Braided Surfaces and Small Exotic 4-Manifolds

Abstract

Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \[ M_K\times S^1\longrightarrow T^2\times S^2 \] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided $(2g+2)$-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over $T^2$. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of $Y_K$ and its twisted double $X_K$, as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to $\CP^2\#k\CPb^2$, $k=3,5$ \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a $k=5$ example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.

26 pages, 3 figures