Set of spheres (topology)

In topology , a branch of mathematics , the theorem of spheres is a fundamental theorem from the theory of 3-dimensional manifolds . He was proven by Christos Papakyriakopoulos in 1957 .

Just like the loop theorem known under the name of Dehn's Lemma , it establishes a connection between the homotopy theory (which can be formulated in algebraic terms) and the geometric topology of 3-manifolds; both theorems form the basis for large parts of the 3-manifold theory.

Set of spheres

If an orientable 3-manifold with ${\displaystyle M}$

${\displaystyle \pi _{2}(M)\not =0}$

then there is an embedding

${\displaystyle f\colon S^{2}\rightarrow M}$

With

${\displaystyle \left[f\right]\not =0\in \pi _{2}(M)}$,

where the 2-sphere and the second homotopy group is from . More generally, if a true subgroup is invariant under the action of on , then there is an embedding with . ${\displaystyle S^{2}}$ ${\displaystyle \pi _{2}(M)}$ ${\displaystyle M}$ ${\displaystyle N\subset \pi _{2}(M)}$ ${\displaystyle \pi _{1}(M)}$ ${\displaystyle \pi _{2}(M)}$ ${\displaystyle f\colon S^{2}\rightarrow M}$ ${\displaystyle \left[f\right]\not \in N}$

meaning

The importance of the set of spheres is that it allows homotopy-theoretical information to be implemented “geometrically” (by means of embedded submanifolds). Elements in are represented by continuous mappings by definition ; In general, however, these do not have to be embedded. The theorem of spheres says that in orientable 3-manifolds with there are always embedded spheres that represent nontrivial elements of . (Note, however, that even under the conditions of the theorem of spheres, not every element of has to be represented by an embedded sphere.) ${\displaystyle \pi _{2}(M)}$ ${\displaystyle f\colon S^{2}\rightarrow M}$ ${\displaystyle \pi _{2}(M)\not =0}$ ${\displaystyle \pi _{2}(M)}$ ${\displaystyle \pi _{2}(M)\setminus \left\{0\right\}}$

Applications

A 3-manifold is called irreducible if there is an edge of an embedded 3-ball in every embedded 2-sphere. Irreducible manifolds are important in 3-dimensional topology because they represent (in addition to bundles via ) the “prime factors” in the decomposition of 3-manifolds , as used in the formulation of the geometry theorem . ${\displaystyle M}$ ${\displaystyle S^{2}}$ ${\displaystyle S^{1}}$

From the theorem of spheres it can be concluded:

An orientable 3-manifold is irreducible if and only if is. ${\displaystyle \pi _{2}(M)=0}$

As a consequence, it follows that orientable, irreducible 3-manifolds with an infinite fundamental group must always be aspherical .

literature

• John Hempel: 3-manifolds. Reprint of the 1976 original. American Mathematical Society, Providence RI 2004, ISBN 0-8218-3695-1 .
• William Jaco : Lectures on three-manifold topology (= Regional Conference Series in Mathematics. 43). American Mathematical Society, Providence RI 1980, ISBN 0-8218-1693-4 .
• Christos D. Papakyriakopoulos : On Dehn's lemma and the asphericity of knots. In: Annals of Mathematics . Series 2, Vol. 66, No. 1, 1957, pp. 1-26, doi : 10.2307 / 1970113 .
• John Stallings : Group theory and three-dimensional manifolds (= Yale Mathematical Monographs. 4, ISSN  0084-3377 ). A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale University Press, New Haven CT et al. 1971.

Web links

Hatcher: Notes on Basic 3-Manifold Topology (PDF; 665 kB)