Contents

# Contents

## Definition

### Finite-dimensional spheres

###### 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 $\mathbb{R}^{n+1}$ consisting of all points $x$ whose distance from the origin is $1$

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

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

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

Topologically, this is equivalent (homeomorphic) to the unit sphere for $r \gt 0$, 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 $\mathbb{R}$^n.

### Infinite dimensional spheres

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

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

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 $(0,\infty)$.

Homotopy theorists define $S^\infty$ 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 $S^n$:

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

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 $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors). Then $S V$ is contractible.

###### Proof

Let $T \colon V \to V$ 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 $(0,\infty)$. Define a homotopy $H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$. 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 $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$. 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 $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$.

## Properties

### Coset space structure

###### Proposition

The n-spheres are coset spaces of orthogonal groups

$S^n \;\simeq\; O(n+1)/O(n) \,.$

Similarly for the corresponding special orthogonal groups

$S^n \;\simeq\; SO(n+1)/SO(n)$

and spin groups

$S^n \;\simeq\; Spin(n+1)/Spin(n)$

and pin groups

$S^n \;\simeq\; Pin(n+1)/Pin(n) \,.$

For fix a unit vector in $\mathbb{R}^{n+1}$. Then its orbit under the defining $O(n+1)$-action on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.

### Branched covers

Every $n$-dimensional PL manifold admits a branched covering of the n-sphere (Alexander 20).

By the Riemann existence theorem, every connected compact Riemann surface admits the structure of a branched cover by a holomorphic function to the Riemann sphere. See there at branched cover of the Riemann sphere.

graphics grabbed from Chamseddine-Connes-Mukhanov 14, Figure 1, Connes 17, Figure 11

For 3-manifolds branched covering the 3-sphere see (Montesinos 74).

All PL 4-manifolds are simple branched covers of the 4-sphere (Piergallini 95, Iori-Piergallini 02).

But the n-torus for $n \geq 3$ is not a cyclic branched over of the n-sphere (Hirsch-Neumann 75)

## 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. This canonically carries the structure of a complex manifold which makes it the Riemann sphere.

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.

## References

### Formalization

Axiomatization of the homotopy type of the 1-sphere (the circle) and the 2-sphere, as higher inductive types, is in

Visualization of the idea of the construction for the 2-sphere is in

### Group actions on spheres

Discussion of free group actions on spheres by finite groups includes

The subgroups of SO(8) which act freely on $S^7$ have been classified in

• J. A. Wolf, Spaces of constant curvature, Publish or Perish, Boston, Third ed., 1974

and lifted to actions of Spin(8) in

• S. Gadhia, Supersymmetric quotients of M-theory and supergravity backgrounds, PhD thesis, School of Mathematics, University of Edinburgh, 2007

Further discussion of these actions is in

where they are related to the black M2-brane BPS-solutions of 11-dimensional supergravity at ADE-singularities.