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The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk,p maps S1 → ℙ1 is an infinite dimensional complex manifold. The loop group LPGL(2,ℂ) acts on Lℙ1. We prove that the group of LPGL(2, ℂ) invariant holomorphic line bundles on Lℙ1 is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of any such line bundle is finite dimensional, and compute the dimension for a generic bundle.
Let N be any closed, Riemannian manifold. In this paper we prove that, for most locally symmetric, nonpositively curved Riemannian manifolds M, and for every continuous map f : N → M, the map f is homotopic to a smooth map with Jacobian bounded by a universal constant, depending (as it must) only on Ricci curvature bounds of N. From this we deduce an extension of Gromov's Volume Comparison Theorem for negatively curved manifolds to (most) nonpositively curved, locally symmetric manifolds.
0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S3,ℝP3 or L(3, 1). Similarly, any triangulation of a compact, orientable, irreducible, ∂-irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal triangulation is defined. It is shown if M is a compact, orientable, irreducible, ∂-irreducible 3-manifold having no essential annuli and distinct from the 3-cell, then \ring M admits an ideal triangulation; furthermore, it is shown that any ideal triangulation of such a 3-manifold can be modified to a 0-efficient ideal triangulation. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is S3 and the triangulation has precisely two vertices. 0-efficient triangulations of 3-manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each boundary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal surface. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition problem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds.