We characterize the representations of the fundamental group of a closed surface to that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces.
For a Legendrian link with or , immersed exact Lagrangian fillings of can be lifted to conical Legendrian fillings
We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible -representations.
The purpose of this paper is to introduce -maps (), which describe higher homotopy normalities, and to study their basic properties and examples. An -map is defined with higher homotopical conditions. It is shown that a homomorphism is an -map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an -map is shown to be an -space if its LS category is not greater than . As an application, we investigate when the inclusions
We provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.
We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups are lifted to homotopy groups of the moduli space of -cobordisms . As a geometrical application, we show that those elements in
We study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension : we prove that their total spaces are rationally inessential if , and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.
We define the Milnor number of a one-dimensional holomorphic foliation as the intersection number of two holomorphic sections with respect to a compact connected component of its singular set. Under certain conditions, we prove that the Milnor number of on a three-dimensional manifold with respect to is invariant by topological equivalences.
In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus with respect to the Weil–Petersson measure on the moduli space . We show that as goes to infinity, a generic surface satisfies asymptotically:
Given an irreducible, end-periodic homeomorphism of a surface with finitely many ends, all accumulated by genus, the mapping torus, , is the interior of a compact, irreducible, atoroidal 3-manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of in terms of the translation length of on the pants graph of . This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.
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