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":

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.

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!

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'.