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Complex K-theory AHSS

Statement

Assume AC. For a finite CW complex $X$ the Atiyah–Hirzebruch spectral sequence of complex topological $K$ -theory has $E_{2}^{p, q} ≅ {\begin{cases} H^{p} (X; Z), & q even, \\ 0, & q odd, \end{cases} d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1},$ and converges to the associated graded of the skeletal filtration of $⨁_{q} K^{p + q} (X)$ ; equivalently $E_{\infty}^{p, q} ≅ F^{p} K^{p + q} (X) / F^{p + 1} K^{p + q} (X)$ . The assumption AC is inherited from the construction of complex $K$ -theory and its coefficients, not from the spectral-sequence machinery.

Facts & Assumptions

[A1]

Assume AC. On finite CW pairs the groups $K^{q}$ form a contravariant two-periodic multiplicative generalized cohomology theory with coefficients $K^{2 k} (*) ≅ Z$ and $K^{2 k + 1} (*) = 0$ (Complex K-theory is a two-periodic generalized cohomology theory).

[A2]

Bott periodicity gives the natural isomorphisms $K^{q} (X) ≅ K^{q - 2} (X)$ used to read the coefficient groups in every degree (Complex Bott periodicity).

[A3]

The cohomological AHSS of a reduced generalized cohomology theory on a finite CW complex has $E_{2}^{p, q} = H^{p} (X; h^{q} (*))$ , differentials of bidegree $(r, 1 - r)$ and stable page the associated graded of the skeletal filtration (Cohomological Atiyah–Hirzebruch spectral sequence).

Proof

technique · direct

Given: Assume AC and let $X$ be a finite CW complex.

1.1

The reduced groups ${\tilde{K}}^{q}$ constructed in [A1] on based finite CW complexes satisfy the reduced generalized-cohomology axioms, and the absolute and relative groups $K^{q}$ in [A1] are the associated pair theory. Thus [A3] applies to $\tilde{K}$ and abuts to the stated skeletal filtration of $K^{q} (X)$ . Its coefficient groups are $K^{2 k} (*) ≅ Z$ and $K^{2 k + 1} (*) = 0$ by [A1], and [A2] supplies the two-periodic identifications in every integer degree.

A1A2A3given

2.1

Substituting $h^{q} (*) = K^{q} (*)$ into $E_{2}^{p, q} = H^{p} (X; h^{q} (*))$ gives $E_{2}^{p, q} = H^{p} (X; Z)$ for even $q$ and $E_{2}^{p, q} = 0$ for odd $q$ , while the differential bidegree and the convergence statement are those of [A3].

A3step 1.1

3.1

Steps 1.1 and 2.1 give the displayed second page, the differential bidegree and the identification of the stable page with the associated graded of the skeletal filtration; the Axiom of Choice enters only through the inherited complex $K$ -theory construction.

step 1.1step 2.1∎

Source notes

Compare Ji, §3.1, printed pp. 10–11, where the parity of the coefficient groups is used to read the $K$ -theory AHSS.

Depends on

Used by

Dependency tree · two levels

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Sources

Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §3.1, printed pp. 10–11 (standard reference, not scraped)