In the Soviet times there was a famous Encyclopedia of Mathematics. I think it is still familiar to every Russian mathematician maybe except very young ones, and yours truly is in possession of all 5 volumes. Browsing it recently (with no real purpose) I came across a certain peculiarity. In the article "Kervaire invariant" by M.A.Shtan'ko there was a claim that the Kervaire invariant is nontrivial in all dimensions $2^k-2$ for $2\le k\le 7$ - yes, including 126. Which, for what I know, is still an open problem 
[Kervaire invariant: Why dimension 126 especially difficult?](https://mathoverflow.net/questions/101031/kervaire-invariant-why-dimension-126-especially-difficult) . In this article, the credit for the $k=6$ and $k=7$ cases (lumped together) was given to M. Barratt, M. Mahowald, and A. Milgram, but with no actual reference.


To be fair, the absence of references is understandable (in the original article, that is) because it was written in 1978 while a complete proof for dimension 62 was only published six years later 
<https://web.math.rochester.edu/people/faculty/doug/otherpapers/barjoma.pdf>
It is possible that back in 1978 the result was just announced. But, what happened to the 126? And to Milgram? The simplest possible explanation is that a proof for 126 was also announced but later retracted. However this is by no means the only possibility, so I am curious what really happened. Besides, those MO folks who know more about the subject then myself might wonder what the attempted proof was like. 


After a bit of search I found a reference which may be relevant (hopefully). In "Some remarks on the Kervaire invariant problem from the homotopy point of view" by M.E.Mahowald (1971) there is Theorem 8 attributed to Milgram and after it the following Remark: "It can be shown that $\theta\_4^2=0$ and thus Milgram's theorem implies $\theta\_6$ exists". If I get it right this indeed means a nontrivial Kervaire invariant in dimension 126, so probably there is a mistake somewhere in this argument. (But even if it is so, damned if I have a clue who has made it: Milgram, Mahowald, or somebody else.) 


I have to admit that a few things about this story look suspicious. To begin with, 
A. Milgram died in 1961 so it should probably be R. J. Milgram if any. In the introduction to "The Kervaire invariant of extended power manifolds " J. Jones stated explicitly that the 62 case is solved by Barratt and Mahowald but not published yet while in the higher dimensions the problem is open, in contradiction to what Shtan'ko wrote the same year. In a couple of papers between 1978 and 1981 I spotted references like [Barratt M. G., Mahowald M., The Arf invariant in dimension 62, to appear] but no traces whatsoever of 126 and Milgram. (Besides this article of Mahowald from almost a decade before.) I am at a loss what to make of all this. It would be nice if someone can set it straight - at the very least, I want to know if Shtan'ko made it up. 


By the way, an English translation of this Encyclopedia article can be found here  

<https://www.encyclopediaofmath.org/index.php/Kervaire_invariant>  

Only, the year is written 1989 instead of 1978 (a second edition, apparently). 



Incorrect information in an old article about the Kervaire invariant
I found the following remark in Zhouli Xu's paper ["The strong Kervaire invariant problem in dimension 62"](https://arxiv.org/abs/1410.6199):



> 
> In [19], R. J. Milgram claims to show that under the same condition as in Theorem 1.1, one has $θ\_{n+2}$ exists. If this were true, then we would have that $\theta\_6$ exists. However, Milgram’s argument fails because of a computational mistake
>  [8].
> 
> 
> 


The paper containing the mistake is



> 
> R. J. Milgram, "Symmetries and operations in homotopy theory" Amer. Math. Soc. Proc.
>  Symposia Pure Math., 22(1971), 203-211
> 
> 
> 


and the other reference is private communication with Robert Bruner.


