach using winding numbers to detect exotic diffeomorphisms on 4-manifolds with $b_2^+ = 2$, providing the first such example on a minimal known manifold.

Findings

Established that $2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2)$ admits exotic diffeomorphisms.

Introduced a method comparing winding numbers of parameter families.

Identified the smallest known 4-manifold supporting exotic diffeomorphisms.

Abstract

While the exotic diffeomorphisms turned out to be very rich, we know much less about the $b^+_2 =2$ case, as parameterized gauge-theoretic invariants are not well defined. In this paper we present a method (that is, comparing the winding number of parameter families) to find exotic diffeomorphisms on simply-connected smooth closed $4$-manifolds with $b^+_2 =2$, and as a result we obtain that $2\mathbb{C}\mathbb{P}^2 \# 10 (-{\mathbb{C}\mathbb{P}^2})$ admits exotic diffeomorphisms. This is currently the smallest known example of a closed $4$-manifold that supports exotic diffeomorphisms.