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September 27, 2005

Literature on Exotic R4s

Posted by Urs Schreiber

Today I received the following question by Florent Dieterlen on exotic ℝ 4\mathbb{R}^4 spaces. Since I am no expert on exotic spaces and cannot readily answer this question I reproduce it here in public (with kind permission). If anyone feels like providing help, please do so.

Hello,

I saw your discussion on a forum about exotic ℝ 4\mathbb{R}^4s. I was brought up as a physicist, but did not practice since, except for dynamical systems. I want to study exotic ℝ 4\mathbb{R}^4s, to be able to construct some, following certain conditions from an application. It is not properly a physics application. The problem is that i don’t have the basics. So my question is: starting from level MSc in physics, how do i get the most efficiently to the level i want:

1) do i have to follow Seiberg-Witten instead of Donaldson? The basics are not completely the same.

2) do you counsel Nash and Sen instead of Nakahara for the basics, if i want something very intuitive?

3) If 1) is yes, what book do you counsel for the study of exotic ℝ 4\mathbb{R}^4s following Seiberg-Witten?

Thanks in advance for your answer,
Best regards,
Florent Dieterlen

Posted at September 27, 2005 6:03 PM UTC

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3 Comments & 0 Trackbacks

Re: Literature on Exotic R4s

I would ask Hendryk Pfeiffer (now in Golm) who has thought about this problem a lot.

You should be warned that this is non-trivial and requires some elaborate theory. And when mathematicians say ‘Seiberg-Witten theory’ it looks completely different from what physicists think its about (N=2 gauge theory). Of course, in the end, it’s the same but that is by no means obvious.

Posted by: Robert on September 27, 2005 10:27 PM | Permalink | Reply to this

Re: Literature on Exotic R4s

Nakahara is sort of the defacto place to start at for an undergrad or first year grad. In fact, much of that book is more or less copied from the classic US diff geometry book by Spivak (a mathematicians primer). (If you don’t understand Nakahara 200% believe me, you have absolutely no chance of understanding Seiberg-Witten theory or Donaldson invariants)

Once you have that background more or less the next place to look at is the book ‘Spin Geometry’ and certain subsections where they introduce some of the concepts.

From there, it starts getting less obvious where to look for and quickly branches off into Physicist vs Mathematician lingo and treatments. I personally had difficulty figuring out what the hell the physicists were trying to do and why it seems important. Only when I had Morse theory and other things in my toolbelt did I really appreciate the whole thing fully.

Sort of the classical very clear treatments that everyone loves are Atiyah papers (that you can google for) as he kind of bridges the gap between the two. Also Wittens original papers on the subject are beautiful as well, and as usual very clear and eloquent.

Posted by: Haelfix on September 28, 2005 7:44 AM | Permalink | Reply to this

Re: Literature on Exotic R4s

I urge that you read this book:

Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society, Providence Rhode Island, 2005.

It’s the best introduction to 4-manifolds and the most efficient path to understanding exotic R^4’s. Also, it has lots of references.


Posted by: John Baez on October 21, 2005 7:10 AM | Permalink | Reply to this