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Omega replacement preserves sequential stable groups

PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites

Statement

Let $X$ be a level-Kan sequential pointed simplicial prespectrum with sphere-first structure maps $K \land X_{n} \to X_{n + 1}$ , $K = Δ^{1} / \partial Δ^{1}$ . For the replacement $Q X$ of the simplicial Omega replacement input, the natural map $G_{k} (X) \to G_{k} (Q X)$ is an isomorphism for every integer $k$ . More precisely, for $i + k \geq 2$ there is a natural isomorphism $π_{i + k} ((Q X)_{i}) ≅ G_{k} (X)$ , compatible with advancement in $i$ .

Statement references

External sequential simplicial Omega replacement

Facts & Assumptions

Given: $X$ , $k$ and $i$ as in the Statement.

Goal: Compute the stable groups of the imported Omega replacement.

[F1]

External sequential simplicial Omega replacement gives $(Q X)_{i} = {colim}_{m} Ω^{m} X_{i + m}$ , its structure maps, and $ι_{R X} = R ι_{X}$ .

[F2]

Cofinal tails and level invariance of stable groups identifies a sequential colimit with any cofinal finite tail.

Proof

1.1givenconstruct

A filtered colimit of Kan simplicial sets is Kan: a horn has finitely many simplices, so a map from it factors through a stage and a filler there gives a filler in the colimit. Homotopy groups commute with this colimit because representatives use finite simplicial spheres and homotopies use their finite products with $Δ^{1}$ . Such maps and homotopies factor through stages, and equalities between finitely many simplices occur at a common later stage. The group operation is preserved by functoriality.

2.1F1F2step 1.1

By step 1.1 and iterated loop–suspension adjunction, $π_{i + k} ((Q X)_{i}) ≅ {colim}_{m} π_{i + k + m} (X_{i + m}) = G_{k} (X)$ . Under $[A, Ω B] ≅ [K \land A, B]$ , each new loop coordinate is inserted on the left. Consequently the transition $Ω^{m} ι$ adds the new inner loop coordinate on the left and corresponds exactly to suspension with $K$ first followed by the structure map. The last equality uses the cofinal tail.

3.1F1F2step 2.1∎

The map $X_{i} \to (Q X)_{i}$ inserts the $m = 0$ term. By $ι_{R X} = R ι_{X}$ , the adjoint structure maps of $Q X$ identify advancement in $i$ with the same transition in the colimit of step 2.1. Thus these isomorphisms are compatible in $i$ , and the map $G_{k} (X) \to G_{k} (Q X)$ becomes the identity under them. This proves both assertions.

Depends on

Used by · 1

Stable spectra and exact triangles

Dependency tree · 3 levels

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