Infinite-dimensional sphere

In algebraic topology, the infinite-dimensional sphere is the inductive limit of all spheres. Although no sphere is contractible, the infinite-dimensional sphere is contractible^[1]^[2] and hence appears as the total space of multiple universal principal bundles.

Definition

With the usual definition $S^{n}=\{x\in \mathbb {R} ^{n+1}|\|x\|_{2}=1\}$ of the sphere with the 2-norm, the canonical inclusion $\mathbb {R} ^{n+1}\hookrightarrow \mathbb {R} ^{n+2},x\mapsto (x,0)$ restricts to a canonical inclusion $S^{n}\hookrightarrow S^{n+1}$ . Hence the spheres form an inductive system, whose inductive limit:^[3]^[4]

S^{\infty }:=\lim _{n\rightarrow \infty }S^{n}

is the infinite-dimensional sphere.

Properties

The most important property of the infinite-dimensional sphere is that it is contractible.^[1]^[2] Since the infinite-dimensional sphere inherits a CW structure from the spheres,^[3]^[5] Whitehead's theorem claims that it is sufficient to show that it is weakly contractible. Intuitively, the homotopy groups of the spheres disappear one by one, hence all do for the infinite-dimensional sphere. Concretely, any map $S^{k}\rightarrow S^{\infty }$ , due to the compactness of the former sphere, factors over a canonical inclusion $S^{n}\hookrightarrow S^{\infty }$ with $k<n$ without loss of generality. Since $\pi _{k}(S^{n})$ is trivial, $\pi _{k}(S^{\infty })$ is also trivial.

Application

$S^{\infty }\twoheadrightarrow \mathbb {R} P^{\infty }$ is the universal principal $\operatorname {O} (1)$ -bundle, hence $\operatorname {EO} (1)\cong S^{\infty }$ . The principal $\operatorname {O} (1)$ -bundle $S^{n}\twoheadrightarrow \mathbb {R} P^{n}$ is then the canonical inclusion $i\colon \mathbb {R} P^{n}\hookrightarrow \mathbb {R} P^{\infty }$ , hence $S^{n}\cong i^{*}S^{\infty }$ .
$S^{\infty }\twoheadrightarrow \mathbb {C} P^{\infty }$ is the universal principal U(1)-bundle, hence $\operatorname {EU} (1)\cong \operatorname {ESO} (2)\cong S^{\infty }$ . The principal $\operatorname {U} (1)$ -bundle $S^{2n+1}\twoheadrightarrow \mathbb {C} P^{n}$ is then the canonical inclusion $j\colon \mathbb {C} P^{n}\hookrightarrow \mathbb {C} P^{\infty }$ , hence $S^{2n+1}\cong j^{*}S^{\infty }$ .
$S^{\infty }\twoheadrightarrow \mathbb {H} P^{\infty }$ is the universal principal SU(2)-bundle, hence $\operatorname {ESU} (2)\cong \operatorname {ESp} (1)\cong S^{\infty }$ . The principal $\operatorname {SU} (2)$ -bundle $S^{4n+3}\twoheadrightarrow \mathbb {H} P^{n}$ is then the canonical inclusion $k\colon \mathbb {H} P^{n}\hookrightarrow \mathbb {H} P^{\infty }$ , hence $S^{4n+3}\cong k^{*}S^{\infty }$ .