Clean intersection

Submanifolds $A,B\subseteq M$ intersect cleanly when $A\cap B$ is an embedded submanifold andat every $x\in A\cap B$. A transverse intersection is the special case in which $T_{x}A+T_{x}B=T_{x}M$.

Submanifolds $A,B\subseteq M$ intersect transversely when $T_{x}A+T_{x}B=T_{x}M$ at every intersection point. The transverse intersection theorem then determines the dimension and smoothness of $A\cap B$.

For complementary-dimensional oriented submanifolds of an oriented manifold meeting transversely in finitely many points, the smooth intersection number is the sum of their local orientation signs. Geometric disjointness implies number zero, but number zero alone need not imply geometric disjointness. Without coherent orientations, use the count modulo $2$.