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I need to understand some of the theory of smooth four manifolds. Eventually I might be interested in learning about Donaldson theory. For the moment I am mainly interested in question such as: if you have a compact, connected, simply-connected four manifold, in how many different ways can I embed a smooth, compact 2-surface in it? A prototype question would be: how many different smooth embeddings of S^2 in the connected sum of CP^2 can I have? I would also like to learn how to calculate self-intersection numbers of 2-surfaces embedded in 4-manifolds.

I have some knowledge of differential geometry and general topology and, at a much lower level, of algebraic topology but my background is in theoretical physics. To give an idea, I feel at home with books like Nakahara "Geometry topology and physics" or "Gravitation" by Misner, Thorne and Wheeler, I like and can follow the more mathematical Naber "Topology, geometry and gauge fields", I find more challenging a book like Bott, Tu "Differential forms in algebraic topology", but I have not worked through it yet (I am more familiar with de Rham theory than anything else in algebraic topology) but e.g. Hatcher "Algebraic Topology" is really though for me.

Given my background, what route would you recommend to learn enough material to allow me to calculate things like those I have pointed out above? I do not want to do research in this area, but be familiar enough with it so that I can use it-see the questions I have raised above for an example of what I mean.

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    $\begingroup$ You will have to struggle through something like Hatcher's book to get much information about these questions. The question about how many smooth embeddings of $S^2$ in $CP^2$, interpreted literally, has an unsatisfying set theoretic answer: uncountably infinitely many. For better interpretations, you'll need to understand the important equivalence relations of topology. How many embeddings up to isotopy? Or if you are willing to drop embeddings: How many immersions up to regular homotopy? How many continuous maps $S^2 \to CP^2$ up to homotopy? And so on... $\endgroup$
    – Lee Mosher
    Commented Mar 14, 2013 at 15:10

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My recommendation would be the book of Freedman and Quinn, Topology of 4-manifolds. It's hands-on, very very good, and suitable I think for a reader of your background. Indeed, I would strongly recommend it to anyone interested in embeddings and immersions of surfaces in 4-manifolds.

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  • $\begingroup$ Definitely a good foundational text. (Although it apparently has some mistakes.) $\endgroup$
    – Jim Conant
    Commented Mar 14, 2013 at 16:27
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I would definitely suggest you to read The Wild World of Four-manifolds of Alexandru Scorpan. It is (IMHO) an excellent introduction to four-manifolds: it is very pleasant to read and very well organized. You can read it at very different levels: there is a main road (which skips lots of technicalities and tries to communicate the beauty and richness of the subject), and many different paths starting from it, to be read just in case some argument fascinates you and you want to understand it more deeply. Of course it speaks a lot about surfaces in four-manifolds.

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For a nice introduction to immersion theoretic topology (leading up, I suppose, to understanding immersions up to regular homotopy, and ultimately to embeddings up to isotopy) you might look at Eliashberg, Y.; Mishachev, N. Introduction to the h-principle. (English summary) Graduate Studies in Mathematics, 48. American Mathematical Society, Providence, RI, 2002. xviii+206 pp. ISBN: 0-8218-3227-1

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It sounds like you want to be able to compute the 2nd homotopy group of a space, and know just enough about bundles to determine when two surfaces have "essentially the same" tubular neighbourhood or not.

So you need a grounding in the 1st and 2nd homotopy groups + covering spaces, as well as homology and cohomology. There are lots of textbooks that cover that material. As mentioned, there's the Hatcher text, and there's also Peter May's text. There's many more, perhaps Bredon's "Geometry and Topology" or Kirk and Davis's book is more suited for your project. Milnor and Stasheff's "Characteristic Classes" would also be helpful.

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