Nakahara Solution of problems in chapter 9.and 10.

[Leeway]

Hey all , I would need help about how to solve the problems in chapters 9 and 10 of Geometry Topology and Physics, Nakahara. 1. Why the Whitney sum of 2 Mobium Strips is a trivial bundle - I found the transition matrices to be the identity, -identity . SO not all of them are identity -> the bundle is not trivial as it has to be. 2. I have a volume element Omega_n of S^n normalized as integral(Omega_n) = 1. Let f be a smooth map : S^2n-1 to S^n and consider the pullback f^*Omega_n. Show that the pullback is closed and written as d(omega_n-1), where omega_n-1 is a n-1 form on S^2n-1. Then show that the Hopf invariant H(f)=Integral{(omega_n-1) /\ d(omega_n-1) } is independent of the choise of omega_n-1. Please help me I will appreciate it very much ! PS: If you knew the solution of the Berry`s phase in chapter 10 pls let me know also .. ;o) Have a nice day.

 

[pellman]

Hi, Leeway. Welcome.

You have availed yourself of a great online resource. There are a lot of smart people here. I'm not one of them though. 210 posts and all of them questions. So its about time for me take a turn at answering one.

For the mobius strip question, picture the whitney sum as like a 2-bladed propeller attached to the line segment (circle) at its hub, each blade of the propellor is one instance of L. For the cylinder it just slides around as is. For the mobius strip, it might rotate 180 degrees at some point as it slides around, but it still looks the same. Therefore, the two cases are identical.

For problem #2, I would recommend working through a simple case, n=2 or 3, in explicit coordinates. Then you will probably see the result clearly and can apply it to the general case.

If you visit us often, learn LaTex, so you can do things like $$\int_{S_n}\Omega=1$$

 

[Leeway]

Thank you a lot for such a long reply, I appreciate it really , anyway I have to have the problems solved quite fast so no time for a cases like n=2,3 in explicit coordinates ;o/ and thank you for the whitney sum advice .. I see .. but the structure group (transition functions) by Mobius strip is G = {e,g}, where g maps the fiber t->-t how is it possible that all the transition functions of the whitney sum are identity maps ? .. and do not worry I know Latex but I did not know that it works here, thanks ;o) ;o)

 

[pellman]

I spent some more time thinking about the mobius strip/Whitney sum problem and I don't really understand it either. If you find out, please come back and share. - Todd