I have just graduated and I have to start thinking about topics for my PhD thesis and areas I am going to specialize in. The thing is that one thing that looks fun to me is classifying smooth manifolds up to diffeomorphism, searching for "differential invariants", which is the way I call invariants that allow to distinguish homeomorphic but not diffeomorphic manifolds. If this has already a standard name, please tell me what is it.

Is this area interesting for doing research?

I've read Milnor's paper on the existance of exotic 7-spheres (which I don't fully understand yet), and it seems like he hasn't any good tool for solving his problem, so he has to biuld his own invariant. Also, all of the homology theories I know so far are invariant under homotopy equivalence, so despite they are sophisticated, they are useless for this purpose, right? These are the reasons why I think this might be interesting.

If so, what should I learn from now on in order to get to the "differential invariants" already known?

Thank you.

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    $\begingroup$ I think the term is "smooth invariants." It sounds to me like you'll be interested in Floer homology. You might also like Scorpan's "The Wild World of 4-Manifolds." But you say you've just graduated. Why do you "have to start thinking about topics for [your] PhD thesis"? It seems to me like you only need to know your general area of interest. Have you been accepted to grad school? $\endgroup$ – Jesse Madnick Jun 28 '13 at 23:43
  • $\begingroup$ My (very limited) understanding is that yes, this is an interesting and active area of research, but it is also very difficult. By "difficult" I mean creating new invariants and distinguishing smooth manifolds is probably a best-case scenario. More likely, I think, one would spend lots of time exploring technical properties of currently known, recently defined invariants. $\endgroup$ – Jesse Madnick Jun 28 '13 at 23:50
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    $\begingroup$ I live in Spain, so things here are a bit different. Once you're graduated, you get into a one year master degree and then you choose a topic for your thesis and start writing it. This should take about three or four years. $\endgroup$ – rf1x Jun 28 '13 at 23:51

I think this the beginning of the subject called "differential topology". The basic tools at here are characteristic classes, various constructions like plumbing and cobordism, handle decomposition and Morse theory, etc. There a great number of books and papers on this subject. Try a google search on introductory books of differential topology.

In dimension $1-3$ it is well known the $Diff$ category coincide with $Top$ for manifolds. So the classification we have for compact surfaces and the geometrizion is "suffice". In dimension $4$, it seems there are uncountablely many inequivalent differential structure available. This is a result by Taubes using Selberg-Witten theory. There is an high level "expository" article available written by Witten, but may not be readable to a beginning graduate student. There are also various treastises on this subject like the one by John Morgan.

The classification of manifolds of dimension higher than 5 is done by Surgery Theory, and there are some notes at here. I audited this class myself. My impression is that you need to read the papers seriously to really understand what is happening, because most of the work is done by constructing explicit examples. However as you wrote the subject is far from complete and we still know little about higher dimensional manifolds. I am sure there will be some real expert at the site happy to answer your question in more detail.


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