3 sphere

The 3-dimensional sphere or 3-sphere for short is an important object in mathematics and physics. Along with Euclidean space, it is the simplest example of a 3-dimensional manifold . ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

definition

A 3-dimensional sphere is understood as a topological manifold that is homeomorphic to the unitary sphere ${\displaystyle \mathbb {R} ^{4}}$ . The latter is denoted by. ${\displaystyle S^{3}}$

The unit sphere is the set of points in 4-dimensional Euclidean space with a distance of one from the origin , i.e. ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {R} ^{4}}$

${\displaystyle S^{3}:=\{x\in \mathbb {R} ^{4}\colon \|x\|_{2}=1\}}$,

where is the Euclidean norm . It can be understood as the edge of the 4- unit sphere and is therefore also referred to as. ${\displaystyle \|\cdot \|_{2}}$ ${\displaystyle B^{4}}$ ${\displaystyle \partial B^{4}}$

properties

Geometric properties

The 3-dimensional hypersurface (3-v) a 3-sphere of radius is ${\displaystyle r}$

${\displaystyle 2\pi ^{2}r^{3}\,}$

and the 4-dimensional hypervolume of a 4-sphere (the 4-volume of the 4-dimensional area within this 3-sphere)

${\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}\pi ^{2}r^{4}.}$

Accordingly, the 4 volume is of . ${\displaystyle {\tfrac {\pi ^{2}}{2}}}$ ${\displaystyle B^{4}}$

Every non-empty intersection of a 3-sphere with a 3-dimensional hyperplane is a 2-sphere or a single point.

The 3-sphere of the radius has the constant, positive cutting curvature . ${\displaystyle r}$ ${\displaystyle {\frac {1}{r^{2}}}}$

Topological properties

The 3-sphere has no edge , is compact and simply connected . Their homology groups are

${\displaystyle H_{i}(S^{3})={\begin{cases}\\\\\end{cases}}}$	${\displaystyle \mathbb {Z} }$,		if   ${\displaystyle i\in \{1,3\}}$
	${\displaystyle \{0\}}$		otherwise.

Each topological space with these homology groups is called the 3- homology sphere .

It is homeomorphic to the one-point compactification of and is the homogeneous space ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle S^{3}\cong SO(4)/SO(3)}$.

Differentiable structure

Like every 3-dimensional manifold, the 3-sphere has a unique differential structure and a unique PL structure according to Moise's theorem .

Round metric

The embedding as a unit sphere in gives the sphere the “round metric” with a constant curvature of 1. In particular, with this metric it becomes a symmetrical space with an isometric group . ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle SO(4)}$

Each metric of constant section curvature is a multiple of the round metric.

The 3-sphere as a Lie group

→ Main article : SU (2)

The 3-sphere is a non-Abelian group . It coincides with the group of the unit quaternions ${\displaystyle S^{3}}$

${\displaystyle \left\{x\in \mathbb {H} \,\mid \,x{\bar {x}}=1\right\}}$

with and . The image ${\displaystyle x=x_{0}+x_{1}\mathrm {i} +x_{2}\mathrm {j} +x_{3}\mathrm {k} }$ ${\displaystyle {\bar {x}}=x_{0}-x_{1}\mathrm {i} -x_{2}\mathrm {j} -x_{3}\mathrm {k} }$

${\displaystyle x_{0}+x_{1}\mathrm {i} +x_{2}\mathrm {j} +x_{3}\mathrm {k} \mapsto {\begin{pmatrix}z_{1}&z_{2}\\-{\overline {z_{2}}}&{\overline {z_{1}}}\end{pmatrix}}}$with and ${\displaystyle z_{1}:=x_{0}+\mathrm {i} _{\mathbb {C} }x_{1}}$ ${\displaystyle z_{2}:=x_{2}+\mathrm {i} _{\mathbb {C} }x_{3}}$

is an isomorphism of the quaternions in the ring of complex 2 × 2 matrices , referring to the subgroup of unitary matrices ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathbb {C} ^{2\times 2}}$ ${\displaystyle S^{3}}$

${\displaystyle \left\{{\begin{pmatrix}z_{1}&z_{2}\\-{\overline {z_{2}}}&{\overline {z_{1}}}\end{pmatrix}}\in \mathbb {C} ^{2\times 2}{\Bigg |}\;|z_{1}|^{2}+|z_{2}|^{2}=1\right\}=:SU(2)}$,

maps. They make up a Lie group that bears the name . ${\displaystyle SU(2)}$

This bijection is at the same time a diffeomorphism

${\displaystyle \quad \mathbb {H} ^{\times }\supset S^{3}\to SU(2)\subset \left(\mathbb {C} ^{2\times 2}\right)^{\times }\quad ;\quad (z_{1},z_{2})\mapsto {\begin{pmatrix}z_{1}&z_{2}\\-{\overline {z_{2}}}&{\overline {z_{1}}}\end{pmatrix}}}$

The 3-sphere is the simplest non-Abelian compact Lie group and is of particular importance in the standard model of elementary particle physics. ${\displaystyle S^{3}=SU(2)}$

Poincaré conjecture

→ Main article : Poincaré conjecture

The 3-sphere is the only simply connected, compact 3-manifold.

Vector fields on the 3-sphere

As a Lie group, the 3-sphere can be parallelized . An example of three linearly independent vector fields on the unit sphere im is ${\displaystyle \mathbb {R} ^{4}}$

${\displaystyle V_{1}(x_{1},x_{2},x_{3},x_{4})=(-x_{2},x_{1},-x_{4},x_{3}),V_{2}(x)=(-x_{3},x_{4},x_{1},-x_{2}),V_{3}(x)=(-x_{4},-x_{3},x_{2},x_{1})}$.

Heegaard decompositions

The 3-dimensional sphere is obtained by gluing the edges of two 3-dimensional spheres to one another, reversing their orientation .

More generally, the 3-sphere has a clear Heegaard breakdown of gender for each . ${\displaystyle g\geq 0}$ ${\displaystyle g}$

Stretching surgeries

→ Main article : Stretching surgery

Any compact 3-manifold can be constructed by surgery on entanglements in the 3-sphere. ${\displaystyle K\subset S^{3}}$

Spherical 3-manifolds

From the geometry program initiated by Thurston and proven by Perelman it follows that all compact 3-manifolds of finite fundamental group are spherical 3-manifolds (or 3-dimensional spherical space forms ), i.e. they are quotient spaces

${\displaystyle M=\Gamma \backslash S^{3}}$

for a finite group of isometrics of the round metric. ${\displaystyle \Gamma \subset SO(4)}$

Examples of 3-dimensional spherical spatial shapes are the lens spaces or the Poincaré homology sphere .

literature

• Nikolai Saveliev: Lectures on the topology of 3-manifolds. An introduction to the Casson invariant. De Gruyter Textbook. Walter de Gruyter, Berlin 1999, ISBN 3-11-016271-7