Branched Covers of Hyperbolic Manifolds and Harmonic Maps
نویسنده
چکیده
Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This conjecture was proved to be false by Farrell and Jones [6] in every dimension in which exotic spheres exist. They constructed examples of homeomorphisms f : M → N between closed negatively curved manifolds for which f is not homotopic to a diffeomorphism. These counterexamples were later obtained also in dimension six by Ontaneda [17] and later generalized by Farrell, Jones and Ontaneda to all dimensions > 5 [8]. In fact, in [17] and [8] examples are given for which f is not even homotopic to a PL homeomorphism. The fact that f is not homotopic to a PL homeomorphism has several interesting strong consequences that imply certain limitations of well known powerful analytic methods in geometry [9], [10], [11], [12] (see [13] for a survey).
منابع مشابه
Geometric Structures on Branched Covers over Universal Links
A number of recent results are presented which bear on the question of what geometric information can be gleaned from the representation of a three-manifold as a branched cover over a fixed universal link. Results about Seifert-fibered manifolds, graph manifolds and hyperbolic manifolds are discussed. Section 0 Introduction. Closed, orientable three-manifolds admit a variety of universal constr...
متن کاملHyperbolic Structures on Branched Covers over Hyperbolic Links
Using a result of Tian concerning deformation of negatively curved metrics to Einstein metrics, we conclude that, for any fixed link with hyperbolic complement, there is a class of irregular branched covering spaces, branched over that link, effectively detectable by their branching indices, which consists entirely of closed hyperbolic manifolds. Section 0 Introduction. One of the elusive compo...
متن کاملA Remark on Khovanov Homology and Two-fold Branched Covers
Examples of knots and links distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover are given. As a result, Khovanov homology does not yield an invariant of two-fold branched covers. Mutation provides an easy method for producing distinct knots sharing a common two-fold branched cover: The mutation in the branch set corresponds to a trivial surger...
متن کامل1 0 Fe b 20 03 Non - left - orderable 3 - manifold groups
We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold bran...
متن کامل