I am coming from a physics background, and reading through different articles on topological classification. In one article (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51), they consider the homotopy groups of the homogeneous space $$ \mathcal{M} = U(\mathcal{H})/DU(\mathcal{H}) $$ where $\mathcal{H}$ is an infinite-dimensional Hilbert space, and $DU$ the space of diagonal, unitary operators. They refer to Kuiper, to conclude that $\pi_k(U(\mathcal{H}))$ is trivial, and hence that $\pi_2(\mathcal{M}) = \pi_1(DU(\mathcal{H}))=\mathbb{Z}^\infty$ which can be calculated, since $DU$ is just an infinite-dimensional torus.
However, in a different classification scheme, (III B of http://iopscience.iop.org/article/10.1088/0031-8949/2015/T168/014001/meta) they conclude that the objects of interest are elements in the complex Grassmannian, $G_{m,m+n}(\mathbb{C}) = U(m+n)/[U(m)\times U(n)]$ and when calculating the relevant homotopy group (which is the second), they conclude that $\pi_2(G_{m,m+n}(\mathbb{C}))=\mathbb{Z}$.
My question is how to understand this in relation to the reference to Kuiper? In his original paper, he stresses that $U(\mathcal{H})$ is NOT the limit of $U(n)$ for $n\to\infty$. But is there an easy way to understand the difference this makes?