Cup product and intersection

This is a handout for my algebraic topology course. The goal is to explain a geometric interpretation of the cup product. Namely, if X is a closed oriented smooth manifold, if A and B are oriented submanifolds of X, and if A and B intersect transversely, then the Poincaré dual of A∩B is the cup product of the Poincaré duals of A and B. As an application, we prove the Lefschetz fixed point formula on a manifold. As a byproduct of the proof, we explain why the Euler class of a smooth oriented vector bundle is Poincaré dual to the zero section. 1 Statement of the result A question frequently asked by algebraic topology students is: “What does cup product mean?” Theorem 1.1 below gives a partial answer to this question. The theorem says roughly that on a manifold, cup product is Poincaré dual to intersection of submanifolds. In my opinion, this is the most important thing to know about cup product. To state the theorem precisely, let X be a closed oriented smooth manifold of dimension n. Let A and B be closed oriented smooth submanifolds of X of dimensions n − i and n − j respectively. Assume that A and B intersect transversely. This means that for every p ∈ A ∩ B, the map TpA ⊕ TpB → TpX induced by the inclusions is surjective. This condition can be obtained by perturbing A or B. Then A ∩B is a submanifold of dimension n− (i+ j), and there is a short exact sequence 0 −→ Tp(A ∩B) −→ TpA⊕ TpB −→ TpX −→ 0. This exact sequence determines an orientation ofA∩B. We will adopt the following convention. We can choose an oriented basis u1, . . . , un−i−j , v1, . . . , vj , w1, . . . , wi for TpX such that u1, . . . , un−i−j , v1, . . . , vj is an oriented basis for TpA and u1, . . . , un−i−j , w1, . . . , wi is an oriented basis for TpB. We then declare that u1, . . . , un−i−j is an oriented basis for Tp(A∩B). If A and B have complementary dimension, i.e. if i + j = n so that A ∩ B is a finite set of points, then a point p is positively oriented if and only if the isomorphism TpA⊕ TpB ' TpX is orientation preserving.