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The first possible complex K-theory AHSS differential is integral Sq-three

Statement

Assume AC. In the complex topological $K$ -theory Atiyah–Hirzebruch spectral sequence of Complex K-theory AHSS, the first two possible differentials are $d_{2} = 0, d_{3} = S q_{Z}^{3} = β_{Z} S q^{2} ρ_{2},$ with Bott-periodic translates of this formula on all even coefficient rows. Here $ρ_{2}$ is reduction modulo two, $S q^{2}$ is the Steenrod square and $β_{Z}$ is the integral Bockstein of $0 \to Z \overset{2}{\to} Z \to Z / 2 \to 0$ ; the operation is defined on integral cohomology by $S q_{Z}^{3} = β_{Z} S q^{2} ρ_{2} : H^{n} (X; Z) \to H^{n + 3} (X; Z)$ .

Facts & Assumptions

[A1]

Assume AC. For a finite CW complex the $K$ -AHSS has $E_{2}^{p, q} = H^{p} (X; Z)$ for even $q$ , $E_{2}^{p, q} = 0$ for odd $q$ , and $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ (Complex K-theory AHSS).

[A2]

Under the $K$ -AHSS, for $p \geq 1$ the differential $d_{3}$ is read from the relevant $k_{p + 3}$ Postnikov layer of the total-degree representing space $k u_{p + q}$ ; after Bott translation to a nonpositive coefficient row, this is the stable operation $β_{Z} S q^{2} ρ_{2}$ (KU representability and the skeletal–Postnikov d-three comparison, The first connective complex K-theory Postnikov invariant is integral Sq-three).

[A3]

Steenrod squares vanish above the degree: for $x \in H^{n} (X; F_{2})$ one has $S q^{k} x = 0$ when $k > n$ , so in particular $S q^{2} (ρ_{2} y) = 0$ for $y \in H^{0} (X; Z)$ (Steenrod normalization, instability, suspension, and top square, Steenrod squares from cup-i).

[A4]

A finite CW complex has finitely many path components, each a connected finite CW complex with $H^{0} (X_{a}; Z) = Z$ . Naturality gives restriction maps of the $K$ -AHSS along the component inclusions (Naturality and edge maps of the AHSS).

[A5]

Bott periodicity gives natural isomorphisms $K^{q} (X) ≅ K^{q - 2} (X)$ and identifies all even coefficient rows with the row $K^{0} (*) = Z$ (Complex Bott periodicity).

Proof

technique · direct

Given: Assume AC, a finite CW complex $X$ , and the $K$ -AHSS of [A1].

1.1A1A5given

The differential $d_{2}$ has bidegree $(2, - 1)$ : it maps the even coefficient row $q$ to the odd row $q - 1$ , which is zero by the coefficient computation; hence $d_{2} = 0$ and $E_{3} = E_{2}$ .

1.2A1A4given

Let $X = ∐_{a = 1}^{m} X_{a}$ be the finite decomposition into connected components. On each $X_{a}$ , every class of $H^{0} (X_{a}; Z) = Z$ is pulled back from the point, so naturality identifies its $d_{3}$ with the pullback of $d_{3}$ on the point; the latter has target $H^{3} (p t; Z) = 0$ . For $y \in H^{0} (X; Z)$ , naturality along $X_{a} ↪ X$ therefore makes every restriction of $d_{3} y \in H^{3} (X; Z)$ zero. Every singular simplex of a disjoint union lies in one component, so restriction gives an isomorphism of singular cochain complexes $C^{*} (X; Z) ≅ \prod_{a = 1}^{m} C^{*} (X_{a}; Z)$ and hence an injective map $H^{3} (X; Z) \to \prod_{a} H^{3} (X_{a}; Z)$ . Thus $d_{3} y = 0$ without assuming a componentwise decomposition of the entire AHSS.

1.3A2A5given

For $p \geq 1$ and any even coefficient row, the comparison and $k$ -invariant lemmas identify $d_{3}$ on $E_{3}^{p, q}$ with $β_{Z} S q^{2} ρ_{2}$ , transported along the Bott identification of the coefficient rows.

2.1A3step 1.2

On the $p = 0$ row the operation $β_{Z} S q^{2} ρ_{2}$ also vanishes, since $S q^{2}$ is zero on classes of degree zero by instability; thus the formula $d_{3} = β_{Z} S q^{2} ρ_{2}$ holds on the $p = 0$ row as well.

3.1A5step 1.1step 1.3step 2.1

Combining steps 1.1, 1.3 and 2.1 gives $d_{2} = 0$ and $d_{3} = β_{Z} S q^{2} ρ_{2}$ on every even coefficient row, with the Bott-periodic translates supplied by [A5]; the sign ambiguity of the long exact sequence convention is immaterial because the operation has order two.

4.1step 3.1∎

step 3.1 proves the asserted vanishing of $d_{2}$ and the identification of $d_{3}$ with $S q_{Z}^{3} = β_{Z} S q^{2} ρ_{2}$ on all even coefficient rows.

Source notes

Compare Ji, Proposition 3.12, printed p. 12, for the statement $d_{3} = {\tilde{S q}}^{3}$ and the description of the operation as the composite of reduction mod two, $S q^{2}$ and the integral Bockstein; the normalization of the operation is supplied by Adams, Proposition 16.6, printed pp. 391–393.

Depends on

Used by

A nonzero d-three in the K-AHSS for RP-two times RP-four Example

Dependency tree · two levels

43 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

Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, Proposition 3.12, printed p. 12 (standard reference, not scraped)
J. F. Adams, Stable Homotopy and Generalised Homology, Proposition 16.6, printed pp. 391–393 (standard reference, not scraped)