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Image filtration for a split tower

Statement

Let $F^{∙} E$ be equivalent as an augmented tower to the tail tower of a family $(G_{j})_{j \geq 0}$ , including an equivalence $E ≃ ⨁_{j \geq 0} G_{j}$ . For every integer $n$ and $q \geq 0$ , the image of $π_{n} F^{q} E \to π_{n} E$ is the image of the canonical tail inclusion $π_{n} (⨁_{j \geq q} G_{j}) \to π_{n} (⨁_{j \geq 0} G_{j})$ . Moreover the quotient of this image by the corresponding image at $q + 1$ is naturally $π_{n} G_{q}$ . This statement does not require identifying homotopy groups of an infinite coproduct with a direct sum.

Facts & Assumptions

Given: The augmented tower equivalence, an integer $n$ , and $q \geq 0$ .

Goal: Identify the image filtration and its successive quotient.

[F1]

Decreasing spectral filtrations and associated graded defines image filtration groups.

[F2]

Graded pieces of a direct-sum tail tower identifies the finite decomposition of each tail.

Proof

1.1F1givenconstruct

The commutative augmentation square in the specified equivalence identifies the two displayed images. The tail inclusion has a retraction induced by identity maps on summands indexed at least $q$ and zero maps on the others; thus its map on homotopy groups is injective.

2.1F2step 1.1∎

By [F2], $T^{q} ≃ T^{q + 1} \oplus G_{q}$ . Applying $π_{n}$ to this finite biproduct gives $π_{n} T^{q} ≅ π_{n} T^{q + 1} \oplus π_{n} G_{q}$ . Under the injection from step 1.1, the quotient of the two successive images is consequently $π_{n} G_{q}$ , naturally in the specified splitting.

Depends on

Used by · 2

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