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

Step 8 of 9: Assembling the proof: non-existence of

\theta_j

for

j \ge 7

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In plain words

One loose end remains before the pieces fit: does $\Omega$ (defined via ordinary homotopy fixed points, easy to compute with) actually agree with $\Omega_{\mathbb{O}}$ 's honest categorical fixed points (harder to compute with directly, but the object the slice machinery understands)? The Homotopy Fixed Point Theorem says yes.

With that identification in hand, everything assembled in the previous steps clicks together: if $\theta_j$ existed for $j\ge 7$ , the Detection Theorem would force a nonzero class inside a group that the Gap and Periodicity theorems jointly prove is zero — a contradiction, so no such $\theta_j$ exists.

j \ge 7 \implies \text{Hurewicz image of }\theta_j\text{ in } \Omega^{2-2^{j+1}}(\mathrm{pt}) = 0

Detailed analysis

The Homotopy Fixed Point Theorem (Theorem 1.10) proves the natural map from the honest fixed-point spectrum of $\Omega_{\mathbb{O}}$ to its homotopy fixed-point spectrum is a weak equivalence, so $\pi_n^{C_8}\Omega_{\mathbb{O}} \cong \pi_n \Omega$ for all $n$ — letting the Gap Theorem's computation of honest $C_8$ -fixed points be reinterpreted as a statement about $\Omega$ itself.

The Periodicity and Gap Theorems together give $\Omega^i(\mathrm{pt}) = 0$ whenever $i \equiv 2 \pmod{256}$ ; since dimension $2^{j+1}-2$ for $j\ge 7$ satisfies exactly this congruence once translated by the period, the group $\Omega^{2-2^{j+1}}(\mathrm{pt})$ that the Detection Theorem targets is zero for every $j\ge 7$ .

If $\theta_j$ existed for such $j$ , the Detection Theorem would force its Hurewicz image in that group to be nonzero — a contradiction. Hence $\theta_j$ does not exist for any $j\ge 7$ , proving Theorem 1.1 (Hill-Hopkins-Ravenel 2009/2016) and, via Browder's reduction from Step 1, that framed manifolds of Kervaire invariant $1$ exist only in the six dimensions $2,6,14,30,62,126$ .

Knowledge used in this step

Homology and cohomology