Let H denote the real Hilbert space 𝓁^{2}(ℤ) with its usual inner product.

If {e_{n} | n ∈ ℤ} denotes its standard orthonormal basis, define the unitary mapping W : H → H via W(e_{n}) = e_{n+1}, extended linearly to all of H.

Let S denote the unit sphere of H: S = {x ∈ H | ||x|| = 1}. Note that W(S) = S.

Then W|S generates an isometric action of the group ℤ on S that is free and properly discontinuous.

*Question*: What is the quotient S/W of S by W ?

*Thoughts*: Since S is contractible, S/W must be a K(ℤ,1). This seems to make S/W into some Riemannian Hilbert manifold that is a fibre bundle over the circle. Is this correct? If so, what is its fibre? And what is its glueing map?