For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism $$\varphi:\Omega\otimes\mathbb{Q}\to R$$ where $R$ is any commutative ring over $\mathbb{Q}$. There are well known genera such as the $L$-genus, the $\hat{A}$-genus, and the ellpitic genus.

Is there a similar notion of genus for algebraic cobordism in use, and if so, is it interesting? Here by algebraic cobordism I mean either in the sense of Levine-Morel or Voevodsky.

  • $\begingroup$ A genus can frequently be lifted to a map of ring spectra. This is reasonably well understood classically and I am guessing that this is the more common perspective motivically (not that it should be but that the people working with these things are more inclined to work with this side. I am probably wrong though.). In this sense, if you are willing to work with $MGL$ which differs slightly from $\Omega^{alg}$ then we "understand" some genera motivically. $\endgroup$ – Sean Tilson May 2 '16 at 9:15

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