# 0708-1300/Homework Assignment 2

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Read sections 6-7 of chapter II and appendix C (on pages 531-534) 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)
1, 2, 3, S4, S5, S6 82
1, 2, S3, S4, 5 86

## Due Date

This assignment is due in class on Thursday October 18, 2007.

## Just for Fun

Prove that the lens space ${\displaystyle L(3,1}$), defined in class and on pages 85-86 of our text, can also be obtained by gluing two solid tori ${\displaystyle D^{1}\times S^{1}}$ using a map ${\displaystyle \varphi :S^{1}\times S^{1}\to S^{1}\times S^{1}}$ which identifies their (toroidal) boundaries. With the boundaries identified as ${\displaystyle S^{1}\times S^{1}=T^{2}={\mathbb {R} }^{2}/{\mathbb {Z} }^{2}}$, can you write a simple formula for ${\displaystyle \varphi }$?

Don't click on the next link if you don't want to see a proposed solution. The proposer of this solution did not derive it from the definition on pages 85-85 of our text but from the one in page 151 and with LOTS of help from M.Watkins . So, there is Double-Fun deriving it directly from those formulas in ${\displaystyle \mathbb {R} ^{6}}$.