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 formu...

This is the note for my talk in Graduate Student Seminar NEU. W. Massey defined Massey product as a higher order cohomology operation, which is a generalization of cup product. A first application of Massey product is in Knot theory showing that Borromean rings are linked with zero linking numbers. As another application, people can use Massey product describe the differentials in spectral sequ...