references / general idea of kervaire invariant problem There's a workshop at MSRI in a couple months on the Kervaire invariant problem that I'd really like to attend.  I saw Hopkins speak about it a while back without understanding much of the talk, but I'm thinking that maybe by now I'm more prepared to see what it's all about.  At least, I'd like to be able to get something out of the workshop, even if I don't understand everything that's going on.  I don't know much more about the problem than the wikipedia article can tell me.  So, I'm looking for:
(a) recommendations for reading to acquaint myself with the necessary background information, and
(b) a sketch of the story -- I can somewhat follow the intro to the paper itself, but seeing some of the details spelled out would help me start to actually understand what's going on.
(Or, unfortunately possibly: (c) a warning that it's probably too difficult to pick up all the material in just a few months and that it's not worth my time to try.)
By the way, wikipedia says that Kervaire used the invariant to create a 10-dimensional PL manifold with no differentiable structure, but I thought that the invariant is zero in dimension 10.  What's the deal there?
 A: Here's a recent survey article by Victor Snaith:
http://chucha.math.cinvestav.mx/morfismos/v13n2/arfsurveyMFMS.pdf
(I think there's also a copy on the arxiv)
A: Snaith's book is good place to look. Also, Hopkins lecture at Atiyah's birthday is awesome. It was caled the doomsday conjecture because of what would happen in the EHP sequence if these elements were not homotopy classes. In this vein, i should point out the "recent" note that appeared online http://www.math.northwestern.edu/~pgoerss/papers/kinoteNov16.pdf by Goerss and Mahowald. both of which understand stable homotopy theory extensively. Mahowald is a major influence on this whole subject and specifically an expert in this area.
As for dylan's comment, I believe he is exactly right. The kervaire invariant does not require a smooth structure. The cohomology of a manifold has a product, and if the manifold is even dimensional then the cup product pairing gives us a quadratic form on the middle dimensional cohomology/homology group. The kervaire invariant is the arf invariant of this quadratic form. SO there is no need for a smooth structure. What is probably the case, is that he showed that the framing of a manifold could be used to show the vanishing of the kervaire invariant. I suggest looking at the work of kervaire and kervaire-milnor, they are excellent writers.
A: Have a look at the MIT-Harvard Seminar and the lecture notes given at this link.
For some background, see Petr. M. Akhmetev's work.
I see that you are starting with the wikipedia article. For very very introductory stuff, such as statement itself, there's the article of Teichner, article in Scientific American, a Simons foundation article with some animations and an article in Nature.
Most of the above are lifted shamelessly from Douglas Ravenel's homepage.
