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 $\R^3$ 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.

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 $\R^3$ homeomorphism arising from a straight line homotopy
quotient of closed unit disk in $\R^2$ by the identification of all points in its boundary with each other, i.e., $D^2/\partial D^2$ via identification with one-point compactification of $\R^2$: the interior of the disk can be identified with $\R^2$, and the boundary point is identified with the point at infinity.
suspension of circle (easy, fill in)

Topological space properties

Property Satisfied? Is the property a homotopy-invariant property of topological spaces? Explanation Corollary properties satisfied/dissatisfied
manifold Yes No Via stereographic projection, we see that the 2-sphere minus any point is homeomorphic to the Euclidean plane. Thus, we can give it an atlas with two charts, each chart obtained by removing a different point and mapping homeomorphically to the Euclidean plane. satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.
path-connected space Yes Yes It is a union of two open subsets homeomorphic to the Euclidean plane (hence path-connected), and with non-empty intersection. Thus, it is path-connected. satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)
simply connected space Yes Yes Special case of n-sphere is simply connected for n greater than 1. Follows from union of two simply connected open subsets with path-connected intersection is simply connected, which is a corollary of the Seifert-van Kampen theorem satisfies: simply connected manifold
rationally acyclic space No Yes The second homology group is isomorphic to the group of integers, hence it is nontrivial and has nontrivial torsion-free part. See homology of spheres dissatisfies: acyclic space, weakly contractible space, contractible space
space with Euler characteristic zero No Yes The Euler characteristic is 2, see homology of spheres
space with Euler characteristic one No Yes The Euler characteristic is 2, see homology of spheres
compact space Yes No Can be realized as a closed bounded subset of $\R^3$ satisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness

Algebraic topology

Homology groups

Further information: homology of 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.

Homology-based invariants

Invariant General description Description of value for sphere $S^n$ Description of value for $S^2$
Betti numbers The $k^{th}$ Betti number $b_k$ is the rank of the $k^{th}$ homology group. For $n = 0$, $b_0 = 2$, all other $b_k$ are $0$; for $n > 0$, $b_0 = b_n = 1$, all other $b_k$s are $0$. $b_0 = b_2 = 1$, all other $b_k = 0$.
Poincare polynomial Generating polynomial for Betti numbers $2$ for $n = 0$, $1 + x^n$ for $n > 0$ $1 + x^2$
Euler characteristic $\sum_{k=0}^\infty (-1)^k b_k$ $2$ for $n$ even, $0$ for $n$ odd. 2

Homotopy groups

Further information: homotopy of spheres

Value of $k$ General name for $\pi_k$ What is $\pi_k(S^2)$?
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.

Manifold-type invariants

Invariant Value Explanation
dimension (we can use any of the dimension definitions since this is a connected manifold) 2
smallest dimension of Euclidean space in which it can be immersed 3 The usual embedding in $\R^3$ as a sphere with a center and radius
smallest dimension of Euclidean space in which it can be embedded 3 The usual embedding in $\R^3$ as a sphere with a center and radius
smallest dimension of Euclidean space in which it can be embedded as a flat submanifold -- It's not possible to embed this as a flat manifold anywhere, because the sphere is intrinsically curved (something to do with Gauss-Bonnet theorem, deeper stuff, Euler characteristic being nonzero, links need to be added)
minimum number of charts needed in an atlas for this manifold 2 the complement of any single point gives a chart mapping to all of $\R^2$, thus, we can construct an atlas by using the complements of two distinct points.