Topological sphere

The sphere is an important object in the mathematical sub-areas of topology and differential geometry . From the point of view of these mathematical areas, the sphere is a manifold . It is important because it is the simplest example of a compact manifold.

Spheres in topology

A topological sphere is understood to be a topological manifold that is homeomorphic to the unitary sphere in R ^{n + 1} . It is designated with . From a topology perspective, the surface of a cube, for example, is also a 2-sphere. The 1-dimensional sphere is also called a circle . ${\ displaystyle S ^ {n}}$

A topological sphere is obtained by gluing the edges of two spheres with one another, reversing their orientation. ${\ displaystyle n}$ ${\ displaystyle n}$

The sphere is also precisely the Alexandroff compactification of and therefore compact . It is also created by gluing the edge of a -dimensional, closed full sphere (here the compactness follows from the fact that the sticking together (as a final topology formation ) is continuous and therefore maps the compact, closed full sphere onto a compact). ${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle n}$

The -sphere of is homeomorphic to the geometric edge of every n-simplex and is in this sense a curved polyhedron . ${\ displaystyle (n-1)}$ ${\ displaystyle S ^ {n-1}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

This is not a subset of a homeomorphic, as is evident from Borsuk's antipodal theorem . This in turn implies the so-called invariance of the dimension . ${\ displaystyle S ^ {n} \ subset \ mathbb {R} ^ {n + 1}}$ ${\ displaystyle \ mathbb {R} ^ {m} (m \ leq n)}$

This is not a retract from . This means that there is no continuous mapping of the n-dimensional unit sphere onto the (n-1) -dimensional sphere , which leaves the points of the fix. This statement is equivalent to the statement of Brouwer's Fixed Point Theorem . ${\ displaystyle S ^ {n-1}}$ ${\ displaystyle B ^ {n} = \ {x \ in \ mathbb {R} ^ {n}: \ | x \ | _ {2} \ leq 1 \}}$ ${\ displaystyle B ^ {n}}$ ${\ displaystyle S ^ {n-1}}$ ${\ displaystyle S ^ {n-1}}$

This can also be understood as the homogeneous space formed as a quotient . The Hair measure of carries over while a rotation invariant measure on the sphere . ${\ displaystyle S ^ {n}}$ ${\ displaystyle SO (n + 1) / SO (n)}$ ${\ displaystyle SO (n + 1)}$ ${\ displaystyle S ^ {n}}$

Differentiable structures

In the area of differential topology , the sphere is equipped with a differentiable structure , so that one can speak of differentiable mappings on the sphere. On a topological manifold it is usually possible to define different, incompatible, differentiable structures. The stereographic projection, for example, induces the differentiable structure most viewed on the sphere. In the case of the sphere, it depends on the dimension whether there are other differentiable structures. The mathematician John Milnor dealt with this topic and showed the existence of so-called exotic spheres .

Statements about spheres

Poincaré conjecture

The Poincaré conjecture reads:

Every closed, simply connected 3-dimensional manifold is homeomorphic to the 3-sphere

In addition, there is a generalization of the conjecture to n -dimensional manifolds in the following form:

Every closed n-manifold with the homotopy type of an n-sphere is homeomorphic to the n-sphere.

For the case n = 3, this generalized conjecture agrees with the original Poincaré conjecture. In the case of Stephen Smale in 1960 , it was proven by Michael Freedman in the 1982 case . The Russian mathematician Grigori Perelman proved the Poincaré conjecture in 2002, for which he was awarded the Fields Medal . However, he refused. ${\ displaystyle n> 4}$ ${\ displaystyle n = 4}$

Exotic spheres

The American mathematician John Milnor discovered in 1956 that there are differentiable manifolds that are homeomorphic to the 7-sphere, but that their differentiable structures are not compatible with one another. Together with the Swiss mathematician Michel Kervaire , he showed that there are 15 different differentiable structures (28 if orientation is taken into account ) for the 7-sphere . ${\ displaystyle S ^ {7}}$

Set of spheres

The mathematicians Harry Rauch , Wilhelm Klingenberg and Marcel Berger were able to show that given certain preconditions for the curvature of compact Riemannian manifolds, these are homeomorphic to the sphere, that is, they are topological spheres. This statement has been tightened. It could even be shown that this Riemannian manifold is diffeomorphic to the sphere with the normal differentiable structure.

Topological groups

The only spheres that simultaneously have a group structure and thus form a topological group are the 0, 1 and 3 spheres. The 0-sphere corresponds to the group , the 1-sphere corresponds to the Lie group U (1) and the 3-sphere corresponds to the Lie group SU (2). ${\ displaystyle \ mathbb {Z} / 2 \ mathbb {Z}}$ ${\ displaystyle S ^ {1}}$ ${\ displaystyle S ^ {3}}$

The 7-sphere is not a topological group, but it is a real Moufang loop , since it can be described by the octonion with the amount 1.

Parallelizability

The 1, 3 and 7 spheres are the only spheres that can be parallelized . From the hedgehog theorem it follows that a sphere with even dimensions cannot be parallelized. The exceptional position of the 1, 3 and 7 spheres is related to the existence of the division algebras.

literature

Lutz Führer: General topology with applications . Vieweg Verlag , Braunschweig 1977, ISBN 3-528-03059-3 .

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 ).

John M. Lee: Introduction to Topological Manifolds . 2nd Edition. Springer-Verlag , New York et al. 2011, ISBN 978-1-4419-7939-1 .

John M. Lee: Introduction to Smooth Manifolds . Springer-Verlag, New York et al. 2003, ISBN 0-387-95495-3 .

Horst Schubert : Topology . 4th edition. BG Teubner Verlag , Stuttgart 1975, ISBN 3-519-12200-6 .

Individual evidence

^ H. Schubert: Topology . 1975, p. 166 .
^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 186 .
↑ L. Guide: General topology with applications . 1977, p. 176 .
^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 158 .
^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 158-159 .

[1] H. Schubert: Topology . 1975, p. 166 .

[2] E. Harzheim: Introduction to combinatorial topology . 1978, p. 186 .

[3] L. Guide: General topology with applications . 1977, p. 176 .

[4] E. Harzheim: Introduction to combinatorial topology . 1978, p. 158 .

[5] E. Harzheim: Introduction to combinatorial topology . 1978, p. 158-159 .