nLab
infinite-dimensional sphere

Contents

Context

Spheres

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

General

There are several ways to make sense of the notion of the n-sphere in the limit (rather: colimit) that nβ†’βˆžn \to \infty. Typically the resulting infinite-dimensional sphere S ∞S^\infty has the remarkable property that it is contractible as a topological space – or at least weakly contractible, in that its map to the point is a weak homotopy equivalence.

In homotopy theory

Due to their weak homotopy equivalence to points, in homotopy theory infinite-dimensional spheres provide nothing new in themselves, but as a source of big contractible spaces they serve as a starting point for making concrete models of classifying spaces.

For example, the 2k-sphere is the total space of the tautological circle group-principal bundle over complex projective k-space, so that in the colimit 2kβ†’βˆž2k \to \infty the infinite-dimensional sphere emerges as a model for the universal principal bundle over the classifying space BU(1)B \mathrm{U}(1). This being contractible relates to important statements such as that the zero-section into the Thom space of the universal line bundle is a weak equivalence.

Realizations

An an infinite spherical cell complex

Realizing the n-sphere as a cell complex of sorts, such that S nβ†ͺS n+1S^n \hookrightarrow S^{n+1} is a relative cell complex-inclusion for all nβˆˆβ„•n \in \mathbb{N}, the infinite-dimensional sphere may be taken to be the colimit over this sequence of inclusions. Since the n+1-sphere is n-connected it follows that S ∞S^\infty is ∞\infty-connected and hence weakly contractible:

Specifically in the category Top of topological spaces, the n-sphere has a standard CW-complex structure with exactly 2-cells in each dimension, obtained inductively by attaching two nn-dimensional hemispheres to the (nβˆ’1)(n-1)-sphere regarded as the equator in the nn-sphere.

Since forming homotopy groups Ο€ k(βˆ’)\pi_k(-) commutes with taking the colimit over these relative cell complex inclusions (by the skeleton filtration), and since the n-sphere has trivial homotopy groups in dimension k<nk \lt n, it follows at once that the homotopy groups of the infinite-dimensional sphere all vanish, and hence that it is weakly equivalent to the point in the classical model structure on topological spaces:

Ο€ k(S ∞) =Ο€ k(lim⟢nβˆˆβ„•S n) =lim⟢nβˆˆβ„•Ο€ k(S n) =lim⟢nβˆˆβ„•Ο€ k(S n+k+1)⏟=* ≃* \begin{aligned} \pi_k \big( S^\infty \big) & =\; \pi_k \big( \underset{ \underset{n \in \mathbb{N}}{\longrightarrow} }{\lim} \; S^n \big) \\ & = \; \underset{ \underset{n \in \mathbb{N}}{\longrightarrow} }{\lim} \; \pi_k \big( S^n \big) \\ & = \; \underset{ \underset{n \in \mathbb{N}}{\longrightarrow} }{\lim} \; \underset{ = \, \ast }{ \underbrace{ \pi_k \big( S^{n+k+1} \big) } } \\ & \simeq \ast \end{aligned}

As the unit spheres in a topological vector space

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

S(V)={x:Vsuch thatβ€–xβ€–=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,∞)(0,\infty).

Homotopy theorists define S ∞S^\infty to be the sphere in the (incomplete) normed vector space (traditionally with the l 2l^2 norm) of infinite sequences almost all of whose values are 00, which is the directed colimit of the S nS^n:

S βˆ’1β†ͺS 0β†ͺS 1β†ͺS 2β†ͺβ‹―S ∞. S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty \,.

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

Theorem

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

Proof

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

It is simplest to start with the non-zero vectors, Vβˆ–{0}V \setminus \{0\}. As TT is injective, it restricts to a map from this space to itself which commutes with the scalar action of (0,∞)(0,\infty). Define a homotopy H:[0,1]Γ—(Vβˆ–{0})β†’Vβˆ–{0}H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\} by H t(v)=(1βˆ’t)v+tTvH_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 TT. To see that it is well-defined, we need to show that H t(v)H_t(v) is never zero. The only place where it could be zero would be on an eigenvector of TT, but as TT is a shift map then it has none.

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

If VV 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,∞)(0,\infty), they descend to the definition of the sphere as the quotient of Vβˆ–{0}V \setminus \{0\}.

Last revised on January 26, 2021 at 00:28:35. See the history of this page for a list of all contributions to it.