0708-1300/Homework Assignment 5

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Announcements go here

In Preparation

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 $M^{n}$ be a manifold. Show that the following definitions for the orientability of $M$ are equivalent:

1. There exists a nowhere vanishing $n$-form on $M$.
2. There exists an atlas $\{(U_{\alpha },\phi _{\alpha }:U_{\alpha }\to {\mathbb {R} }^{n})\}$ for $M$, so that $\det(\phi _{\alpha }\phi _{\beta }^{-1})>0$ wherever that makes sense.

Problem 2. Show that the tangent space $TM$ of any manifold $M$ is orientable.

Problem 3.

1. Show that if $M$ and $N$ are orientable then so is $M\times N$.
2. Show that if $M$ and $M\times N$ are orientable then so is $N$.

Problem 4. Show that $S^{n}$ is always orientable.

Problem 5. Recall that a form is called closed if it is in the kernel of $d$ and exact if it is in the image of $d$. Show that every exact form is closed.

Due Date

This assignment is due in class on Thursday December 6, 2007.