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Omega towers and the suspension index

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

Statement

Let $Y = (Y_{n} ≃ Ω Y_{n + 1})_{n \geq 0}$ be a coherent Omega tower of pointed spaces and let $Y$ be its corresponding object of $S p$ . Put $π_{k} (Y) = π_{0} {Map}_{S p} (S [k], Y)$ . For every integer $k$ and $n \geq 0$ with $n + k \geq 2$ there are compatible natural isomorphisms $π_{k} (Y) ≅ π_{n + k} (Y_{n}) ≅ G_{k} (Y)$ . The tail $(Y_{n + 1})_{n \geq 0}$ represents $Y [1]$ , and $π_{k} (Y [1]) ≅ π_{k - 1} (Y)$ .

Facts & Assumptions

Given: $Y$ , $k$ and $n$ as stated. Here $[1]$ is suspension and $[- 1]$ its inverse; $Map$ denotes the mapping space in $S p$ .

Goal: Compare tower and enhanced groups and determine the suspension index.

[F1]

The stable enhancement gives the tower equivalence, $Ω^{\infty} (Y [n]) ≃ Y_{n}$ , stability and the adjunction $Σ^{\infty} ⊣ Ω^{\infty}$ , with $S = Σ^{\infty} S^{0}$ . External stable enhancement and smash product of spectra

[F2]

Cofinal tails of sequential group systems have canonically isomorphic colimits. Cofinal tails and level invariance of stable groups

Proof

1.1F1given

Stability makes suspension and loops inverse. Hence $Map (S [k], Y) ≃ Map (S, Y [- k]) ≃ Ω^{\infty} (Y [- k])$ . Since $Y [- k] ≃ Ω^{n + k} (Y [n])$ , taking components gives $π_{k} (Y) ≅ π_{n + k} (Y_{n})$ . These are group isomorphisms, with the abelian group law detected in degree $n + k \geq 2$ . Naturality of the adjunction makes them compatible with the tower maps.

2.1F1F2step 1.1

Each bonding map in the tower is an isomorphism on the relevant groups because $Y_{n} ≃ Ω Y_{n + 1}$ . Therefore the colimit is canonically any sufficiently late stage group, compatibly with step 1.1. This proves $π_{k} (Y) ≅ G_{k} (Y)$ .

3.1F1step 1.1step 2.1∎

Evaluation gives $Ω^{\infty} ((Y [1]) [n]) ≃ Ω^{\infty} (Y [n + 1]) ≃ Y_{n + 1}$ compatibly with the loop maps, so the tail represents $Y [1]$ . Applying step 1.1 to the tail identifies its $k$ th group with the $(k - 1)$ st group of $Y$ . This proves the suspension formula.

Depends on

Used by · 2

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