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

Step 4 of 9: The norm functor and the

C_8

-spectrum

MU^{((C_8))}

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In plain words

Take four copies of the Real bordism spectrum and smash them together, then let a cyclic group of order $8$ permute the four copies — with a twist: going all the way around once applies complex conjugation. This single move, HHR's "multiplicative norm" construction, upgrades a $C_2$ -spectrum into a genuinely $C_8$ -equivariant one with far richer symmetry to exploit.

The point of building something so elaborate is that $C_8$ (rather than the smaller $C_2$ ) is exactly the amount of symmetry needed to make the Gap and Periodicity theorems of the next steps true; smaller cyclic groups simply don't leave enough room.

MU^{((C_8))} = MU_{\mathbb{R}} \wedge MU_{\mathbb{R}} \wedge MU_{\mathbb{R}} \wedge MU_{\mathbb{R}}, \quad C_8 : (a,b,c,d) \mapsto (\bar d, a, b, c)

Detailed analysis

HHR's key new construction (§2.2.3, "the multiplicative norm functor") builds, from a $C_2$ -spectrum such as $MU_{\mathbb{R}}$ , a genuinely $C_{2^k}$ -equivariant commutative ring spectrum by an equivariant smash-power construction; applied to produce a four-fold power, this gives $MU^{((C_8))} = MU_{\mathbb{R}} \wedge MU_{\mathbb{R}} \wedge MU_{\mathbb{R}} \wedge MU_{\mathbb{R}}$ with $C_8$ acting by cyclically permuting the four factors and applying the $C_2$ -action (complex conjugation) once per full cycle.

Very roughly, $MU^{((C_8))}$ can be thought of as the cobordism theory of stably almost complex manifolds equipped with a $C_8$ -action whose restriction to the subgroup $C_2\subset C_8$ recovers a Real structure — a direct $C_8$ -analogue of how $MU_{\mathbb{R}}$ itself classifies $C_2$ -equivariant ("Real") manifolds.

This $C_8$ -equivariant spectrum, and a companion periodicity class $D$ built from an equivariant refinement of Bott periodicity, are exactly the raw materials from which $\Omega_{\mathbb{O}}$ — and hence the detecting theory $\Omega$ of the previous step — is assembled.

Terms in this step

Norm functor: A construction in equivariant homotopy theory that builds a $G$ -equivariant spectrum from a spectrum for a subgroup $H \le G$ by an equivariant smash-power indexed by the coset space $G/H$ , generalising the classical restriction-induction adjunction multiplicatively.
$C_8$ -equivariant spectrum: A spectrum equipped with an action of the cyclic group of order $8$ , encoding symmetry data beyond a single space; the natural home for cohomology theories sensitive to $C_8$ -actions on manifolds.

Knowledge used in this step

Homology and cohomology