  Let $f:V^n\looparrowright M^m$ be a smooth generic immersion. Then the set of
points, that have at least $k$ preimages is an image of a (non-generic)
immersion. If the manifolds $V^n$ and $M^m$ are oriented and $m-n$ is even,
then the manifold of $k$-fold points is also oriented. In this paper we compute
the oriented bordism class of the manifold of $k$-fold points in terms of the
differential $df$, provided the tangent bundle of the manifold $M^m$ has a
nowhere zero cross-section.
