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Complex K-AHSS for spheres

Example

Assume AC. For $n > 0$ the reduced complex $K$ -groups of the sphere are ${\tilde{K}}^{0} (S^{n}) ≅ {\begin{cases} Z, & n even, \\ 0, & n odd, \end{cases} {\tilde{K}}^{1} (S^{n}) ≅ {\begin{cases} 0, & n even, \\ Z, & n odd, \end{cases}$ and the sphere $K$ -AHSS has no nonzero differential and no nontrivial extension.

Facts & Assumptions

[A1]

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

[A2]

The coefficient groups of complex $K$ -theory are Bott-periodic: $K^{2 k} (*) ≅ Z$ and $K^{2 k + 1} (*) = 0$ , with all $K^{q} (*)$ obtained by shifting $K^{0} (*) = Z$ (Complex Bott periodicity, Complex K-theory AHSS).

[A3]

For $n > 0$ , $H^{0} (S^{n}; Z) ≅ H^{n} (S^{n}; Z) ≅ Z$ , and all other reduced cohomology groups of the sphere vanish; hence the $K$ -AHSS of $S^{n}$ is supported in the two columns $p = 0$ and $p = n$ (Complex K-theory of spheres and the universal coefficient comparison of Complex K-theory AHSS).

[A4]

The published sphere computation gives ${\tilde{K}}^{0} (S^{2 m}) ≅ Z$ and ${\tilde{K}}^{0} (S^{2 m + 1}) = 0$ , with $K^{1}$ of the opposite parity (Complex K-theory of spheres).

[A5]

Collapse determines only the associated graded and not the extensions (AHSS collapse generally determines only the associated graded object).

Verification

technique · direct

Given: Assume AC, $n > 0$ , and the $K$ -AHSS of the sphere $S^{n}$ with its standard CW structure having one $0$ -cell and one $n$ -cell.

1.1A1A2A3

By [A3] the $E_{2}$ page has $E_{2}^{0, q} = K^{q} (*)$ and $E_{2}^{n, q} = K^{q} (*)$ for every even $q$ , and vanishes in all other positions.

2.1A1A2step 1.1

For $n$ even, consider a differential $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ with source in an even coefficient row $q$ at a nonzero column $p \in {0, n}$ . If $r$ is even then the target row $q - r + 1$ is odd, so the target lies in a vanishing coefficient row. If $r$ is odd then the target column $p + r$ is odd, hence is neither $0$ nor $n$ when $p = 0$ (the number $n$ being even) and exceeds $n$ when $p = n$ ; in both cases the target column lies outside the support ${0, n}$ of $H^{*} (S^{n}; Z)$ , and the parity of the target row is irrelevant. In either case the target vanishes, so every differential is zero.

2.2A1A2A4step 1.1

Suppose $n$ is odd. If $n = 1$ , every differential has $r \geq 2$ by [A1], so a source in column $0$ or $1$ has target column $p + r > 1$ ; hence every target is zero. Now let $n \geq 3$ . The only possibly nonzero differentials are $d_{n} : E_{n}^{0, q} \to E_{n}^{n, q - n + 1}$ on even rows $q$ . There is no incoming differential at column $0$ , so the surviving subgroup there is $\ker d_{n}$ . On the other hand, the edge quotient $E_{\infty}^{0, q} = F^{0} K^{q} (S^{n}) / F^{1} K^{q} (S^{n})$ identifies with the image of restriction to the basepoint. For even $q$ , [A4] and Bott periodicity give $F^{1} = {\tilde{K}}^{q} (S^{n}) = 0$ , while pullback along $S^{n} \to *$ splits restriction, so that image is all of $K^{q} (*) = Z$ . Thus $\ker d_{n} = Z$ inside its source $Z$ , forcing $d_{n} = 0$ .

3.1A4A5step 2.1step 2.2

Steps 2.1 and 2.2 show that all differentials vanish and $E_{2} = E_{\infty}$ . If $n$ is odd, each total-degree diagonal has only one nonzero term, so there is no extension problem. If $n$ is even, an even total degree $t$ has two graded pieces, at $p = 0$ and $p = n$ , and the filtration gives $0 \to F^{n} K^{t} (S^{n}) ≅ Z \to K^{t} (S^{n}) \to K^{t} (*) ≅ Z \to 0$ . The quotient map is restriction to the basepoint and is split by pullback along $S^{n} \to *$ , since the composite $* \to S^{n} \to *$ is the identity. Thus $K^{t} (S^{n}) ≅ Z \oplus Z$ in even degree, with the reduced summand $F^{n} ≅ {\tilde{K}}^{t} (S^{n}) ≅ Z$ from [A4]; odd total degrees vanish. Hence the only two-piece extension is split, while the odd-dimensional cases have a single graded piece.

4.1step 3.1∎

Steps 2.1, 2.2 and 3.1 verify the displayed reduced groups and show that the sphere $K$ -AHSS has no nonzero differential and no nontrivial extension.

Source notes

Compare Ji, Theorem 3.1 and §3.2.1, printed pp. 9–11, for the sphere coefficient computation and its placement in the $K$ -AHSS.

Depends on

Used by

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Sources

Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §3.2.1 and Theorem 3.1, printed pp. 9–11 (standard reference, not scraped)