# 0708-1300/Homework Assignment 3

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Read sections 8-10 of chapter II 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.

Also, read section 12 of chapter I of Bredon's book, but you can be a little less careful here.

## 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, 3, S4, S5 88
S1, 2, 3, S4, 5 89

Note that these problems largely concern with material that we will not cover in class.

Problem 4 from page 88 needs a minor variation to be completely precise. See Brendon's book errata for more detail or much better find all the details your self.

## Due Date

This assignment is due in class on Thursday November 1, 2007.

## Just for Fun

• Trace the proof of the Whitney embedding theorem to find an embedding of the two dimensional real projective plane, ${\displaystyle {\mathbb {R} }{\mathbb {P} }^{2}=S^{2}/(p=-p)}$, inside ${\displaystyle {\mathbb {R} }^{5}}$. Do not do anything explicitly; just convince yourself that indeed you can find a small atlas (how small?), use it to embed ${\displaystyle {\mathbb {R} }{\mathbb {P} }^{2}}$ in some large ${\displaystyle {\mathbb {R} }^{N}}$ (how large?), and figure out how many times you will need to use Sard's theorem before you're down to the target, ${\displaystyle {\mathbb {R} }^{5}}$.
• Now see if you can come up with some cleverer way of viewing ${\displaystyle {\mathbb {R} }{\mathbb {P} }^{2}}$, that will allow you to explicitly embed it in ${\displaystyle {\mathbb {R} }^{5}}$.