arXiv subjects
Publications and source records attributed to Max Lipyanskiy.
We prove that Floer cohomology of cyclic Lagrangian correspondences is invariant under transverse and embedded composition of Lagrangians under a general set of assumptions. In the Corrigendum, we introduce an additional assumption of monotonicity for cylinders which is needed to avoid bubbling at the Y-end.
The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately addressed. In particular, our goal is to construct a chain level theory based on smooth mappings of manifolds with corners. In addition, we will avoid using the fact that smooth manifolds with can be triangulated. As we shall see, transversality and intersections play a major role in setting up this theory. From a pedagogical viewpoint, having intersection theory arguments available from the start facilitates simple and intuitive computations.
We provide a short proof that an $L^2_1$ and $J$-holomorphic curve is in fact smooth. As an application, we deduce a removal of singularity theorem for curves of finite energy.
We construct a version of Hamiltonian Floer Homology based on the notion of a semi-infinite cycle. As an application, we provide a new proof for the existence of critical points of the action functional.
We develop an analytic framework that combines the compactness theory of Gromov for J-holomorphic curves with that of Uhlenbeck for ASD connections. Compactness, regularity and removal of singularity theorems are discussed.
This is the first of a series of papers on foundations of Floer theory. We give an axiomatic treatment of the geometric notion of a semi-infinite cycle. Using this notion, we introduce a bordism version of Floer theory for the cotangent bundle of a compact manifold M. Our construction is geometric and does not require the compactness and gluing results traditionally used to setup Floer theory. Finally, we prove a bordism version of Viterbo's theorem relating Floer bordism of the cotangent bundle to the ordinary bordism groups of the free loop space of M.
A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated to one of the families. It is conjectured that there are exactly 6 exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea's census of closed hyperbolic 3-manifolds, associated to 5 other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated to the seventh family we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields.