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

Step 7 of 9: The Periodicity Theorem:

256

-fold periodicity

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In plain words

A general fact about commutative ring spectra, going back to Bott periodicity for $K$ -theory, says that inverting a suitable "orientation-like" class produces periodic behaviour. HHR carry out an equivariant version of this using an explicit class built from an equivariant refinement of Bott periodicity.

A modest direct computation in the $RO(C_8)$ -graded slice spectral sequence for $\Omega_{\mathbb{O}}$ then pins the exact period down to $256$ — a number forced by the specific choice of $C_8$ and the four-fold norm construction of the earlier step.

\Omega^*(X) \approx \Omega^{*+256}(X)

Detailed analysis

Inverting a class $D: S^{\ell\rho_8} \to MU^{((C_8))}$ , built from an equivariant refinement of the classical Bott periodicity class, produces the $C_8$ -spectrum $\Omega_{\mathbb{O}} = D^{-1}MU^{((C_8))}$ ; $\Omega$ is defined as its $C_8$ -homotopy fixed-point spectrum.

A direct computation in the $RO(C_8)$ -graded slice spectral sequence for $\Omega_{\mathbb{O}}$ (§9, using that $\Omega_{\mathbb{O}}$ is an equivariant commutative ring spectrum) shows $\Omega^*(X) \approx \Omega^{*+256}(X)$ for every $X$ — the Periodicity Theorem (Theorem 1.7).

Alternatively, general nilpotence-technology arguments show $\pi_*\Omega$ is periodic with some power-of- $2$ period without pinning down the exact number; the explicit slice-spectral-sequence computation is needed only to nail the period at exactly $256$ , matching the Gap Theorem's width.

Terms in this step

$RO(G)$ -graded cohomology: A refinement of ordinary $\mathbb{Z}$ -graded cohomology in which degrees are indexed by actual real representations of $G$ , not just integers; necessary because equivariant spectra "see" more structure than a single integer grading can record.

Knowledge used in this step

Homology and cohomology