nLab C2-equivariant homotopy groups of spheres

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Note: C2-equivariant homotopy groups of spheres and C2-equivariant homotopy groups of spheres both redirect for "C_2-equivariant homotopy groups of spheres".
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Idea

C 2 C_2 -equivariant homotopy groups of spheres are particularly amenable to study because they are computed in a range by \mathbb{R}-motivic homotopy groups via real Betti realization with its Galois action;

this is special to C 2C_2 among finite groups, since the Artin-Schreier theorem implies that it is the only nontrivial finite group occurring as an absolute Galois groups. (…)

RO(G)\mathrm{RO}(G)-graded vs Mackey functor homotopy groups

In general, for all real orthogonal virtual GG-representations, we have a cofiber sequence of GG-spectra

S(V) +D(V) +S V; S(V)_+ \rightarrow D(V)_+ \rightarrow S^V;

In the case V=σV = \sigma, we find that S(V)=C 2,+S(V) = C_{2,+}; hence there is a natural long exact sequence in stable homotopy in the motivic grading π s,c C 2Xπ c+(sc)σ C 2X\pi_{s,c}^{C_2}X \coloneqq \pi_{c + (s-c)\sigma}^{C_2} X, this reads as

In particular, the five lemma combined with the equivariant Whitehead theorem then proves the following (see Prop 3.2 of Guillou-Isaksen 24).

Proposition

A map of C 2C_2-spectra induces an isomorphism on RO(C 2)\mathrm{RO}(C_2)-graded homotopy groups if and only if it induces an isomorphism on \mathbb{Z}-graded mackey functor homotopy groups.

The corresponding theorem is seldom true for other groups, since every other group attains two dimensional irreducible real orthogonal GG-representations, whose compactifications will not be cofibers of GG-sets. For instance, a counterexample due to Clover May in the case G=C 3G = C_3 appears as Ex. 3.1 of Guillou-Isaksen 24.

Relation to motivic homotopy groups

The following diagram appears on page 3 of Guillou-Isaksen 24.

In this section, we will explain the regions. Explicitly:

  • The region c>12s5c \gt \frac{1}{2}s - 5 is described in Theorem

  • (…)

The region c>12s5c \gt \frac{1}{2}s - 5

In Heller-Ormsby 14, given a finite Galois extension L/kL/k with Galois group GG, a symmetric monoidal functor

c / *:Sp GSH k c_{\mathbb{C}/\mathbb{R}}^*:\mathrm{Sp}_{G} \rightarrow \mathrm{SH}_k

is constructed, which when evaluated on the endomorphism of the unit, yields the ring homomophism A(G)GW(k)A(G) \rightarrow GW(k) of Dress relating the Burnside ring to the Grothendieck-Witt ring?.

The following theorem is Theorem 1.1 of Heller-Ormsby 17 (with image shown in Heller-Ormsby 14).

Theorem

If kk is a real closed field and L=k[i]L = k[i] its algebraic closure, then the functor

c L/k *:Sp C 2SH k c_{L/k}^*:\mathrm{Sp}_{C_2} \rightarrow \mathrm{SH}_k

is a fully faithful symmetric left adjoint whose image is generated under tensor products and colimits by finite etale kk-algebras.

This yields the corollary that π s,c π s,c C 2\pi^{\mathbb{R}}_{s,c} \rightarrow \pi^{C_2}_{s,c} is an isomorphism in degrees satisfying scs \geq c and c23s5/3c \geq \frac{2}{3}s - 5/3. (Thm 3.11 Heller-Ormsby).

By relating the \mathbb{R}-motivic and C 2C_2-equivariant Adams E E_\infty-pages, Belmont-Guillou-Isaksen 21 improved this to the following to the following.

Theorem

The map π s,c (𝕊 ) 2,η π s,c(𝕊 C 2) 2 \pi^{\mathbb{R}}_{s,c}(\mathbb{S}_{\mathbb{R}})^{\wedge}_{2,\eta} \rightarrow \pi_{s,c}(\mathbb{S}_{C_2})_2^{\wedge} is

  1. an injection if s=2c+5s = 2c + 5, and

  2. an isomorphism if s<2c+5s \lt 2c + 5 and (s,c)(0,2)(s,c) \neq (0,-2).

The region s0,c2s \geq 0, c \leq -2

Due to Bredon 67, as recalled e.g. in section 3.3 of Guillou-Isaksen 24, when c2c \leq -2 and s0s \geq 0, there are periodicity isomorphisms which recognize negative coweight C 2C_2-equivariant homotopy groups as ρ\rho-power torsion subgroups of positive-coweight groups. (…)

The region (s,c)(1,1)(s,c) \leq (-1,-1)

The spectral Mackey functor theorem quickly shows that any map S nmσS 0S^{-n - m\sigma} \rightarrow S^0 is nullhomotopic when n,m >0n,m \in \mathbb{Z}_{\gt 0}.

The region 1cs52-1 \leq c \leq \frac{s-5}{2} for small ss

(…)

References

Last revised on April 30, 2024 at 19:09:31. See the history of this page for a list of all contributions to it.