arXiv · 0706.2247
This paper has been withdrawn by the author due to an error in the proof of Theorem 1.
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Emmanuel Wagner. 2009-03-30. A refinement of Khovanov-Rozansky link homology. https://arxiv.org/abs/0706.2247
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We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.
In this article, we demonstrate that there exists an infinite family of lens spaces L(p,q) that cannot be obtained from the 3-sphere by two integral surgeries. The first such lens space in our family is L(177,29).
Here we study the problem of whether a given codimension one immersion $f$ of a closed and connected manifold into Euclidean space can be extended to an equidimensional immersion $F$ of some compact manifold $M$ whose restriction to the boundary parameterizes the image $S$ of $f$. Such an extension $F$ exists if and only if (i) simple conditions on orientations and wrapping numbers are satisfied and (ii) a distinguished finite subset of the first cohomology of $S$ with coefficients in a certain sheaf of finite groups contains the preferred point.