Sphere

Definition

As a subset of Euclidean space

The unit $n$-sphere $S^{n}$ is defined as the subset of Euclidean space $\mathbb {R} ^{n+1}$ comprising those points whose distance from the origin is $1$.

$S^{n}=\{(x_{0},x_{1},\dots ,x_{n})\mid x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=1\}$

Particular cases

$n$	sphere $S^{n}$
0	$S^{0}$ -- discrete two-point space
1	circle
2	2-sphere
3	3-sphere

Equivalent spaces

Space	How strongly is it equivalent to the circle?
boundary of the $(n+1)$-hypercube	homeomorphic; not diffeomorphic because of sharp edges
boundary of the $(n+1)$-simplex	homeomorphic; not diffeomorphic because of sharp edges
ellipsoid in $\mathbb {R} ^{n+1}$	equivalent via affine transformation
one-point compactification of $\mathbb {R} ^{n}$	homeomorphic via stereographic projection
for $n\geq 2$: universal cover of real projective space $\mathbb {R} \mathbb {P} ^{n}$, which is the space of lines in $\mathbb {R} ^{n+1}$	homeomorphic, diffeomorphic, also isometric if we choose the natural metric.

Algebraic topology

Homology groups

With coefficients in $\mathbb {Z}$, the $n$-sphere $S^{n}$ has $H_{0}(S^{n})\cong H_{n}(S^{n})\cong \mathbb {Z}$ and $H_{k}(S^{n})=0$ for $k\notin \{0,n\}$. In particular, the $n$-sphere is $(n-1)$-connected.

Interpretations in terms of various homology theories:

Fill this in later

With coefficients in any $R$-module $M$ for a ring $R$, the $n$-sphere $S^{n}$ has $H_{n}(S^{n})=M$ and $H_{k}(S^{n})=0$ for all $k\neq n$.

Cohomology groups and cohomology ring

With coefficients in $\mathbb {Z}$, the $n$-sphere $S^{n}$ has $H^{0}(S^{n})\cong H^{n}(S^{n})\cong \mathbb {Z}$ and $H^{k}(S^{n})=0$ for $k\notin \{0,n\}$. In particular, the $n$-sphere is $(n-1)$-connected.

With coefficients in any $R$-module $M$ for a ring $R$, the $n$-sphere $S^{n}$ has $H^{n}(S^{n})=M$ and $H^{k}(S^{n})=0$ for all $k\neq n$.

The cohomology ring is isomorphic to $\mathbb {Z} [x]/(x^{2})$, where $x$ is a generator of the <amth>n^{th}</math> cohomology.

Homotopy groups

Further information: n-sphere is (n-1)-connected

For $k<n$, the homotopy group $\pi _{k}(S^{n})$ is the trivial group. $\pi _{n}(S^{n})\cong \mathbb {Z}$, with the identity map $S^{n}\to S^{n}$ being a generator.

The cases $k>n$ are discussed below:

Case	What can we say?
$n=1$	$\pi _{k}(S^{n})$ is trivial for all $k>1$
$k=2n-1$	$\pi _{k}(S^{n})\cong \mathbb {Z}$
$k>n,k\neq 2n-1$	$\pi _{k}(S^{n})$ is a finite abelian group