Abstract: Borel conjecture is a central conjecture in geometric topology (and still open for higher dimension): any homotopy equivalent between two closed aspherical manifolds (i.e. universal cover is contractable) is homotopic to a homeomorphism (this is false for general manifold).
There are a lot of crucial works in this topic. I will introduce some backgrounds first. Then I will focus on Gromov’s approach in the case of negative curvature (in this case, Borel Conjecture was completely proved by Farrell-Jones later) from the point of view of geodesic dynamics: let M and N be two closed negatively curved manifolds, if M is homotopy equivalent to N, then the geodesic foliations G(M) and G(N) are homeomorphic (i.e. there is a homeomorphism between unit tangent bundles S(M) —> S(N) sending leaves from G(M) into leaves from G(N)).
In the end, I will try to replace the requirement “negatively curved” by weaker one “with geodesic flow of Anosov type” , and see what can we say: do this kind of manifolds satisfy Gromov’s approach or even the Borel Conjecture?
