Preprint A168/2002
Geometrical versus Topological Properties of Manifolds and a Remark on PoincarĂ© Conjecture

Krerley Oliveira | Matheus, Carlos

**Keywords: **
Immersions | Hausdorff dimension | finite geometrical type

Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ we prove that if
the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere
$S^n$. A consequence of our main theorems is a conjecture which is
equivalent to Poincaré Conjecture.
Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces are topologically the sphere minus a finite number of
points. A characterization of the $2n$-catenoid is obtained.