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Multiplicative AHSS for a multiplicative generalized theory

Statement

Let $\tilde{h}$ be a reduced generalized cohomology theory with a specified unital, associative, graded-commutative coherent external product compatible with suspension and satisfying the two relative cofiber-boundary Leibniz identities. Then for a finite CW complex $X$ and a cellular approximation $Δ : X \to X \times X$ of the diagonal the cohomological AHSS of Cohomological Atiyah–Hirzebruch spectral sequence is multiplicative from its second page on: the relative products of skeletal pairs and $Δ$ pair the skeletal exact couple with itself, that pairing is a Leibniz pairing for $d_{1}$ and descends to $E_{2}$ , and for every $r \geq 2$ the page $E_{r}$ is a unital associative graded-commutative bigraded ring on which $d_{r}$ is a derivation of total degree one, $d_{r} (x y) = d_{r} (x) y + (- 1)^{p + q} x d_{r} (y) (x \in E_{r}^{p, q}),$ with $E_{r + 1} ≅ H (E_{r}, d_{r})$ an isomorphism of bigraded rings. The product on $E_{2}$ corresponds to the graded cup product under the natural isomorphism $E_{2}^{p, q} ≅ H^{p} (X; h^{q} (*))$ , so it is independent of the chosen cellular approximation of the diagonal. The skeletal filtration is multiplicative, $F^{p} F^{q} \subseteq F^{p + q}$ , and $E_{\infty} ≅ {gr}_{F} h^{*} (X)$ as graded rings. The first page carries the pairing and its Leibniz rule but is not a ring in general: Pairings of skeletal exact couples induce multiplicative AHSS records an admissible cellular approximation whose first-page pairing is not associative. A ring prespectrum is one source of such external-product data, not a hypothesis imposed by the theorem: any data satisfying the displayed properties qualify.

Facts & Assumptions

[A1]

The external product data assumed in the statement comprise the natural bilinear relative pairings, the unit, associativity, graded commutativity, suspension compatibility and the two Leibniz identities (Pairings of skeletal exact couples induce multiplicative AHSS).

[A2]

The finite-CW cohomological AHSS has the skeletal exact couple of Cohomological Atiyah–Hirzebruch spectral sequence, and its stable page is the associated graded of the skeletal filtration.

[A3]

The skeletal exact couple paired with itself through $Δ$ carries the typed first-page pairing of the pairing lemma, a Leibniz rule for $d_{1}$ , the descent of the pairing to $E_{2}$ ; from $E_{2}$ on the page products make each page a unital associative graded-commutative bigraded ring on which the differential is a derivation, with $E_{r + 1} ≅ H (E_{r}, d_{r})$ as rings; the product on $E_{2}$ is the graded cup product under $E_{2}^{p, q} ≅ H^{p} (X; h^{q} (*))$ , so homotopic cellular approximations of the diagonal agree from the second page on and the first page is not in general a ring; the filtration is multiplicative and the stable product is the associated-graded product (Pairings of skeletal exact couples induce multiplicative AHSS).

Proof

technique · direct

Given: A reduced generalized cohomology theory $\tilde{h}$ with the external product data of [A1], a finite CW complex $X$ , and a cellular approximation $Δ$ of the diagonal.

1.1

The assumed data satisfy exactly the hypotheses of the pairing lemma [A3]: the relative products are natural and bilinear, the unit, associativity and graded-commutativity hold, the two relative boundary Leibniz identities are assumed, and $Δ$ is a cellular approximation of the diagonal of the finite complex.

A1given

1.2

The skeletal exact couple, its first page, its second page and its stable identification are those of [A2], so the pairing lemma applies to this couple and to this diagonal approximation.

A2given

2.1

Applying [A3] gives the typed first-page pairing, its Leibniz rule for $d_{1}$ and its descent to $E_{2}$ ; from the second page on it gives the ring products, the derivation identity $d_{r} (x y) = d_{r} (x) y + (- 1)^{p + q} x d_{r} (y)$ for $r \geq 2$ , the ring comparison $E_{r + 1} ≅ H (E_{r}, d_{r})$ and the identification of the $E_{2}$ product with the graded cup product, hence its independence from the chosen cellular diagonal from $E_{2}$ onward; and it gives the inclusion $F^{p} F^{q} \subseteq F^{p + q}$ together with the identification of $E_{\infty}$ with the associated graded.

A3step 1.1step 1.2

3.1

Step 2.1 is exactly the asserted multiplicative structure from the second page on, together with the first-page pairing and its Leibniz rule and the exclusion of a general first-page ring; no representability or ring-spectrum hypothesis is used, since only the listed product data and the pairing lemma's construction enter.

step 2.1∎

Source notes

Compare Miller, Lecture 29, printed pp. 100–101, for the product structure on each page from the second page on, the derivation property and the associated-graded ring statement in the ordinary-cohomology case; and Dugger, Multiplicative structures on homotopy spectral sequences II, §3.1 with Theorem 3.4, printed pp. 4–5, for the pairing of filtering towers and the identification of the second-page product with the graded cup product on which the independence from the diagonal approximation rests.

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Haynes Miller, MIT 18.906 Algebraic Topology II, Lecture 29, printed pp. 100–101 (standard reference, not scraped)
Daniel Dugger, Multiplicative structures on homotopy spectral sequences II, §3.1 and Theorem 3.4, printed pp. 4–5 (standard reference, not scraped)