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^n_{dR}(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 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^2_{dR}(S^2). Deduce further that \dim H^2_{dR}(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:
L11a193.png 0708-1300-LinkComplementExample1.png 0708-1300-LinkComplementExample2.png
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.