TITLE: Kervaire invariant: Why dimension 126 especially difficult?
QUESTION [35 upvotes]: Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\theta_6$ exists in the "$128$-stem"), i.e., why the celebrated Hill-Hopkins-Ravenel 
proof technique fails in this last remaining case?
I read a delightful exposition by Erica Klarreich ("Mathematicians solve 45-year-old Kevaire Invariant Puzzle,"
The Best Writing on Mathematics, 2010, 373ff, Simons Foundation link), which piqued my interest.
But Michael Hopkins' online presentation slides announcing the result
in 2009 ("Applications of algebra to a problem in topology") are beyond my ken.
Perhaps this an area too abstruse for all but the experts?
I'd appreciate pointers to expositions.  Thanks!

REPLY [17 votes]: More detail on Jones's 30 manifold and analogous constructions can be found in my  Extended powers of manifolds and the Adams spectral sequence , Cont. Math. 271, 41--51.  The basic idea in it dates from 1979 or so (immediately upon seeing Jones's construction) but didn't make it into print for another 20 years.  There are also comments about the homotopy theoretic approach to the problem in   the talk I gave in Edinburgh  in 2011 at the workshop on the Kervaire invariant.
The main point is that there is a small set of homology classes in low dimension (dimension 2 for Jones' construction) which one wants to realize as the fundamental class of a manifold with tangent bundle realized by a permutation representation of $\pi_1$.  To get the 30 dimensional Kervaire manifold from $S^7$ one only needs a two manifold and an appropriate representation in $S_4$.  ($D_8$ will do:  only the Sylow 2-subgroup matters.)  To get from $S^7$ to the 62 dimensional Kervaire manifold requires a 6-manifold with a representation of $\pi_1$ in $S_8$.  Unfortunately, no such manifold exists.  In dimension 126, one would need a 14 manifold, and presumably such also fails to exist. So, one needs to find another approach that relaxes the input data needed.