Hopf's theorem

The Hopf theorem is a proposition from the mathematical branch of algebraic topology . It goes back to an important work by the mathematician Heinz Hopf , which appeared in volume 96 of the Mathematische Annalen in 1927. The sentence is sometimes referred to as Brouwer-Hopf's sentence because Heinz Hopf achieved his sentence as an extension of an earlier result by Luitzen Egbertus Jan Brouwer .

In the context of the Thom - Pontryagin theory it is shown that Hopf's theorem follows as a special case from a superordinate theorem .

Formulation of the sentence

In modern terms, the sentence can be stated as follows:

For each coherent , oriented , closed , differentiable n-manifold     (   ) is mapping degree a Homotopieinvariante of pictures in the n-sphere in such a way that two continuous maps  which the diversity     in the n-sphere     map, if and homotopic are, if they have the same degree of mapping     . ${\displaystyle M}$ ${\displaystyle n\in \mathbb {N} }$   ${\displaystyle f_{1},f_{2}}$ ${\displaystyle M}$ ${\displaystyle S^{n}\subset {\mathbb {R} }^{n+1}}$ ${\displaystyle D(f_{1})=D(f_{2})}$

Because every whole number can be implemented     as a degree of mapping of a suitably chosen continuous mapping of the given manifold     into the n-sphere, the following even applies: ${\displaystyle \gamma \in \mathbb {Z} }$ ${\displaystyle M}$

Is     the quantity system of homotopy classes of continuous maps    , the mediated mapping degree feature   a bijection  , through to each   exactly one homotopy class     to     hear. ${\displaystyle [M,S^{n}]}$ ${\displaystyle f\colon \,M\to S^{n}}$   ${\displaystyle D}$   ${\displaystyle {\overline {D}}\colon \,[M,S^{n}]\to \mathbb {Z} }$ ${\displaystyle \gamma \in \mathbb {Z} }$   ${\displaystyle [f]\in [M,S^{n}]}$ ${\displaystyle {\overline {D}}([f])=D(f)=\gamma }$

The general theorem for dimension 2

The theorem for n = 2 is essentially the result that Brouwer presented in his work in Volume 71 of the Mathematische Annalen in 1912.

The special phrase for the sphere

The main application of Hopf's theorem is in the following case   : ${\displaystyle M=S^{n}}$

Two continuous mappings of the n-sphere in themselves are homotopic if and only if their degrees of mapping match.

It turns out that the above bijection, mediated by the degree of mapping, even conveys a group isomorphism of the n-th homotopy group of the n-sphere onto the group of whole numbers . ${\displaystyle deg:(\pi _{n}S^{n},*)\simeq (\mathbb {Z} ,+)}$

Furthermore, i. V. m. the multiplication rule for the degree of mapping is the following corollary :

For two continuous mappings of     the n-sphere in itself, the linked functions   and are     always homotop. ${\displaystyle f_{1},f_{2}}$   ${\displaystyle f_{1}\circ f_{2}}$ ${\displaystyle f_{2}\circ f_{1}}$

literature

Original work

• Heinz Hopf: mapping classes of n-dimensional manifolds . In: Math. Ann . tape 96 , 1927, pp. 209–224 ( maths.ed.ac.uk [PDF; 1.5 MB ]).
• Luitzen Egbertus Jan Brouwer: Illustration of manifolds . In: Math. Ann . tape 71 , 1912, pp. 97–115 ( digizeitschriften.de ).

Monographs

• Glen E. Bredon : Topology and Geometry (=  Graduate texts in mathematics . Volume 139 ). Springer-Verlag, New York [u. a.] 1993, ISBN 3-540-97926-3 .
• Theodor Bröcker , Tammo tom Dieck : Kobordism Theory (=  Lecture Notes in Mathematics . Volume 178 ). Springer-Verlag, Berlin [a. a.] 1970, ISBN 3-540-05341-7 .
• James Dugundji : Topology . 8th edition. Allyn and Bacon, Boston 1973.
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Tammo tom Dieck : Topology . 2., completely reworked. and exp. Edition. De Gruyter, Berlin [a. a.] 2000, ISBN 3-11-016236-9 .

Individual evidence

1. ↑ Heinz Hopf: mapping classes of n-dimensional manifolds . In: Math. Ann . tape 96 , p. 209 ff .
2. ↑ Schubert: S. 289-290.
3. ↑ The so-called Pontryagin-Thom construction is used as an essential tool ; see. Chapter II, Section 16 in Bredon: p. 118 ff. And Chapter III in Bröcker / tom Dieck: p. 24 ff.
4. ↑ tom Dieck: pp. 284-285.
5. ^ Dugundji: p. 352.
6. ↑ Harzheim: p. 169.
7. ^ Bredon: p. 124.
8. ↑ Harzheim: pp. 168, 136.
9. ↑ Harzheim: p. 169.