# Understanding four manifolds (more details inside)

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.

• 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... – Lee Mosher Mar 14 '13 at 15:10