On assignment 5:
I realized not everyone might be familiar with principal G-bundles so let me spell out what the hypotheses in the assignment mean in some more detail:
That S^3 -> M -> S^2 x S^2 is a principal S^3-bundle means that there is a pullback square
M----------------> ES^3
| |
v v
S^2 x S^2------> BS^3
where S^3 -> ES^3 -> BS^3 is the universal principal S^3-bundle.
For the assignment, the only thing you need to know about this is that the total space ES^3 is contractible, and hence that the loop space of BS^3 may be identified with S^3 up to homotopy.
That the Euler class is non-zero means that the induced map
H^4(BS^3;Q) -> H^4(S^2 x S^2;Q)
is non-zero.
You may use these facts in your solution.
Do not hesitate to write to me if you have questions about the assignment!
Alexander
I realized not everyone might be familiar with principal G-bundles so let me spell out what the hypotheses in the assignment mean in some more detail:
That S^3 -> M -> S^2 x S^2 is a principal S^3-bundle means that there is a pullback square
M----------------> ES^3
| |
v v
S^2 x S^2------> BS^3
where S^3 -> ES^3 -> BS^3 is the universal principal S^3-bundle.
For the assignment, the only thing you need to know about this is that the total space ES^3 is contractible, and hence that the loop space of BS^3 may be identified with S^3 up to homotopy.
That the Euler class is non-zero means that the induced map
H^4(BS^3;Q) -> H^4(S^2 x S^2;Q)
is non-zero.
You may use these facts in your solution.
Do not hesitate to write to me if you have questions about the assignment!
Alexander