Homology Intersection

When two cycles have a transversal intersection  on a smooth manifold , then  is a cycle. Moreover, the homology class that  represents depends only on the homology class of  and . The sign of  is determined by the orientations on , , and .

For example, two curves can intersect in one point on a surface transversally, since

The curves can be deformed so that they intersect three times, but two of those intersections sum to zero since two intersect positively and one intersects negatively, i.e., with the manifold orientation of the curves being the reverse orientation of the ambient space.

On the torus illustrated above, the cycles intersect in one point.

The binary operation of intersection makes homology on a manifold into a ring. That is, it plays the role of multiplication, which respects the grading. When  and , then . In fact, intersection is the dual to the cup product in Poincaré duality. That is, if  is the Poincaré dual to  and  is the dual to  then  is the dual to .

Without the notion of transversal intersection, intersections are not well-defined in homology. On a more general space, even a manifold with singularities, the homology does not have a natural ring structure.