  The cobordism group $N(M^n)$ of codimension-one immersions in the
$n$-manifold $M^n$ has a natural filtration induced by any cellular
decomposition. The problem addressed in this paper is the explicit computation
of the graded group $gr^*N(M^n)$. We introduce some new invariants for
immersions enlightening the Atiyah-Hirzebruch spectral sequence associated to
$N(M)$, which are of combinatorial-geometric nature. Explicit computations are
developed for $n\leq7$, and the group structure is also investigated for
orientable 4-manifolds.

Geometric Topology