0708-1300/Homework Assignment 2: Difference between revisions
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Prove that the lens space <math>L(3,1</math>), defined in class and on pages 85-86 of our text, can also be obtained by gluing two solid tori <math>D^1\times S^1</math> using a map <math>\varphi:S^1\times S^1\to S^1\times S^1</math> which identifies their (toroidal) boundaries. With the boundaries identified as <math>S^1\times S^1=T^2={\mathbb R}^2/{\mathbb Z}^2</math>, can you write a simple formula for <math>\varphi</math>? |
Prove that the lens space <math>L(3,1</math>), defined in class and on pages 85-86 of our text, can also be obtained by gluing two solid tori <math>D^1\times S^1</math> using a map <math>\varphi:S^1\times S^1\to S^1\times S^1</math> which identifies their (toroidal) boundaries. With the boundaries identified as <math>S^1\times S^1=T^2={\mathbb R}^2/{\mathbb Z}^2</math>, can you write a simple formula for <math>\varphi</math>? |
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Don't click on the next link if you don't want to see a [[0708-1300/proposed solution|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 |
Don't click on the next link if you don't want to see a [[0708-1300/proposed solution|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 [http://www.maths.ex.ac.uk/~mwatkins/lensspaces.pdf M.Watkins] . So, there is Double-Fun deriving it directly from those formulas in <math>\mathbb{R}^6</math>. |
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Latest revision as of 21:35, 6 October 2007
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Reading
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 ), defined in class and on pages 85-86 of our text, can also be obtained by gluing two solid tori using a map which identifies their (toroidal) boundaries. With the boundaries identified as , can you write a simple formula for ?
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 .
