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

Step 3 of 9: Strategy: reduce to a detection problem in a new cohomology theory

In plain words

Rather than fighting the Adams spectral sequence differentials head-on for every $j$ , Hill, Hopkins and Ravenel (2009) look for a "detector": a new cohomology theory $\Omega$ , engineered to be sensitive to exactly the classes $\theta_j$ , so that if $\theta_j$ existed it would have to leave a visible mark somewhere inside $\Omega$ .

Three separate structural facts about $\Omega$ then do all the remaining work: a Detection Theorem (any surviving $\theta_j$ leaves a nonzero mark), a Periodicity Theorem ( $\Omega$ repeats itself every $256$ steps), and a Gap Theorem (there is simply no room for a mark near the bottom of that $256$ -step window) — combine the three and the mark $\theta_j$ was supposed to leave has nowhere left to go.

\theta_j \in \pi_{2^{j+1}-2}S^0 \text{ exists} \implies \text{Hurewicz image of } \theta_j \text{ in } \Omega^{2-2^{j+1}}(\mathrm{pt}) \ne 0

Detailed analysis

HHR construct a genuinely new multiplicative cohomology theory $\Omega$ (§1.2-1.3, arXiv:0908.3724), modeled on Atiyah's $C_2$ -equivariant Real $K$ -theory but built instead from $C_2$ -equivariant complex cobordism $MU_{\mathbb{R}}$ (the Real bordism spectrum of Landweber and Fujii), together with a companion honestly-equivariant spectrum $\Omega_{\mathbb{O}}$ of which $\Omega$ is defined to be the $C_8$ -homotopy fixed points.

The Detection Theorem (Theorem 1.6) states that if $\theta_j \in \pi_{2^{j+1}-2}S^0$ exists (for $j>2$ ), its image under the unit map $S^0 \to \Omega$ is nonzero in the group $\Omega^{2-2^{j+1}}(\mathrm{pt})$ — turning the topological existence question for $\theta_j$ into an algebraic non-vanishing question inside $\Omega$ .

The remaining steps build $\Omega$ precisely enough to prove the two companion facts — periodicity and a gap — that, combined with the Detection Theorem, force that algebraic group to vanish once $j\ge 7$ .

Terms in this step

Real bordism spectrum ( $MU_{\mathbb{R}}$ ): The $C_2$ -equivariant refinement of complex cobordism, classifying "Real manifolds" (stably almost complex manifolds equipped with a compatible conjugate-linear $C_2$ -action), due to Landweber and Fujii.
Hurewicz image: The image of a stable homotopy class under the natural map from stable homotopy groups of spheres to the homotopy (or cohomology) groups of a given cohomology theory, obtained by composing with that theory's unit map.

Knowledge used in this step

Homology and cohomology