Timeline for What should be taught in a 1st course on smooth manifolds?
Current License: CC BY-SA 2.5
11 events
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| Dec 19, 2018 at 8:59 | comment | added | Lennart Meier | The (English edition of the) book Introduction to Differential Topology by T. Bröcker and K. Jänich does the Ehresmann fibration theorem. In general, this is quite a good book, though sometimes the arguments might be a little short for students with little experience. | |
| Mar 27, 2011 at 3:00 | vote | accept | jlk | ||
| Mar 21, 2011 at 13:39 | comment | added | Emerton | Dear jlk, You're welcome. Note though that my comment about differential topology courses not emphasizing "morphisms as families" may have been a bit hasty, since this is one of the focuses of Morse theory. So from an algebraic geometer's perspective (and for other reasons too), Morse theory is another excellent possibility to consider. Regards, Matthew | |
| Mar 20, 2011 at 20:44 | comment | added | Emerton | ... target (i.e. zero dimensional fibres) ties in with covering space theory, which the students already know (presumably --- if not, then that might be a better topic than Ehresmann). I'm sorry that I can't give a more interesting answer. Best wishes, Matthew | |
| Mar 20, 2011 at 20:42 | comment | added | Emerton | Dear jlk, I think that Ehresmann's theorem is the natural point at which properness appears. I think that the "morphisms as families" point of view is not usually emphasized in courses in differential topology the way it is in algebraic geometry, but I don't see why you couldn't discuss it. (And thus explain that it is important to have a notion --- i.e. proper submersion --- which captures the idea of a smooth family of compact manifolds without requiring that the base or total space themselves be proper.) Note also that the case of Ehresmann's theorem with equidimensional source and ... | |
| Mar 20, 2011 at 0:31 | comment | added | jlk | @Emerton: Do you have suggestions for applications of properness in a first course in manifold theory? (As an algebraic geometer, I know a lot of applications to algebraic geometry, but fewer to manifold theory.) | |
| Mar 19, 2011 at 20:14 | comment | added | Emerton | I agree; for students who are going on to other fields (e.g. algebraic geometry) where differential topology plays a role (both as techincal background in some situations, and as more general motivational background), this is one of the most useful results to take away from a manifolds course. Also, although it is not particularly a differential topology course, just teaching the definition of proper map would be helpful (and greatly appreciated by students going on to algebraic geometry!). | |
| Mar 19, 2011 at 19:19 | comment | added | Pete L. Clark | +1: this one result makes the whole subject nicer and easier to understand. I am always confused when basic structural results like this are omitted in introductory courses. A close cousin of this phenomenon is the practice in algebraic geometry courses of defining a vector bundle to be a certain kind of sheaf (so that the Serre-Swan theory is reduced to an unmotivated definition!). | |
| Mar 19, 2011 at 6:14 | comment | added | Ryan Budney | IMO it makes a fine homework problem in an introductory manifolds course, right around when one learns the proof of the tubular neighbourhood theorem. | |
| Mar 18, 2011 at 23:53 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
added two references
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| Mar 18, 2011 at 23:29 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |