Homework 5 is now available from the course web page.
We did not quite have to time to finish the discussion about the relation between Postnikov towers and minimal models in the last lecture but we will do this on Tuesday.
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