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Recently I'm reading Donaldson's Geometry of four manifolds. It seems to me that the book requires a lot for background. Additionally, the proof in the book is too sketchy without too much detail. I had a really hard time to digest the content in the book. Do we have other textbook demonstrating the same topic with more detail and background?

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Please do not ignore the other author, Peter Kronheimer. Based on all of the material I've read, I do not agree with your belief about the book. I think it is more detailed than you will find elsewhere which covers all of that material. Here are some useful alternatives, though:

Take that book and replace the structure group $SU(2)$ by $SO(3)$. This route is taken in Petrie-Randall's "Connections, definite forms, and four-manifolds".

If you care more for easy digestion and less about whether it is sketchy, the closest thing I can think of to Donaldson-Kronheimer's book is Freed-Uhlenbeck's "Instantons and four-manifolds".

For rigorous yet digestive background, I recommend Booss-Bleecker's "Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic physics". Other relevent material, which has a lot of explanations/comments but not complete and not always rigorous, are Gompf-Stipsicz' "4-manifolds and Kirby calculus" and Scorpan's "The wild world of 4-manifolds".

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  • $\begingroup$ Kronheimer is really a good writer! $\endgroup$ – Bombyx mori Sep 10 '17 at 1:16
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It's been a while but I remember that I found John Morgan's "An introduction to gauge theory" quite helpful when I was first trying to read about 4-manifolds and gauge theory. While it doesn't go nearly as far as Donaldson-Kronheimer, it does give a gentle introduction to gauge theory. Here's the complete reference:

Morgan, John W., An introduction to gauge theory, Friedman, Robert (ed.) et al., Gauge theory and the topology of four-manifolds. Lectures of the graduate summer school. Providence, RI: American Mathematical Society. IAS/Park City Math. Ser. 4, 53-143 (1998). ZBL0911.57024.

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A quite "low-brow" book you might find enjoyable reading is The Wild World of Four manifolds, which surveyed the whole theory beginning from handle body decomposition. It has a long list of reference books you can use for further reading. The material may be out of date by now, though. If you can provide more detailed feedback on where you got stuck, I am sure the experts in the site can provide more help.

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