Differential Topology: Proving Integral of f*dw=0 and Line Bundle over RPn

[babak_812]

Hi
I would appreciate any help with these problems. Thanks in Advance!

1.Suppose X is the boundary of a manifold W, W is compact, and f : X → Y is a smooth map. Let w be a closed k-form on Y ,

where k = dimX. Prove that if f extends to all of W then integral of f* dw = 0. Assume that
W is orientable.

what I mean by f* is ofcourse the cotangent of f

2.Over RPn = Sn/A, (where A is the antipodal map and Sn is the n-sphere) define a line bundle v where for
every x ∈ Sn the fiber over the equivalence class of x is is the line {tx : t ∈ R}. Show
that v really is a bundle (by defining a sensible bundle atlas) and that it is nontrivial. (for n>=1)

3.Let p : E → S1 be the Mobius band (a real rank 1 bundle over S1).
(a) By constructing two nowhere zero sections of E ⊕ E, prove that E ⊕ E is a trivial
bundle.
(b) Prove that E ⊕ . . . ⊕ E (k summands) is trivial if and only if k is even.
(c) Show that every real rank 1 bundle over S1 over S1 is either trivial or is isomorphic
to the Mobius band.

 

[jvicens]

Hi there,

I would be happy to help with these problems. For the first problem, we can use the fact that f extends to all of W to show that the integral of f*dw is zero. Since W is compact, we can use the fact that the integral of a closed form over a compact manifold is equal to the integral of its boundary. Since X is the boundary of W, we can write the integral as the sum of integrals over X and the interior of W.

Now, since f extends to all of W, we can use the pullback property of integrals to write the integral over W as the integral over X. This means that the integral over the interior of W is equal to zero. Since w is a closed k-form on Y, it is also closed on the interior of W, so the integral of f*dw over the interior of W is equal to zero. Therefore, the integral over X must also be equal to zero, proving that the integral of f*dw is equal to zero.

For the second problem, we can define a sensible bundle atlas for v by considering two open sets on RPn, one containing the north pole and one containing the south pole. On each of these open sets, we can define a local trivialization of v by projecting the line {tx : t ∈ R} onto the open set. These two local trivializations will overlap on the equator of RPn, but since A maps points on the equator to antipodal points, the two local trivializations will agree on the overlap. Therefore, we have a well-defined bundle atlas for v, showing that it is indeed a bundle.

To show that v is nontrivial, we can consider the section s : RPn → v defined by s(x) = [x,1], where [x,1] is the equivalence class of x in v. This section is clearly nonzero at every point on RPn, so v is nontrivial.

For the third problem, we can use the fact that the Mobius band is a nontrivial real rank 1 bundle over S1 to prove the statements. For part (a), we can consider the sections s1(x) = [x,1] and s2(x) = [x,-1] of E ⊕ E. These sections are nonzero at every point on S1, so they form a basis for E ⊕ E, showing that it is