Infinite-dimensional sphere

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: : 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. 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 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. == Literature ==