Worked solution: Hill-Hopkins-Ravenel proof via equivariant stable homotopy theory (2009)

William Browder's 1969 breakthrough "The Kervaire invariant of a framed manifold" transformed a geometric problem of differential topology into an algebraic survival question in stable homotopy theory. The classical Kervaire invariant, introduced by Michel Kervaire in 1960 to construct the first topological manifold with no smooth structure, associates an element of $\mathbb{Z}/2$ to any $(4k+2)$-dimensional framed manifold by evaluating a quadratic refinement on its middle-dimensional homology; the problem asks in which dimensions $n$ a framed manifold of Kervaire invariant $1$ can exist.

Browder proved two structural facts. First, the invariant vanishes identically whenever $n$ is not of the form $2^{j+1}-2$: in any other dimension, surgery on middle-dimensional homology classes can always be done without obstruction, killing all non-zero quadratic values and leaving a framed sphere. Second, in the remaining dimensions $n = 2^{j+1}-2$, existence of a Kervaire-invariant-$1$ manifold is equivalent to the survival of a specific algebraic class in the classical Adams spectral sequence: the element $h_j^2 \in \operatorname{Ext}_{\mathcal{A}}^{2,2^{j+1}}(\mathbb{F}_2, \mathbb{F}_2)$ over the mod-$2$ Steenrod algebra $\mathcal{A}$ must be a permanent cycle, surviving all differentials $d_r$ ($r \ge 2$) to represent a non-trivial element $\theta_j \in \pi_{2^{j+1}-2}^S$.

Browder's reduction therefore confines the problem to the sparse list of dimensions $n \in \{2, 6, 14, 30, 62, 126, 254, \ldots\}$ and shifts the challenge from constructing exotic smooth manifolds to computing differentials in the Adams spectral sequence — setting the stage for the low-dimensional constructions of Step 2 and the non-existence theorem of Hill, Hopkins, and Ravenel.