0708-1300/Homework Assignment 6

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Reading

At your leisure, read your class notes over the break, and especially at some point right before classes resume next semester. Here are a few questions you can ask yourself while reading:

• Do you understand pullbacks of differential forms?
• Do you think you could in practice integrate any differential form on any manifold (at least when the formulas involved are not too messy)?
• Do you understand orientations and boundaries and how they interact?
• Why is Stokes' theorem true? Both in terms of the local meaning of $d$, and in terms of a formal proof.
• Do you understand the two and three dimensional cases of Stokes' theorem?
• Do you understand the Hodge star operator $\star$?
• How did we get $d\star dA=J$ from the least action principle?
• Do you understand how Poincare's lemma entered the derivation of Maxewell's equations?
• Do you understand the operator $P$? (How was it used, formally derived, and what is the intuitive picture behind it?)
• What was $H_{dR}$ and how did it relate to pullbacks and homotopy.

Doing

Solve the following problems and submit your solutions of problems 1, 3 and 4. This is a very challenging collection of problems; I expect most of you to do problem 2 with no difficulty (it is a repeat of an older problem), problem 1 with some effort, and I hope each of you will be able to do at least one further problem. It will be great if some of you will do all problems!

Problem 1. If $M$ is a compact orientable $n$-manifold with no boundary, show that $H_{dR}^{n}(M)\neq 0$.

Problem 2. The standard volume form on $S^{2}$ is the form $\omega$ given by $\omega ={\frac {1}{4\pi }}\left(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\right)$. Show that $\int _{S^{2}}\omega =1$.

Problem 3. Show that if $\omega \in \Omega ^{2}(S^{2})$ satisfies $\int _{S_{2}}\omega =0$, then $\omega$ is exact. Deduce that if $w_{1}\in \Omega ^{2}(S^{2})$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_2\in\Omega^2(S^2)} satisfy $\int _{S_{2}}\omega _{1}=\int _{S_{2}}\omega _{2}$, then $[\omega _{1}]=[\omega _{2}]$ as elements of $H_{dR}^{2}(S^{2})$. Deduce further that $\dim H_{dR}^{2}(S^{2})=1$.

Problem 4. A "link" in ${\mathbb {R} }^{3}$ is an ordered pair $\gamma =(\gamma _{1},\gamma _{2})$, in which $\gamma _{1}$ and $\gamma _{2}$ are smooth embeddings of the circle $S^{1}$ into ${\mathbb {R} }^{3}$, whose images (called "the components of $\gamma$") are disjoint. Two such links are called "isotopic", if one can be deformed to the other via a smooth homotopy along which the components remain embeddings and remain disjoint. Given a link $\gamma$, define a map $\Phi _{\gamma }:S^{1}\times S^{1}\to S^{2}$ by $\Phi _{\gamma }(t_{1},t_{2}):={\frac {\gamma _{2}(t_{2})-\gamma _{1}(t_{1})}{||\gamma _{2}(t_{2})-\gamma _{1}(t_{1})||}}$. Finally, let $\omega$ be the standard volume form of $S^{2}$, and define "the linking number of $\gamma =(\gamma _{1},\gamma _{2})$" to be $l(\gamma )=l(\gamma _{1},\gamma _{2}):=\int _{S^{1}\times S^{1}}\Phi _{\gamma }^{\star }\omega$. Show

1. If two links $\gamma$ and $\gamma '$ are isotopic, then their linking numbers are the same: $l(\gamma )=l(\gamma ')$.
2. If $\omega '$ is a second 2-form on $S^{2}$ for which $\int _{S^{2}}\omega '=1$ and if $l'(\gamma )$ is defined in the same manner as $l(\gamma )$ except replacing $\omega$ with $\omega '$, then $l(\gamma )=l'(\gamma )$. (In particular this is true if $\omega '$ is very close to a $\delta$-function form at the north pole of $S^{2}$).
3. Compute (but just up to an overall sign) the linking number of the link L11a193, displayed below:

The links L11a193, $\gamma _{3}$ and $\gamma '_{3}$.

Just for Fun

Prove that the two (3-component) links $\gamma _{3}$ and $\gamma '_{3}$ shown above are not isotopic, yet their complements are diffeomorphic. (See more at Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5)

Due Date

This assignment is due in class on Thursday January 10, 2007.