  It is conjectured since long that for any convex body $K \subset
\mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at
least $2n$ normals from different points on the boundary of $K$. The conjecture
is known to be true for $n=2,3,4$.
  Motivated by a recent results of Y. Martinez-Maure, and an approach by A.
Grebennikov and G. Panina, we prove the following: Let a compact smooth
$m$-dimensional manifold $M^m$ be immersed in $ \mathbb{R}^n$. We assume that
at least one of the homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k<m$
vanishes. Then under mild conditions, almost every normal line to $M^m$
contains an intersection point of at least $\beta +4$ normals from different
points of $M^m$, where $\beta$ is the sum of Betti numbers of $M^m$.
