Transversal Intersection

Two submanifolds  and  in an ambient space  intersect transversally if, for all ,

where the addition is in , and  denotes the tangent map of . If two submanifolds do not intersect, then they are automatically transversal. For example, two curves in  are transversal only if they do not intersect at all. When  and  meet transversally then  is a smooth submanifold of the expected dimension .

In some sense, two submanifolds "ought" to intersect transversally and, by Sard's theorem, any intersection can be perturbed to be transversal. Intersection in homology only makes sense because an intersection can be made to be transversal.

Transversality is a sufficient condition for an intersection to be stable after a perturbation. For example, the lines  and  intersect transversally, as do the perturbed lines , and they intersect at only one point. However,  does not intersect  transversally. It intersects in one point, while  intersects in either none or two points, depending on whether  is positive or negative.

When , then a transversal intersection is an isolated point. If the three spaces have an vector space orientation, then the transversal condition means it is possible to assign a sign to the intersection. If  are an oriented basis for  and  are an oriented basis for , then the intersection is  if  is oriented in  and  otherwise.

More generally, two smooth maps  and  are transversal if whenever  then .