Kuiper's Theorem - Bott's Unitary Group

Bott's Unitary Group

There is another infinite-dimensional unitary group, of major significance in homotopy theory, that to which the Bott periodicity theorem applies. It is certainly not contractible. The difference from Kuiper's group can be explained: Bott's group is the subgroup in which a given operator acts non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis {e_i}, for some N, being the identity on the remaining basis vectors.