Exotic Dehn twists on 4-manifolds

Abstract

We initiate the study of exotic Dehn twists along 3-manifolds $\ne S^3$ inside $4$-manifolds, which produces the first known examples of exotic diffeomorphisms of contractible 4-manifolds, more generally of definite 4-manifolds, and exotic diffeomorphisms of 4-manifolds with $\ne S^3$ boundary that survive after one stabilization. We also construct the smallest closed 4-manifold known to support an exotic diffeomorphism. These exotic diffeomorphisms are the Dehn twists along certain Seifert fibered 3-manifolds. As a consequence, we get loops of diffeomorphisms of 3-manifolds that topologically extend to some 4-manifolds $X$ but not smoothly so, implying the non-surjectivity of ${\pi }_1(\mathrm{Diff}(X))\to {\pi }_1(\mathrm{Homeo}(X))$. Our method uses $2$-parameter families Seiberg-Witten theory over ${\mathbb{RP}}^2$, while known methods to detect exotic diffeomorphisms used $1$-parameter families gauge-theoretic invariants. Using a similar strategy, we construct a new kind of exotic diffeomorphisms of 4-manifolds, given as commutators of diffeomorphisms.

29 pages, 1 figure. Minor improvements in some of the main theorems. We also thank the anonymous referee for an insightful report