0708-1300/Homework Assignment 5
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The information below is preliminary and cannot be trusted! (v)
Reading
Read sections 1-3 of chapter V of Bredon's book three times:
- First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
- Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
- And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.
Also, do the same with your own class notes - much of what we do for this part of the class is not in the textbook!
Doing
Solve all of the following problems, but submit only your solutions of problems *,* and *:
Problem 1. Let [math]\displaystyle{ M^n }[/math] be a manifold. Show that the following definitions for the orientability of [math]\displaystyle{ M }[/math] are equivalent:
- There exists a nowhere vanishing [math]\displaystyle{ n }[/math]-form on [math]\displaystyle{ M }[/math].
- There exists an atlas [math]\displaystyle{ \{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\} }[/math] for [math]\displaystyle{ M }[/math], so that [math]\displaystyle{ \det(\phi_\alpha\phi^{-1}_\beta)\gt 0 }[/math] wherever that makes sense.
Problem 2. Show that the tangent space [math]\displaystyle{ TM }[/math] of any manifold [math]\displaystyle{ M }[/math] is orientable.
Problem 3.
- Show that if [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] are orientable then so is [math]\displaystyle{ M\times N }[/math].
- Show that if [math]\displaystyle{ M }[/math] and [math]\displaystyle{ M\times N }[/math] are orientable then so is [math]\displaystyle{ N }[/math].
Problem 4. Show that [math]\displaystyle{ S^n }[/math] is always orientable.
Problem 5. Recall that a form is called closed if it is in the kernel of [math]\displaystyle{ d }[/math] and exact if it is in the image of [math]\displaystyle{ d }[/math]. Show that every exact form is closed.
Due Date
This assignment is due in class on Thursday December 6, 2007.
