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

Example

Assume AC. In the complex $K$ -theory Atiyah–Hirzebruch spectral sequence for ${R P}^{2} \times {R P}^{4}$ , let $u$ and $v$ be the degree-one mod-two generators and let $z = β_{Z} (u v) \in H^{3} (-; Z)$ . Then $d_{3} (z) = β_{Z} (u v^{4}) \neq 0;$ equivalently $d_{3}$ agrees with $β S q^{2} ρ_{2}$ on this class. No complex $K$ -theory Künneth theorem or group-order argument is used.

Facts & Assumptions

[A1]

Assume AC. In the complex $K$ -AHSS, $d_{2} = 0$ and $d_{3} = β_{Z} S q^{2} ρ_{2}$ on all even coefficient rows (The first possible complex K-theory AHSS differential is integral Sq-three).

[A2]

Assume AC. For $u, v$ the degree-one mod-two generators one has $β_{Z} S q^{2} ρ_{2} (β_{Z} (u v)) = β_{Z} (u v^{4}) \neq 0$ (A Bockstein class on RP-two times RP-four has nonzero integral Sq-three).

Verification

technique · direct

Given: Assume AC, the space ${R P}^{2} \times {R P}^{4}$ , its $K$ -AHSS, and $z = β_{Z} (u v) \in H^{3} (-; Z)$ .

1.1

The class $z$ lies in $E_{3}^{3, - 2}$ : the coefficient row $- 2$ is even and $H^{3}$ is the integral cohomology degree of the target, and by [A1] the differential $d_{3}$ on this row is the operation $β_{Z} S q^{2} ρ_{2}$ .

A1given

2.1

By [A2] the value of that operation on $z = β_{Z} (u v)$ is $β_{Z} (u v^{4}) \neq 0$ .

A2step 1.1

3.1

Therefore $d_{3} (z) = β_{Z} (u v^{4})$ is nonzero, which exhibits a nonzero $d_{3}$ and shows that the $K$ -AHSS does not collapse for this space.

A1step 2.1

4.1

Steps 1.1 and 3.1 verify the displayed nonzero value of $d_{3}$ without using a $K$ -theory Künneth theorem.

step 3.1∎

Source notes

Compare Ji, §3.2.4, Figure 2 and Proposition 3.12, printed pp. 11–12, for the nonzero $d_{3}$ on ${R P}^{2} \times {R P}^{4}$ .

Depends on

Used by

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Sources

Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §3.2.4, Figure 2 and Proposition 3.12, printed pp. 11–12 (standard reference, not scraped)