  For an integer $r\ge 2$, the space of $r$-immersions of $M$ in $\mathbb R^n$
is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that at
most $r-1$ points of $M$ are mapped to the same point in $\mathbb R^n$. The
space of $r$-immersions lies "between" the embeddings and the immersions. We
calculate the connectivity of the layers in the homological Taylor tower for
the space of $r$-immersions in $\mathbb R^n$ (modulo immersions), and give
conditions that guarantee that the connectivity of the maps in the tower
approaches infinity as one goes up the tower. We also compare the homological
tower with the homotopical tower, and show that up to degree $2r-1$ there is a
"Hurewicz isomorphism" between the first non-trivial homotopy groups of the
layers of the two towers.
