nLab
sphere

Definition

The $n$-dimensional unit sphere , or simply $n$-sphere, is the topological space given by the subset of the $(n+1)$-dimensional Cartesian space ${ℝ}^{n+1}$ consisting of all points $x$ whose distance from the origin is $1$

$$S^n=\{x:{ℝ}^{n+1}\mid \parallel x\parallel =1\}.$$S^n = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = 1 \} \,.

The $n$-dimensional sphere of radius $r$ is

$$S_r^n=\{x:{ℝ}^{n+1}\mid \parallel x\parallel =r\}.$$S^n_r = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = r \} .

Topologically, this is equivalent to the unit sphere for $r>0$, or a point for $r=0$.

This is naturally also a smooth manifold of dimension $n$, with the smooth structure induced from the standard sooth structure on $ℝ$^n.

Infinite dimensional spheres

One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) normed vector space $V$:

$$S(V)=\{x:V\text{such that}\parallel x\parallel =1\}.$$S(V) = \{ x: V \;\text{such that}\; {\|x\|} = 1 \} .

If a locally convex topological vector space admits a continuous linear injection into a normed vector space, this can be used to define its sphere. If not, one can still define the sphere as a quotient of the space of non-zero vectors under the scalar action of $(0,\mathrm{\infty })$.

Homotopy theorists define $S^{\mathrm{\infty }}$ to be the sphere in the (incomplete) normed vector space (traditionally with the $l^2$ norm) of infinite sequences almost all of whose values are $0$, which is the directed colimit of the $S^n$:

$$S^{-1}\hookrightarrow S^0\hookrightarrow S^1\hookrightarrow S^2\hookrightarrow \cdots S^{\mathrm{\infty }}.$$S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty .

In themselves, these provide nothing new to homotopy theory, as they are at least weakly contractible and usually contractible. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of classifying spaces.

If the vector space is a shift space, then contractibility is straightforward to prove.

Theorem

Let $V$ be a shift space of some order. Let $SV$ be its sphere (either via a norm or as the quotient of non-zero vectors). Then $SV$ is contractible.

Proof

Let $T:V\to V$ be a shift map. The idea is to homotop the sphere onto the image of $T$, and then down to a point.

It is simplest to start with the non-zero vectors, $V\setminus \{0\}$. As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\mathrm{\infty })$. Define a homotopy $H:[0,1]\times (V\setminus \{0\})\to V\setminus \{0\}$ by $H_t(v)=(1-t)v+tTv$. It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$. To see that it is well-defined, we need to show that $H_t(v)$ is never zero. The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a shift map then it has none.

As $T$ is a shift map, it is not surjective and so we can pick some $v_0$ not in its image. Then we define a homotopy $G:[0,1]\times (V\setminus \{0\})\to V\setminus \{0\}$ by $G_t(v)=(1-t)Tv+t{v}_0$. As $v_0$ is not in the image of $T$, this is well-defined on $V\setminus \{0\}$. Combining these two homotopies results in the desired contraction of $V\setminus \{0\}$.

If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of $(0,\mathrm{\infty })$, they descend to the definition of the sphere as the quotient of $V\setminus \{0\}$.

Properties

• The $n$-sphere is the boundary of the $(n+1)$-ball.

• These spheres, or rather their underlying topological spaces or simplicial sets, are fundamental in (ungeneralised) homotopy theory. In a sense, Whitehead's theorem says that these are all that you need; no further generalised homotopy theory (in a sense dual to Eilenberg–Steenrod cohomology theory) is needed.

Low dimensions

• The $(-1)$-sphere is the empty space.
• The $0$-sphere is the disjoint union of two points.
• The $1$-sphere is the circle.
• The $2$-sphere is usual sphere from ordinary geometry.

Note that this violates the convention that a $1$-foo is a foo; instead the ruling convention being used is that an $n$-foo has dimension $n$. One could follow both by saying ‘$n$-circle’ instead, although this might get confused with the $n$-torus.

Related concepts

• homotopy groups of spheres