Hopf theorem

The set of Hopf is a theorem from the mathematical branch of algebraic topology. It goes back to an important work of the mathematician Heinz Hopf, which appeared in Volume 96 of Mathematische Annalen in 1927. The set is locally referred to as a set of Brouwer - Hopf because Heinz Hopf scored his sentence in extension of an earlier result of Luitzen Egbertus January Brouwer.

Under the Thom - Pontryagin theory it is shown that the set of Hopf follows as a special case of a parent theorem.

• 2.1 Original Articles
• 2.2 monographs

Wording of the sentence

The set can be specified in a modern formulation as follows:

Because every integer can be realized as a mapping degree of a suitably chosen continuous mapping of the given manifold into the n- sphere, then even applies:

The general rate for the dimension 2

The rate of n = 2 is essentially the one result which Brouwer has presented his work in the band 71 of the Mathematische Annalen in 1912.

The special rate for the sphere

The main application is the set of Hopf in the case:

This shows that the above -mediated by the imaging provides an even level of bijection of the nth group isomorphism homotopy n- sphere to the group of integers.

Furthermore, there is conjunction. the multiplication rule for the mapping degree the following corollary: