Cup product

In algebraic topology, a branch of mathematics, the cup-product defines a multiplicative structure on a cohomology. This provides on the cohomology a ring structure, which is referred to as a cohomology. A similar product for homologies do not exist.

For topological spaces and natural numbers, the cup-product defines a product

With the properties

Definition

The following three definitions for the cup-product are presented. The definition of the cup-product for the singular cohomology is the most common of the three and includes the definitions for the de Rham and simplicial cohomology.

De Rham cohomology

This definition assumes that a differentiable manifold is.

In the de Rham cohomology Kohomologieklassen are represented by differential forms. For the exterior product of differential forms the Leibniz rule. It may therefore be the cup-product of the Kohomologieklassen and represented by

Define and receives because of the Leibniz rule, a well-defined image of the cohomology groups.

Simplicial cohomology

This definition assumes that a simplicial complex is.

In the simplicial cohomology Kohomologieklassen are represented by homomorphisms, where the- th chain group, ie the free abelian group on the set of - simplices is the simplicial complex. For a simplex will be denoted by and the plane spanned by the first and last corners Untersimplizes. For two homomorphisms, we define by

This link satisfies the Leibniz rule, so you get a well-defined image of the cohomology groups by the cup-product of Kohomologieklassen defined by and as the cohomology class of.

Singular cohomology

This definition works for arbitrary topological spaces, in the case of differentiable manifolds and simplicial complexes, the so -defined ring structure on the singular cohomology is isomorphic to the above-defined ring structures on the de Rham cohomology and simplicial.

Be a ring and the singular cohomology with coefficients in. Kohomologieklassen are represented by homomorphisms, where the- th singular chain group, ie the free abelian group on the set of all continuous maps of the standard - is simplex after. One designated or the inclusions of the standard - or -simplex as a " front - dimensional face " or " rear - dimensional face " in the standard - Simplex. For a singular simplex and cochains, we define

This link satisfies the Leibniz rule, so you get a well-defined image of the cohomology groups by the cup-product of Kohomologieklassen defined by and as the cohomology class of.

The cup product defines an additional multiplicative structure on the cohomology groups. You may use this multiplicative structure are sometimes different spaces whose cohomology groups are isomorphic as (additive ) abelian groups.

Sectional shape and signature

For a closed, orientable -dimensional manifold exists an isomorphism. The cup-product thus defines a symmetric bilinear form

The so-called sectional shape.

The signature of is by definition the signature of the symmetric bilinear form. The Hirzebruch signature theorem says that one can represent the signature as a polynomial in the Pontryagin classes.

Simply connected differentiable 4 -manifolds (but not Diffeomorphie ) classified up to homeomorphism by their sectional shape. For the classification simply connected topological 4 -manifolds we still needed in addition to the sectional shape of the Kirby - Siebenmann invariant.