Incorrect information in an old article about the Kervaire invariant 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?  . 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).  
 A: I'll try to give more precise detail soon, but here's my understanding of this history.   The 'proof' Peter May mentions is the 'standard mistake' in the subject. 
Consider elements in a spectral sequence coming from the homotopy exact couple of a tower.  If you have a geometric construction that the boundary of $x$ is $y$, and you can show that $y$ is itself null-homotopic, it is tempting to think you have shown that $x$ survives to a non-zero class.   However, $x$ is one of the reasons that $y$ is null-homotopic, so you haven't really observed anything about $x$ from knowing only that $y$ is null-homotopic.   What you need is that $y$ was already null-homotopic before $x$ got there to kill it.   In other words, you need that $y$ is null-homotopic in an appropriately high stage of the tower, not just in the $0^{th}$ term.  For an example related to this case, see pp. 38-39 of
http://www.rrb.wayne.edu/papers/fin_conj_handout.pdf
The mistake Milgram made in "Symmetries and operations" was just a miscalculation in the $\Sigma_4$ extended power of the $30$-sphere, or  perhaps of $S^{30} \cup_2 e^{31}$.   It did seem like this would give $\theta_6$ in the $126$-stem, given what was then known about $\theta_4$, until the mistake was noticed, and apparently Shtan'ko wrote his report during this burst of enthusiasm.
Shtan'ko's 'A. Milgram' was just a mistake.  Surely he meant R. J. Milgram'.
A: I found the following remark in Zhouli Xu's paper "The strong Kervaire invariant problem in dimension 62":

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.
A: I'm feeling mischievous.  The history of the Kervaire invariant problem is strewn with false proofs, Jim Milgram's is only one of many.  A student of mine (who I will leave nameless) had a preprint (around 1980?) that solved the problem but that Mark Mahowald quickly shot down.   The most recent example I know of is that of a Russian mathematician (who I will also leave nameless).  See Math Reviews MR2590025 (2010k:55031) 
Differentials of the Adams spectral sequence and the Kervaire invariant  (Russian) Dokl. Akad. Nauk 427 (2009), no. 5, 601–604; translation in Dokl. Math. 80 (2009), no. 1, 573–576.  From the text (translated from the Russian): "In this paper, we study the differentials of the Adams spectral sequence for stable homotopy groups of spheres and solve the Kervaire invariant one problem for n-dimensional manifolds when $n=2^i-2$, $i\geq 6$."  That is a four page paper.  Would that it were so simple!
