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Tim Perutz
Tim Perutz
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This paper by Volker Braun shows that the orientable 8-manifold $X=\mathbb{RP}^3\times \mathbb{RP}^5$ gives an example. One has $$K^0(X) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/4\oplus (\mathbb{Z}/2)^2$$ and $$H^{ev}(\mathbb{RP}^3\times \mathbb{RP}^5) \cong \mathbb{Z}^2 \oplus (\mathbb{Z}/2)^5.$$$$H^{ev}(X) \cong \mathbb{Z}^2 \oplus (\mathbb{Z}/2)^5.$$ Braun does the calculations using the Künneth formulae for the two theories, with the discrepancy in size arising frombecause the Tor terms, whose order depends onof the tensor product of finite abelian groups is sensitive to their structure (not, not just their order) of the K-theory/cohomology groups of the factors.

One another remark is that Atiyah and Hirzebruch told us the $d_3$-differential in their spectral sequence. It's the operation $Sq^3 \colon H^i(X;\mathbb{Z})\to H^{i+3}(X;\mathbb{Z})$ given by $Sq^3 := \beta\circ Sq^2 \circ r$, where $r$ is reduction mod 2 and $\beta$ the Bockstein. As you say, Dan, if this is non-vanishing, K-theory has smaller torsion. This happens iff there's a mod 2 cohomology class $u$ such that $u$ admits an integral lift but $Sq^2 (u) $ does not. Can someone think of a nice example where this occurs?

This paper by Volker Braun shows that the orientable 8-manifold $X=\mathbb{RP}^3\times \mathbb{RP}^5$ gives an example. One has $$K^0(X) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/4\oplus (\mathbb{Z}/2)^2$$ and $$H^{ev}(\mathbb{RP}^3\times \mathbb{RP}^5) \cong \mathbb{Z}^2 \oplus (\mathbb{Z}/2)^5.$$ Braun does the calculations using the Künneth formulae for the two theories, with the discrepancy in size arising from the Tor terms, whose order depends on the structure (not just order) of the K-theory/cohomology groups of the factors.

One another remark is that Atiyah and Hirzebruch told us the $d_3$-differential in their spectral sequence. It's the operation $Sq^3 \colon H^i(X;\mathbb{Z})\to H^{i+3}(X;\mathbb{Z})$ given by $Sq^3 := \beta\circ Sq^2 \circ r$, where $r$ is reduction mod 2 and $\beta$ the Bockstein. As you say, Dan, if this is non-vanishing, K-theory has smaller torsion. This happens iff there's a mod 2 cohomology class $u$ such that $u$ admits an integral lift but $Sq^2 (u) $ does not. Can someone think of a nice example where this occurs?

This paper by Volker Braun shows that the orientable 8-manifold $X=\mathbb{RP}^3\times \mathbb{RP}^5$ gives an example. One has $$K^0(X) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/4\oplus (\mathbb{Z}/2)^2$$ and $$H^{ev}(X) \cong \mathbb{Z}^2 \oplus (\mathbb{Z}/2)^5.$$ Braun does the calculations using the Künneth formulae for the two theories, with the discrepancy in size arising because the order of the tensor product of finite abelian groups is sensitive to their structure, not just their order.

One another remark is that Atiyah and Hirzebruch told us the $d_3$-differential in their spectral sequence. It's the operation $Sq^3 \colon H^i(X;\mathbb{Z})\to H^{i+3}(X;\mathbb{Z})$ given by $Sq^3 := \beta\circ Sq^2 \circ r$, where $r$ is reduction mod 2 and $\beta$ the Bockstein. As you say, Dan, if this is non-vanishing, K-theory has smaller torsion. This happens iff there's a mod 2 cohomology class $u$ such that $u$ admits an integral lift but $Sq^2 (u) $ does not. Can someone think of a nice example where this occurs?

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Tim Perutz
Tim Perutz
  • 13.2k
  • 1
  • 53
  • 79

This paper by Volker Braun shows that the orientable 8-manifold $X=\mathbb{RP}^3\times \mathbb{RP}^5$ gives an example. One has $$K^0(X) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/4\oplus (\mathbb{Z}/2)^2$$ and $$H^{ev}(\mathbb{RP}^3\times \mathbb{RP}^5) \cong \mathbb{Z}^2 \oplus (\mathbb{Z}/2)^5.$$ Braun does the calculations using the Künneth formulae for the two theories, with the discrepancy in size arising from the Tor terms, whose order depends on the structure (not just order) of the K-theory/cohomology groups of the factors.

One another remark is that Atiyah and Hirzebruch told us the $d_3$-differential in their spectral sequence. It's the operation $Sq^3 \colon H^i(X;\mathbb{Z})\to H^{i+3}(X;\mathbb{Z})$ given by $Sq^3 := \beta\circ Sq^2 \circ r$, where $r$ is reduction mod 2 and $\beta$ the Bockstein. As you say, Dan, if this is non-vanishing, K-theory has smaller torsion. This happens iff there's a mod 2 cohomology class $u$ such that $u$ admits an integral lift but $Sq^2 (u) $ does not. Can someone think of a nice example where this occurs?