Totaro proved that the cycle class map for an algebraic variety factors through complex cobordism. In other words, we have a factorization

$$CH^i(X) \rightarrow MU^{2i}(X) \rightarrow H^{2i}(X; \mathbb{Z}).$$

This is supposed to given a conceptual explanation for the Atiyah-Hirzebruch topological obstructions to the integral Hodge conjecture. More precisely, the claim is that any odd degree cohomology operation on integral cohomology should lift to $MU$ (where it has to be 0), and therefore kill the cycle class of an algebraic variety. (See top of 2nd page of this paper.)

The point I don't understand is: why must a cohomology operation on $H\mathbb{Z}$ lift to $MU$?

  • 3
    $\begingroup$ I'd perhaps refer you to Landweber's papers On the Complex Bordism of Eilenberg-Mac Lane Spaces and Connective Coverings of BU" and "Steenrod Representability of Stable Homotopy" in which he computes the image of the Thom map $MU_*(K(\mathbb{Z},q))\rightarrow H^*(K(\mathbb{Z},q);\mathbb{Z})$. If i recall, the Thom map is surjective in many situations. $\endgroup$
    – Tyrone
    Nov 17, 2016 at 11:22


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