On a certain cup product.

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 formu...

The Uniqueness of a Certain Type of Subdirect Product

We introduce the "$type{lffs}$ subdirect product" and show that every ring is uniquely a $type{lffs}$ subdirect product of a family of $simple{basicls}$ rings. Also we show some applications.

Extended higher cup-product Chern-Simons theories

It is well known that the proper action functional of (4k+ 3)-dimensional U(1)-Chern-Simons theory including the instanton sectors is given on gauge equivalence classes of fields by the fiber integration of the cup product square of classes in degree-(2k + 2) differential cohomology. We first refine this statement from gauge equivalence classes to the full higher smooth moduli stack of fields, ...

On Linear Independence of a Certain Multivariate Infinite Product

A Short Proof of De Shalit’s Cup Product Formula

We give a short proof of a formula of de Shalit, expressing the cup product of two vector valued one forms of the second kind on a Mumford curve in terms of Coleman integrals and residues. The proof uses the notion of double indices on curves and their reciprocity laws.