0708-1300/Homework Assignment 6
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The information below is preliminary and cannot be trusted! (v)
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Reading
At your leisure, read your class notes over the break, and especially at some point right before classes resume after the break.
Doing
Solve and submit your solutions of the following problems:
Problem 1. If is a compact orientable n-manifold with no boundary, show that .
Problem 3. The "standard volume form on S^2" is the form given by . Show that .
Problem 3. Show that if satisfies , then is exact, and therefore, if and satisfy , then as elements of .
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 homotopy along which the components 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:
Due Date
This assignment is due in class on Thursday January 10, 2007.
Just for Fun
Prove that the following two links are not isotopic, yet their complements are diffeomorphic:
(See more at Classes: 2004-05: Math 1300Y - Topology: Homework Assignment 5)