In algebraic topology, the infinite-dimensional sphere is the inductive limit of all spheres. Although no sphere is contractible, the infinite-dimensional sphere is contractibleHatcher 2002, p. 19, Exercise 16tom Dieck 2008, (8.4.5) Example and hence appears as the total space of multiple universal principal bundles.
Definition With the usual definition of the sphere with the 2-norm, the canonical inclusion restricts to a canonical inclusion . Hence the spheres form an inductive system, whose inductive limit:Hatcher 2002, p. 7tom Dieck 2008, p. 222
is the infinite-dimensional sphere.
Properties The most important property of the infinite-dimensional sphere is that it is contractible. Since the infinite-dimensional sphere inherits a CW structure from the spheres,tom Dieck 2008, p. 306 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 , due to the compactness of the former sphere, factors over a canonical inclusion with without loss of generality. Since is trivial, is also trivial.
Application
is the universal principal -bundle, hence . The principal -bundle is then the canonical inclusion , hence . is the universal principal U(1)-bundle, hence . The principal -bundle is then the canonical inclusion , hence . is the universal principal SU(2)-bundle, hence . The principal -bundle is then the canonical inclusion , hence .
Literature
References
External links
infinite-dimensional sphere at the nLab