4-Manifolds and Kirby Calculus by Gompf.

MathematicalPhysicist

What should one know before reading this textbook?
I have taken a course in differential topology (it was called Analysis on manifolds), and a course in topology which covered also introduction to algebriac topology, and I am reading Spanier's textbook, will that suffice for this textbook?

zhentil

Not even close. Gompf's book is for researchers in the field of 4-mfld topology. I could potentially see it being used for an advanced topics course, but not likely.

MathematicalPhysicist

So what do you need to know before you read it?

wofsy

Some familiarity with algebraic geometry, gauge theory (but you are a physicist so you likely know this already), and the theory of fiber bundles.

You might want to look at other references of handle bodies that preceded Kirby calculus. If you are interested in this tangent I can give you references

slider142

I would be interested in these references as well. :D

MathematicalPhysicist

Yes, wofsy, any references will be terrific by me.

wofsy

for handle body theory there are many references. The ones I have looked at are Intro to Piecewise-Linear Topology by Rourke and Sanderson and Lectures on the H-Cobordism Theorem by Milnor. These are tough books but well worth the pain.

For a quick intro to modern gauge theory from the mathematician's view point, there is an appendix in Milnor's Characteristic Classes that will get you started. This wonderful exposition derives Chern and Pontryagin classes from the curvature 2-form in only a few pages. The view of gauge theory is that it is differential geometry without a metric, only a connection. But the interplay of this more general form of differential geometry with standard Riemannian geometry leads to profound results. For instance, if the connection is compatible with a metric one always has an Euler class derived from the curvature form. However, there are connections which are not compatible with any metric and then the Euler class may not be expressed in term of curvature. Milnor gives an example of a flat connection - zero curvature 2 form - on a 2 plane bundle over a surface that has non zero Euler class!

I do not know this stuff well and would be glad to read it with you.

zhentil

Wofsy, I agree that Milnor's book is a fantastic reference, but in the interest of full disclosure one should mention that the list of prerequisites for that book is even longer. I would say that Milnor's book on Morse Theory, Milnor's book on Characteristic classes, some exposure to algebraic topology (say at the level of Hatcher or Bott & Tu), and decent books on differential topology (Milnor's book may be good enough, but I'll say Hirsch to be safe) and Riemannian geometry are necessary.

This is coming from the point of view of a geometer/topologist, of course. IMO, the only way to start learning 4-manifold topology is to understand why h-cobordism doesn't work in dimension four. Even the hand-waviest explanation of this is that you can't embed 2-disks generically in dimension 4. If one has to take even this on faith...

wofsy

The appendix requires little background - it is true that you must know the universal definition of characteristic classes - but their proofs are secondary to this section.