2-sphere

Definition

The 2-sphere, denoted $S^2$, is defined as the sphere of dimension 2. Below are some explicit definitions.

As a subset of Euclidean space

The 2-sphere in ${\mathbb{R}}^3$ with center $(x_0,y_0,z_0)$ and radius $r>0$ is defined as the following subset of ${\mathbb{R}}^3$:

$\{(x,y,z)\mid (x-x_0{)}^2+(y-y_0{)}^2+(z-z_0{)}^2=r^2\}$

In particular, the unit 2-sphere centered at the origin is defined as the following subset of ${\mathbb{R}}^3$:

$\{(x,y,z)\mid x^2+y^2+z^2=1\}$

Note that all 2-spheres are equivalent up to translations and dilations, and in particular, they are homeomorphic as topological spaces.

Equivalent spaces

Space	How it is equivalent to the 2-sphere viewed geometrically
complex projective line $\mathbb{C}{\mathbb{P}}^1$ or ${\mathbb{P}}^1(\mathbb{C})$	Stereographic projection; hence homeomorphic and diffeomorphic
one-point compactification of the Euclidean plane	Stereographic projection; hence homeomorphic and diffeomorphic
double cover (and hence also universal cover) of the real projective plane $\mathbb{R}{\mathbb{P}}^2$ or ${\mathbb{P}}^2(\mathbb{R})$	Identification of antipodal points gives the double cover from $S^2$ to $\mathbb{R}{\mathbb{P}}^2$
boundary of 3-simplex	homeomorphism arising from a straight line homotopy
hollow cube in ${\mathbb{R}}^3$	homeomorphism arising from a straight line homotopy

Algebraic topology

Homology groups

Further information: homology computation for spheres

The homology groups with coefficients in $\mathbb{Z}$ are as follows: $H_0(S^2)\cong H_2(S^2)\cong \mathbb{Z}$, and all other homology groups are zero. The reduced homology groups with coefficients in $\mathbb{Z}$ are as follows: ${\tilde{H}}_2(S^2)\cong \mathbb{Z}$, and all other reduced homology groups are zero.

More generally, for homology with coefficients in any module $M$ over any commutative unital ring $R$, $H_0(S^2;M)\cong H_2(S^2;M)\cong M$ and all other homology groups are zero. For reduced homology, ${\tilde{H}}_2(S^2;M)\cong M$, and all other reduced homology groups are zero.

Cohomology groups

Further information: cohomology computation for spheres

The cohomology groups with coefficients in $\mathbb{Z}$ are as follows: $H^0(S^2)\cong H^2(S^2)\cong \mathbb{Z}$, and all other cohomology groups are zero. The cohomology ring is $\mathbb{Z}[x]/(x^2)$, where $x$ is an additive generator of $H^2(S^2)$.

More generally, for coefficients in any commutative unital ring $R$, $H^0(S^2;R)\cong H^2(S^2;R)\cong R$, and the other cohomology groups are zero. The cohomology ring is $R[x]/(x^2)$, where $x$ is a generator of $H^2(S^2)$ as a $R$-module.

Homotopy groups

Further information: homotopy computation for spheres

Value of $n$	General name for ${\pi }_n$	What is ${\pi }_n(S^2)$?
0	set of path components	one-point set; so $S^2$ is a path-connected space
1	fundamental group	trivial group; so $S^2$ is a simply connected space
2	second homotopy group	$\mathbb{Z}$, i.e., the group of integers. The identity map from $S^2$ to itself is a generator for this group.
3	third homotopy group	$\mathbb{Z}$, i.e., the group of integers. The generating element of this is termed the Hopf fibration and the fibers of the map are all homeomorphic to the circle $S^1$.
4	fourth homotopy group	$\mathbb{Z}/2\mathbb{Z}$ -- Fill this in later

Algebraic and coalgebraic structure

Algebraic structure

The 2-sphere is not a H-space, i.e., it cannot be given a multiplicative structure satisfying the properties of identity and associativity up to homotopy. In particular, it does not arise from a topological monoid or a topological group.

Coalgebraic structure

Further information: comultiplication on spheres

The 2-sphere has a natural choice of comultiplication, i.e., if we choose $p$ as a basepoint, there is a map:

$(S^2,p)\to (S^2,p)\vee (S^2,p)$

where $\vee$ denotes the wedge sum and the map is a continuous based map, i.e., a continuous map preserving basepoint. This map is cocommutative and coassociative up to homotopy, and it is used to give an abelian group structure to the set of homotopy classes from the based 2-sphere to any based topological space. This group is termed the second homotopy group.