The -dimensional sphere of radius is
For fix a unit vector in . Then its orbit under the defining -action on is clearly the canonical embedding . But precisely the subgroup of that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to , hence .
One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) normed vector space :
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 .
Homotopy theorists define to be the sphere in the (incomplete) normed vector space (traditionally with the norm) of infinite sequences almost all of whose values are , which is the directed colimit of the :
In themselves, these provide nothing new to homotopy theory, as they are at least weakly contractible and usually contractible. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of classifying spaces.
If the vector space is a shift space, then contractibility is straightforward to prove.
Let be a shift space of some order. Let be its sphere (either via a norm or as the quotient of non-zero vectors). Then is contractible.
Let be a shift map. The idea is to homotop the sphere onto the image of , and then down to a point.
It is simplest to start with the non-zero vectors, . As is injective, it restricts to a map from this space to itself which commutes with the scalar action of . Define a homotopy by . It is clear that, assuming it is well-defined, it is a homotopy from the identity to . To see that it is well-defined, we need to show that is never zero. The only place where it could be zero would be on an eigenvector of , but as is a shift map then it has none.
As is a shift map, it is not surjective and so we can pick some not in its image. Then we define a homotopy by . As is not in the image of , this is well-defined on . Combining these two homotopies results in the desired contraction of .
If 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 , they descend to the definition of the sphere as the quotient of .
These spheres, or rather their underlying topological spaces or simplicial sets, are fundamental in (ungeneralised) homotopy theory. In a sense, Whitehead's theorem says that these are all that you need; no further generalised homotopy theory (in a sense dual to Eilenberg–Steenrod cohomology theory) is needed.
Note that this violates the convention that a -foo is a foo; instead the ruling convention being used is that an -foo has dimension . One could follow both by saying ‘-circle’ instead, although this might get confused with the -torus.
Visualization of the idea of the construction for the 2-sphere is in