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

Step 6 of 9: The Gap Theorem: no room near the bottom

In plain words

Once the slices of $MU^{((C_8))}$ are known explicitly (previous step), a short computation — the "Cell Lemma" — shows certain equivariant homotopy groups of these building blocks simply vanish in a narrow range near degree $0$ , because the relevant orbit spaces turn out to be simply connected.

Stacking this vanishing across every layer of the slice tower shows the homotopy groups of the whole detecting theory $\Omega$ vanish in that same narrow range — a "gap" with no room for anything to survive.

\Omega^i(\mathrm{pt}) = 0 \quad (0 < i < 4)

Detailed analysis

The Cell Lemma (Lemma 1.14) computes that for $G = C_{2^n}$ and any slice cell $\widehat S$ , the equivariant homotopy groups $\pi_k^G H\underline{\mathbb{Z}} \wedge \widehat S$ vanish for $-4 < k < 0$ ; this reduces to the elementary fact that the orbit space $S^{m\rho_G}/G$ is simply connected, being the suspension of a connected space.

Combining the Cell Lemma with the Slice Theorem's explicit description of the slices of $MU^{((C_8))}$ (and hence of $\Omega_{\mathbb{O}}$ ), HHR show the $C_8$ -fixed-point homotopy groups $\pi_i^{C_8}\Omega_{\mathbb{O}}$ vanish for $-4<i<0$ .

The Homotopy Fixed Point Theorem of a later step identifies these groups with $\pi_i \Omega$ itself, giving the Gap Theorem: $\Omega^i(\mathrm{pt}) = 0$ for $0<i<4$ — exactly the vanishing range the final assembly step needs.

Terms in this step

Cell Lemma: A short computation showing that certain low-degree equivariant homotopy groups of Eilenberg-Mac Lane spectra smashed with slice cells vanish, because the relevant orbit space is simply connected.

Knowledge used in this step

Homology and cohomology