Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

Question

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such understanding of Freedman's work, not even enough to be able to identify what I don't know. Where can I begin? I understand algebraic topology on about the level of Hatcher (I realize this isn't very much).

Answer 1

Start by reading Milnor's "Lectures on h-cobordism theorem", since Freedman's proof is a very difficult variation on Smale's proof in higher dimensions.

Answer 2

Edit 11/21/2023: As was pointed out in the comments, there is now a really wonderful reference for the Disk Embedding Theorem, a main ingredient in Freedman's proof. See The Disc Embedding Theorem, published in 2021 by Oxford University Press. It is the result of many hundreds of hours of work and many dedicated authors who put an incredible amount of effort into understanding Freedman's arguments and carefully explicating them.

Original answer:

That's a tall order! I would wager that very few people understand Freedman's proof completely. I found this series of video lectures that he gave to be quite helpful. In particular, it will give you an idea about the sort of mathematics that is involved.

Also, Freedman and Quinn's book on $4$-manifold topology is great! Even if you aren't to the point where you can follow it in detail, you can still benefit from skimming through it.