0708-1300/Homework Assignment 5: Difference between revisions
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* Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail. |
* Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail. |
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* 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. |
* 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. |
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'''Also,''' do the same with your own class notes - much of what we do for this part of the class is '''not''' in the textbook! |
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==Doing== |
==Doing== |
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Solve the following problems |
Solve all of the following problems, but submit only your solutions of problems *,* and *: |
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{|align=center border=1 cellspacing=0 cellpadding=5 |
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'''Problem 1.''' Let <math>M^n</math> be a manifold. Show that the following definitions for the orientability of <math>M</math> are equivalent: |
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|- align=center |
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# There exists a nowhere vanishing <math>n</math>-form on <math>M</math>. |
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!problems |
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# There exists an atlas <math>\{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\}</math> for <math>M</math>, so that <math>\det(\phi_\alpha\phi^{-1}_\beta)>0</math> wherever that makes sense. |
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!on page(s) |
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|- align=center |
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'''Problem 2.''' Show that the tangent space <math>TM</math> of any manifold <math>M</math> is orientable. |
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|S1, S2 |
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|100-101 |
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'''Problem 3.''' |
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|- align=center |
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# Show that if <math>M</math> and <math>N</math> are orientable then so is <math>M\times N</math>. |
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|S1, S2, 3 |
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# Show that if <math>M</math> and <math>M\times N</math> are orientable then so is <math>N</math>. |
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|264 |
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|} |
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'''Problem 4.''' Show that <math>S^n</math> is always orientable. |
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'''Problem 5.''' Recall that a form is called closed if it is in the kernel of <math>d</math> and exact if it is in the image of <math>d</math>. Show that every exact form is closed. |
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==Due Date== |
==Due Date== |
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Revision as of 20:34, 21 November 2007
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In Preparation
The information below is preliminary and cannot be trusted! (v)
Reading
Read 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.
Also, do the same with your own class notes - much of what we do for this part of the class is not in the textbook!
Doing
Solve all of the following problems, but submit only your solutions of problems *,* and *:
Problem 1. Let be a manifold. Show that the following definitions for the orientability of are equivalent:
- There exists a nowhere vanishing -form on .
- There exists an atlas for , so that wherever that makes sense.
Problem 2. Show that the tangent space of any manifold is orientable.
Problem 3.
- Show that if and are orientable then so is .
- Show that if and are orientable then so is .
Problem 4. Show that is always orientable.
Problem 5. Recall that a form is called closed if it is in the kernel of and exact if it is in the image of . Show that every exact form is closed.
Due Date
This assignment is due in class on Thursday December 6, 2007.
