The Relation between Cup Product and Wedge Product

[lavinia]

On a smooth triangulation of a manifold differential forms can be viewed as real cochains by integration. The wedge product of two forms gives another real cochain. So does their cup product.

- are they cohomologous?

- Is there a limiting process that relates them?

 

[mathwonk]

isn't this the content of theorem 14.28 of Bott-Tu, and the assertions on page 192?

I.e. they say that the deRham cohomology with wedge products, is isomorphic as an algebra to the Cech cohomology with cup products, and they seem to say that this also holds for singular and Cech cohomologies with cup products.

Isn't it more or less Fubini's theorem? i.e. to compare the integration action of a wedge product on a cell, with the cup product value?

Have you tried these "obvious" approaches and run into difficulties checking them out?

 

[lavinia]

isn't this the content of theorem 14.28 of Bott-Tu, and the assertions on page 192?

I.e. they say that the deRham cohomology with wedge products, is isomorphic as an algebra to the Cech cohomology with cup products, and they seem to say that this also holds for singular and Cech cohomologies with cup products.

Isn't it more or less Fubini's theorem? i.e. to compare the integration action of a wedge product on a cell, with the cup product value?

Have you tried these "obvious" approaches and run into difficulties checking them out?

My thought was that one approximates the integral by the product of the integrals on the faces of the cube. Then subdivision should improve the approximation. The integral of the wedge product would be the limit.

Before taking any limits, are the wedge and cup products - viewed as smooth real valued cochains- cohomologous?

I don't know anything about Czech cohomology. Thanks for the info.

 

[mathwonk]

well i am embarrassed to realize i know diddle about cech cohomology myself after all these years, but i want to recommend it very much to you as a beautiful and natural theory.

start from an open cover, in particular look at a polyhedron like an icosahedron. to each vertex associate the open faces (and edges) adjacent to that vertex (the "star" of the vertex). this is the canonical open cover. each open set is thought of as a vertex.

then intersect two such open sets and consider that as a 1 simplex, i.e. an edge.
notice that in our example the intersection of two open sets gives the open faces adjacent to the edge joining the two vertices defining the original open stars.

intersecting three such open stars associated to the vertices of a face, gives that open face. thus abstractly, a k simplex is the intersection of any k+1 open sets of the original open cover.this is a beautiful construction since it allows one to consider functions defined on the open sets associated to the k simplices. this allows a natural way to use say smooth functions as coefficients for ones cohomology rather than just integers.

since one can consider constant functions as well as smooth functions as coefficients, this allows cech cohomology to be used as a bridge between singular and derham cohomology. this is the approach in bott tu.

 

[blue_raver22]

The cup product and wedge product are two important operations in differential geometry that play a crucial role in understanding the topology and geometry of a manifold. Both operations are defined on differential forms, which can be viewed as real cochains on a smooth triangulation of the manifold. The wedge product of two forms is a real cochain that represents the exterior product of the two forms, while the cup product represents the intersection product.

One question that often arises is whether the cup product and wedge product are cohomologous. In other words, can one be obtained from the other by a homotopy or deformation? The answer to this question is not always straightforward. In general, the cup product and wedge product are not always cohomologous, but there are certain cases where they can be related through a homotopy. For example, on a compact oriented manifold, the cup product and wedge product are cohomologous.

Another way to relate the cup product and wedge product is through a limiting process. This is known as the de Rham limit and it allows us to express the cup product in terms of the wedge product. The de Rham limit is defined as the limit of the cup product as the degree of the forms involved goes to infinity. In this limit, the cup product can be expressed as a combination of wedge products and exterior derivatives. This relationship is particularly useful in studying the topology of a manifold, as it allows us to compute the cup product using only the wedge product.

In conclusion, the cup product and wedge product are both important operations in differential geometry that can be used to study the topology and geometry of a manifold. While they are not always cohomologous, there are cases where they can be related through a homotopy. Additionally, the de Rham limit provides a useful way to express the cup product in terms of the wedge product, allowing for efficient computations in certain cases.