0708-1300/Homework Assignment 4

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Announcements go here

Reading

Read section 11 of chapter II and 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.

Doing

Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":

problems	on page(s)
S1, S2	100-101
S1, S2, 3	264

Also, solve and submit the following question:

Question 6.

1. Show that if $n\neq m$ then ${\mathbf {R} }^{n}$ is not diffeomorphic (homeomorphic via a smooth map with a smooth inverse) to ${\mathbf {R} }^{m}$.
2. Show that if $n\neq m$ then ${\mathbf {R} }^{n}$ is not homeomorphic to ${\mathbf {R} }^{m}$.

Note that a priori the second part of this question is an order of magnitude harder than the first. I am not sure how to do it with our current techniques, though later on it will become an easy consequence of "homology theory".

Due Date

This assignment is due in class on Thursday November 22, 2007.

Just for Fun

Find a geometric interpretation to the formula

$d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y])$.

(Of course, you have to first obtain a geometric understanding of $[X,Y]$, and this in itself is significant and worthwhile).

Dror's notes above / Student's notes below

Look at the story of Barnie the polar bear.

If $n\neq m$ then ${\mathbf {R} }^{n}$ is not homeomorphic to ${\mathbf {R} }^{m}$.