Let F be a family of compact convex sets in Rd. We say
that F has a \emph{topological ρ-transversal of index (m,k)}
(ρ<m, 0<k≤d−m) if there are, homologically, as many transversal
m-planes to F as m-planes containing a fixed ρ-plane in
Rm+k.
Clearly, if F has a ρ-transversal plane, then F
has a topological ρ-transversal of index (m,k), for ρ<m and k≤d−m. The converse is not true in general.
We prove that for a family F of ρ+k+1 compact convex sets in
Rd a topological ρ-transversal of index (m,k) implies an
ordinary ρ-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