Künneth

Künneth is a concept in mathematics, specifically in algebraic topology, named after the mathematician Walter Künneth. It refers to a theorem that provides a way to compute the homology or cohomology groups of a product space from the homology or cohomology groups of the individual spaces. The Künneth theorem is a fundamental tool in the study of topological spaces and their algebraic invariants.

The Künneth theorem states that for two topological spaces X and Y, the homology groups of the product space X × Y are related to the homology groups of X and Y through a tensor product and a torsion product. Specifically, the theorem provides a short exact sequence that relates the homology groups of the product space to the tensor product of the homology groups of the individual spaces and the torsion product, which measures the failure of the tensor product to be exact.

The Künneth theorem has several important applications in algebraic topology. It is used to compute the homology groups of product spaces, which are common in many areas of mathematics. For example, it is used in the study of fiber bundles, where the total space can be viewed as a product of the base space and the fiber. The theorem is also used in the study of cohomology rings, which are important in the classification of manifolds and other topological spaces.

One of the key features of the Künneth theorem is that it provides a way to compute the homology or cohomology groups of a product space without explicitly constructing the product space. This is particularly useful when the product space is too large or complex to construct explicitly. The theorem also provides a way to understand the relationship between the homology or cohomology groups of the individual spaces and the product space.

The Künneth theorem has been generalized and extended in various ways. For example, there are versions of the theorem for cohomology with coefficients in a ring, and for other algebraic invariants such as homotopy groups. The theorem has also been used to study the cohomology of Lie groups and other algebraic structures. These generalizations and extensions have further increased the applicability and importance of the Künneth theorem in mathematics.