Question for experts in topology: What is the second homotopy group of O(4)/O(2)xO(2) ?

Question

... and the same question about SO(4)/SO(2)xSO(2).

How can one visualize the manifold O(4)/O(2)xO(2) in terms of spheres?

What about the second homotopy group of
O(2N)/O(N)xO(N) for N>2?

Where can I read about homotopy groups of Grassmannians?

O(n) and SO(n) are the orthogonal group O(n,R) and the special orthogonal group SO(n,R) of degree n over the field R of real numbers, respectively.
See
http://en.wikipedia.org/wiki/Orthogonal_group
for details.

Answer 1

I am not sure on your notation. Is O for orientable surface? So O(4) is a torus with 4 holes? What is SO? The notation tends to vary between the different topology books.

Okay, I had just learned about those in group theory seminar. I really haven't seen these in conjuction with algebraic groups. I know that Intro to topological Manifolds by John M. Lee covers homotopies in great detail and that we worked with groups (but not those). I don't want to give you a wrong answer.