0708-1300/Homework Assignment 5

0708-1300/Navigation Panel   Add your name / see who's in! # Week of... Links Fall Semester 1 Sep 10 About, Tue, Thu 2 Sep 17 Tue, HW1, Thu 3 Sep 24 Tue, Photo, Thu 4 Oct 1 Questionnaire, Tue, HW2, Thu 5 Oct 8 Thanksgiving, Tue, Thu 6 Oct 15 Tue, HW3, Thu 7 Oct 22 Tue, Thu 8 Oct 29 Tue, HW4, Thu, Hilbert sphere 9 Nov 5 Tue,Thu, TE1 10 Nov 12 Tue, Thu 11 Nov 19 Tue, Thu, HW5 12 Nov 26 Tue, Thu 13 Dec 3 Tue, Thu, HW6 Spring Semester 14 Jan 7 Tue, Thu, HW7 15 Jan 14 Tue, Thu 16 Jan 21 Tue, Thu, HW8 17 Jan 28 Tue, Thu 18 Feb 4 Tue 19 Feb 11 TE2, Tue, HW9, Thu, Feb 17: last chance to drop class R Feb 18 Reading week 20 Feb 25 Tue, Thu, HW10 21 Mar 3 Tue, Thu 22 Mar 10 Tue, Thu, HW11 23 Mar 17 Tue, Thu 24 Mar 24 Tue, HW12, Thu 25 Mar 31 Referendum,Tue, Thu 26 Apr 7 Tue, Thu R Apr 14 Office hours R Apr 21 Office hours F Apr 28 Office hours, Final (Fri, May 2) Register of Good Deeds Errata to Bredon's Book

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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.

Problem 6. Let $f:{\mathbb {R} }_{t}\to S^{1}\subset {\mathbb {C} }$ be given by $f(t)=e^{it}$.

1. Show that there exists a unique $\omega \in \Omega ^{1}(S^{1})$ such that $f^{\star }\omega =dt$.
2. Show that $\omega$ is closed but not exact.

Problem 7. Show, directly from the definitions, that every closed 1-form on ${\mathbb {R} }^{2}$ is exact.

Problem 8. Compute the integral $\int _{S^{2}}zdx\wedge dy$ twice:

1. Using Stokes' theorem.
2. Directly from the definition, by using a one- or two-chart atlas for $S^{2}$.

(Repeat 1 and 2 until they stop giving different answers).

Problem 9. Show that the form $\omega =xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in \Omega ^{2}({\mathbb {R} }_{x,y,z}^{3})$ is invariant under rigid orientation-preserving rotations of ${\mathbb {R} }^{3}$. That is, if $A$ is such a rotation matrix ($AA^{T}=I$ and $\det A=1$) considered also as a linear transformation $A:{\mathbb {R} }^{3}\to {\mathbb {R} }^{3}$, then $A^{\star }\omega =\omega$.

Due Date

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