I have another question regarding homotopy theory and winding numbers (or degrees).
In Manton and Sutcliffe they state the following theorem:
$\pi_2(G/H)=\pi_1(H)$
provided $G$ is a compact, connected and simply connected group, and $H$ denotes the subgroup of $G$. They don't really elaborated about what this theorem exactly entails and my question is:
Does this theorem means that the winding number of the R.H.S. is equal to the winding number of the L.H.S.?
In the beginning I understood this theorem as implying that the winding number belongs to the same group, but they are not necessary equal to each other. I.e. if the winding number is, for example, always an integer for the R.H.S., then it must also be an integer for the L.H.S. But looking at how they use this theorem later in the book, it seems that the winding numbers must also equal each other. Could anybody confirm this for me?
Thanks in advance,
Hunter