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 , 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 ?
- How did we get from the least action principle?
- Do you understand how Poincare's lemma entered the derivation of Maxewell's equations?
- Do you understand the operator ? (How was it used, formally derived, and what is the intuitive picture behind it?)
- What was 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 is a compact orientable -manifold with no boundary, show that .
Problem 2. The standard volume form on is the form given by . Show that .
Problem 3. Show that if satisfies , then is exact. Deduce that if and satisfy , then as elements of . Deduce further that .
Problem 4. A "link" in is an ordered pair , in which and are smooth embeddings of the circle into , whose images (called "the components of ") 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 , define a map by . Finally, let be the standard volume form of , and define "the linking number of " to be . Show
- If two links and are isotopic, then their linking numbers are the same: .
- If is a second 2-form on for which and if is defined in the same manner as except replacing with , then . (In particular this is true if is very close to a -function form at the north pole of ).
- Compute (but just up to an overall sign) the linking number of the link L11a193, displayed below:
The links L11a193, and . |
Just for Fun
Prove that the two (3-component) links and 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.