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 $\mathbb{R}{\mathbb{P}}^2$

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