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I've been reading about the Arf invariant and came across the following conjecture (see here):

Each framed bordism class contains a manifold which admits a (possibly different) framing with zero Arf invariant.

Is this conjecture still open? I did a google search but it turned up nothing useful.

Also: are there any online lists available of open problems involving the Arf invariant?

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This might be a question better suited for MathOverflow. That site is dedicated to research questions. – Kris Williams Sep 2 '12 at 10:55
Thanks, but it's not research, I'm an undergrad. – Rudy the Reindeer Sep 2 '12 at 11:15
Snaith has a paper on the arXiv you might be interested in: He also has a book - alas, he released it just before HHR announced their solution... – Juan S Sep 3 '12 at 9:50
up vote 6 down vote accepted

As I understand the case of dimension $126$ is still open, but otherwise the problem was recently solved (negatively, the manifolds do not exist) by Hill, Hopkins, and Ravenel. Their paper is here. My understanding (which should not be taken too seriously) is that the problem was reduced by Browder to showing whether certain classes in the Adams spectral sequence were permanent cycles, i.e. whether certain homotopy classes existed. H, H, and R apparently constructed a very complicated spectrum to show that those homotopy classes could not. If you google "Kervaire invariant problem" you should find more material on the subject.

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II think you've answered the question posted for Arf invariant =1. The question about Arf invariant=0 is answered positively in the paper that Matt links to. – user17786 Sep 3 '12 at 0:58
@user17786: It's the same thing right? It's classic that the Kevaire invariant is zero unless the dimension is $2^k-2$. The existence of manifolds of Kevaire invariant 1 is known for $n=2,6,14,30,62$. HHR showed that for $k \ge 8$ the that the invariant is zero in dimension $2^k-2$. Thus leaving the case $n=126$ as the last remaining possibility – Juan S Sep 3 '12 at 9:47
@juan In the paragraph after the conjecture Nigel Ray says that the purpose of that paper is to prove tha conjecture, so i would conclude that that conjecture is closed. In a given framed bordism class there may be framed manifolds with kervaire invariant 0 and framed manifolds with kervaire invariant 1. The conjecture in the question asserts the existence of manifolds of Kervaire invariant 0, and there is a different conjecture about the existence of manifolds of Kervaire invariant 1, which is what the answer is about. – user17786 Sep 4 '12 at 10:54
@user17786 - Ahh, I see what you mean now! Matt should be interested in the Kervaire invariant 1 problem anyway! :) (And apologies for the repeated misspellings in my previous post!) – Juan S Sep 4 '12 at 11:48
Haynes Miller gave a Séminaire Bourbaki report on HHR which looks very interesting. – t.b. Sep 7 '12 at 0:07

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