Difference between revisions of "07081300/Homework Assignment 4"
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+  __NOTOC__  
{{07081300/Navigation}}  {{07081300/Navigation}}  
−  
==Reading==  ==Reading==  
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100101  100101  
 align=center   align=center  
−  S1, S2,  +  S1, S2, 3 
264  264  
}  }  
+  
+  Also, solve and submit the following question:  
+  
+  '''Question 6.'''  
+  # Show that if <math>n\neq m</math> then <math>{\mathbf R}^n</math> is not diffeomorphic (homeomorphic via a smooth map with a smooth inverse) to <math>{\mathbf R}^m</math>.  
+  # <strike>Show that if <math>n\neq m</math> then <math>{\mathbf R}^n</math> is not homeomorphic to <math>{\mathbf R}^m</math>.</strike>  
+  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==  ==Due Date==  
−  This assignment is due in class on Thursday November  +  This assignment is due in class on Thursday November 22, 2007. 
==Just for Fun==  ==Just for Fun==  
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Look at the story of [[07081300/Barnie the polar bear  Barnie the polar bear]].  Look at the story of [[07081300/Barnie the polar bear  Barnie the polar bear]].  
+  
+  If <math>n\neq m</math> then <math>{\mathbf R}^n</math> is [[07081300/not homeomorphicnot homeomorphic]] to <math>{\mathbf R}^m</math>. 
Latest revision as of 13:49, 19 November 2007

Reading
Read section 11 of chapter II and sections 13 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  100101 
S1, S2, 3  264 
Also, solve and submit the following question:
Question 6.
 Show that if then is not diffeomorphic (homeomorphic via a smooth map with a smooth inverse) to .

Show that if then is not homeomorphic to .
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
(Of course, you have to first obtain a geometric understanding of , 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 then is not homeomorphic to .