Sphere

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

$$S^n = \{ (x_0,x_1,\dots,x_n) \mid x_0^2 + x_1^2 + \dots + x_n^2 = 1 \}$$

Inductive definition
Inductively, $$S^0$$ is defined as a discrete two-point space, and for any natural number $$n$$, $$S^n$$ is defined as the defining ingredient::suspension of $$S^{n-1}$$.

This definition is illuminative because many of the results about spheres, particularly those involving algebraic topology and the computation of homology and cohomology, are easily derived from corresponding results about suspensions.

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:

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 $$k \ne n$$.

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 $$k \ne n$$.

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

Homotopy groups
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: