Fibrations and the topology of low-dimensional manifolds

该奖项由2009年美国复苏和再投资法案（公法111-5）资助。通过与映射类群的联系，Lefschetz纤维化和4-流形上的相关结构为研究辛和近辛4-流形提供了群论手段。这个项目将使用Lefschetz纤维化与来自Floer同源的不变量一起研究几个相关的问题。首先，PI已经证明了4-流形上的几个几何运算可以实现为Lefschetz纤维化中的单值替换的实例，来自映射类群中的新关系。PI将继续这一研究方向，目标是找到奇异4-流形的新构造。第二，PI将使用相对Ozsváth-Szabó不变量技术（PI和S. Jabuka）产生新的例子接触3流形承认无限多拓扑等价，但顺利不同的斯坦填充，扩展以前的联合工作。接下来是Ozsváth-Szabó不变量本身的拓扑意义：例如，Ozsváth-Szabó不变量是否对给定的4-流形所支持的Lefschetz结构的种类提供约束？相反，我们可以从莱夫谢茨纤维化的单值表示中计算出它的Ozsváth-Szabó不变量吗？最后，他将继续发展扰动Heegaard Floer同调理论，这将扩大Heegaard Floer“包的实用性。自从爱因斯坦将力学和电动力学描述为固有的四维理论以来，所观察到的宇宙通常被认为是一个光滑的四维流形：也就是说，一个光滑表面的四维模拟，如平面或球体。一个基本的问题是：它是什么流形？做一个简单的类比，当一个不经意的观察者看到地球表面时，地球表面通常是“平的”，但推断它是平面的是错误的。在适当的距离尺度上，宇宙同样是“大部分平坦的”，但它的整体拓扑或“形状”尚不清楚。因此，四维拓扑学家的一个目标是描述宇宙底层结构的可能性列表，类似于相对容易理解的可能表面列表，像地球表面这样的“一般平坦”物体可以从中选择（球体，环面等）。也许并非巧合的是，光滑四维流形的理论比任何其他维度的类似理论都要复杂得多。事实上，令人惊讶的是，很少有人知道关于光滑4-流形的重要和基本的存在性和唯一性问题。这项资助支持的工作将通过利用4维流形上的Lefschetz纤维化（或类似的几何结构）以及最近开发的基于规范理论物理学思想的数学工具提供的3维和4维流形的不变量来解决这些问题。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Through their connection with mapping class groups, Lefschetz fibrations and related structures on 4-manifolds provide a group-theoretic means for studying symplectic and near-symplectic 4-manifolds. This project will use Lefschetz fibrations in conjunction with invariants coming from Floer homology to study several related questions. First, the PI has shown that several geometric operations on 4-manifolds can be realized as instances of monodromy substitutions in Lefschetz fibrations, coming from new relations in the mapping class group. The PI will pursue this line of inquiry with the goal of finding new constructions of exotic 4-manifolds. Second, the PI will use the technology of relative Ozsváth-Szabó invariants (developed in joint work of the PI and S. Jabuka) to produce new examples of contact 3-manifolds admitting infinitely many topologically equivalent but smoothly distinct Stein fillings, extending previous joint work. Next is the topological meaning of the Ozsváth-Szabó invariants themselves: for example, do the Ozsváth-Szabó invariants provide constraints on the sorts of Lefschetz structures supported by a given 4-manifold? Conversely, can one calculate the Ozsváth-Szabó invariants of a Lefschetz fibration from its monodromy representation? Finally, he will continue to develop the theory of perturbed Heegaard Floer homology, which will expand the utility of the Heegaard Floer "package."Since Einstein's description of mechanics and electrodynamics as inherently a four-dimensional theory, the observed universe has generally been conceived as a smooth four-dimensional manifold: that is, a four-dimensional analog of a smooth surface such as a plane or sphere. A fundamental question is then: what manifold is it? To pose a simplified analogy, the surface of the earth is generally ``flat'' when viewed by a casual observer, but it is a mistake to infer that it is planar. The universe is similarly "mostly flat" on an appropriate distance scale, but its global topology or "shape" is not known. A goal of the 4-manifold topologist, then, is to describe the list of possibilities for the underlying structure of the universe, in analogy with the relatively easy-to-understand list of possible surfaces from which a "generally flat" object like the surface of the earth can select (sphere, torus, etc.). Perhaps not coincidentally, the theory of smooth 4-dimensional manifolds is vastly more complicated than the analogous theory in any other dimension. Indeed, surprisingly little is known regarding important and basic existence and uniqueness questions for smooth 4-manifolds. The work supported by this grant will approach several of these questions by making use of a Lefschetz fibration (or similar geometric structure) on a 4-manifold, together with the invariants of 3- and 4-dimensional manifolds provided by recently-developed mathematical tools that are based on ideas from gauge-theoretic physics.