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

Step 5 of 9: The Slice Theorem: identifying the layers of

MU^{((C_8))}

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In plain words

Just as an ordinary space can be studied one Postnikov stage at a time, HHR filter their equivariant spectrum by "slices" — an equivariant analogue of the Postnikov tower, indexed not just by an integer but adapted to the group action.

The Slice Theorem is the single computational engine behind everything else: it says each layer of this filtration for $MU^{((C_8))}$ is built purely from ordinary (non-equivariant) integer cohomology, glued together according to a completely explicit recipe — turning a hard equivariant question into bookkeeping with ordinary cohomology.

P_n^n MU^{((C_8))} \simeq H\underline{\mathbb{Z}} \wedge W_n \quad (\text{slice tower})

Detailed analysis

The slice filtration (§4) associates to any equivariant spectrum $X$ a tower of "slice sections" $P^n X$ , built by attaching cells that kill all maps from cells of dimension greater than $n$ ; the layers $P_n^n X$ (the "slices") assemble into the slice spectral sequence computing $\pi_* X$ .

The Slice Theorem (Theorem 1.13, proved via a Reduction Theorem in §6-7) identifies the slices of $MU^{((C_8))}$ explicitly: $P_n^n MU^{((C_8))}$ is a wedge of copies of the equivariant Eilenberg-Mac Lane spectrum $H\underline{\mathbb{Z}}$ (for the constant Mackey functor $\underline{\mathbb{Z}}$ ) smashed with specific "slice cells" of dimension $n$ built from induced representation spheres.

Because every slice is this explicit, the slice spectral sequence for $MU^{((C_8))}$ (and, after further work, for the detecting theory $\Omega$ ) becomes a concrete, computable object — the basis for both the Gap Theorem (next step) and the Periodicity Theorem.

Terms in this step

Slice filtration: An equivariant analogue of the Postnikov tower that filters a $G$ -spectrum by "slice cells" built from induced representation spheres, rather than by ordinary dimension alone.
Eilenberg-Mac Lane spectrum ( $H\underline{\mathbb{Z}}$ ): The equivariant spectrum representing ordinary Bredon cohomology with coefficients in a Mackey functor, here the constant integer Mackey functor $\underline{\mathbb{Z}}$ ; the simplest possible building block a slice can be made from.

Knowledge used in this step

Homology and cohomology