Branched covers of hyperbolic manifolds and harmonic maps

Abstract

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 [21] and later generalized by Farrell, Jones and Ontaneda to all dimensions > 5 [8]. In fact, in [21] 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 ethods in geometry [9], [10], [11], [12] (see [13] for a survey). In all the examples mentioned above, one of the manifolds is always a closed hyperbolic manifold. Hence, both manifolds M and N have the homotopy type of a closed hyperbolic manifold (hence the homotopy type of a closed locally symmetric space). We call these examples of hyperbolic homotopy type. In [2], Ardanza also gave counterexamples to the Lawson–Yau conjecture. In his examples, the manifolds M and N are not homotopy equivalent to a closed locally symmetric space; in particular, they are not homotopy