  Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say
that $\mathcal F $ has a \emph{topological $\rho$-transversal of index $(m,k)$}
($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal
$m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in
$\mathbb R^{m+k}$.
  Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$
has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq
d-m$. The converse is not true in general.
  We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in
$\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an
ordinary $\rho$-transversal. We use this result, together with the
multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann
category of the Grassmannian, and different versions of the colorful Helly
theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences.
