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

Step 1 of 9: Browder's reduction: existence only in exponentially sparse dimensions

In plain words

The Kervaire invariant is a simple yes/no measurement (a single bit, $0$ or $1$ ) you can compute for certain framed manifolds using a quadratic-form trick borrowed from surgery theory. Browder discovered in 1969 that this bit can only ever come out $1$ in a very restrictive, rapidly thinning list of dimensions.

More precisely, he showed the question boils down to whether a single specific class $h_j^2$ , sitting inside a purely algebraic gadget called the Adams spectral sequence, manages to "survive" all the way through to represent a genuine homotopy class $\theta_j$ . Everything in the rest of this proof is really about deciding the fate of that one class, for each $j$ .

n = 2^{j+1} - 2, \quad \theta_j \in \pi_{2^{j+1}-2}^S \ \text{detected by} \ h_j^2 \in \operatorname{Ext}_{\mathcal{A}}^{2,2^{j+1}}(\mathbb{F}_2, \mathbb{F}_2)

Detailed analysis

William Browder (1969) proved that a smooth closed framed manifold with Kervaire invariant $1$ exists in dimension $n$ only if $n = 2^{j+1} - 2$ for some $j$ , and only if the class $h_j^2$ in the Adams $E_2$ -page $\operatorname{Ext}_{\mathcal{A}}^{2,2^{j+1}}(\mathbb{F}_2, \mathbb{F}_2)$ (a mod- $2$ Steenrod algebra computation) is a permanent cycle, surviving all differentials to represent a genuine class $\theta_j$ in the stable homotopy group of spheres $\pi_{2^{j+1}-2}^S$ .

This reduces a geometric existence question (does such a manifold exist?) to a purely algebraic survival question in homotopy theory (does this one specific class die under a differential, or not?) — and moreover confines the search to the sparse, doubly-exponential list of dimensions $n = 2^{j+1}-2$ , since Browder also shows the invariant is automatically zero in every other dimension.

Terms in this step

Kervaire invariant: A $\mathbb{Z}/2$ -valued invariant of a framed manifold of dimension $4k+2$ , defined via a quadratic refinement of the middle-dimensional intersection form; it detects whether the manifold can be simplified (surgered) all the way down to a homotopy sphere.
Adams spectral sequence: A computational tool that starts from a purely algebraic object built from the Steenrod algebra (its $E_2$ -page) and, through a sequence of "differentials", converges to the stable homotopy groups of spheres; a class that survives every differential is called a permanent cycle.

Knowledge used in this step

Homology and cohomology Homological algebra