2-sphere

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

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

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

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

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

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

Homology groups
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
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.

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
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.