Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

Let $\gamma$ be a smooth generic curve in $\mathbb RP^3$. Denote by $C$ the number of its flattening points, and by $T$ the number of planes tangent to $\gamma$ at three distinct points. Consider the osculating planes to $\gamma$ at the flattening points. Let $N$ denote the total number of points where $\gamma$ intersects these osculating plane transversally. Then $T\equiv[N+\theta(\gamma)C]/2\pmod2$, where $\theta(\gamma)$ is the number of noncontractible components of $\gamma$. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in $\mathbb R^3$ has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.