When two cycles have a transversal intersection X_1 intersection X_2 = Y on a smooth manifold M, then Y is a cycle. Moreover, the homology class that Y represents depends only on the homology class of X_1 and X_2. The sign of Y is determined by the orientations on M, X_1, and X_2. For example, two curves can intersect in one point on a surface transversally, since dim X_1 + dim X_2 = 1 + 1 = 2 = dim M - 0.

codimension | cup product | homology | manifold | manifold orientation | transversal intersection | vector space orientation