nLab infinite-dimensional sphere





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

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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 B U ( 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.


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.


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.


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 March 5, 2024 at 00:05:44. See the history of this page for a list of all contributions to it.