# What is the quotient of the unit sphere by the bilateral shift on infinite-dimensional separable real Hilbert space?

Let H denote the real Hilbert space 𝓁2(ℤ) with its usual inner product.

If {en | n ∈ ℤ} denotes its standard orthonormal basis, define the unitary mapping W : H → H via W(en) = en+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?

• Two Hilbert manifolds are homotopy equivalent iff they are diffeomorphic, so the resulting space is just $S^{1} \times H$. May 15 at 15:40
• Yes. I'd like to know its Riemannian structure. May 15 at 15:45