Prerequisites to understand Immersion conjecture proof by Cohen (1985)

Question

I would like to understand the proof by Cohen of the Immersion Conjecture, but since it is a relatively recent work I probably need lots of prerequisites: my background is composed by an introduction to differential geometry (essentially Do Carmo topics up to global differential geometry), to differential topology (Milnor's "Topology from a differentiable viewpoint") and to algebraic topology (we partially covered Hatcher's book).

May someone recommend to me what are the most important topics (and where to find them) I have to cover to get at least a partial understanding of Cohen's theorem?

EDIT: it has been proposed to me as subject for the final dissertation of my bachelor -- not a full exposition of it, but an overview on the history of the conjecture and its proof, with focus on some parts of it by my choice.

Answer 1

The "prerequisites" for the full argument are quite extensive. Here are some of them:

• Hirsch-Smale theory.
• Massey and Brown-Peterson's work on the ideal $I_n$ of relations satisfied by normal Stiefel-Whitney classes of $n$-manifolds.
• The Thom spectrum $MO$; the splitting of $MO/I_n$ into Brown-Gitler spectra (braid Thom spectra).
• The Brown-Peterson programme for the immersion conjecture: their construction of $BO/I_n$ and its lifting properties.
• Cohen's proof of the deThomification $BO/I_n \to BO(n-\alpha(n))$.

The problem is that each item in this list would itself have prerequisites of its own, and then even after amassing all the prerequisites the actual arguments are still very complicated. I'd suggest taking a look at the survey articles:

• Cohen-Tillmann's "Lectures on immersion theory"
• and if you can read French, Lannes' Bourbaki article "La conjecture des immersions"

and going from there.