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  • 7
    $\begingroup$ Similar to 1: The Dirac operator defines a class in K-homology. Apply now the Chern-Connes character to go to de Rham homology. What you will end up with is the Poincare dual of the \hat{A}-class. $\endgroup$
    – AlexE
    Commented Feb 8, 2016 at 20:29
  • $\begingroup$ FWIW, in the context of Atiyah-Singer, I liked to think of the Chern character as encoding analytic properties, and the Todd class as encoding geometrical properties. $\endgroup$
    – Steve Huntsman
    Commented Feb 8, 2016 at 20:50
  • 3
    $\begingroup$ @SebastianGoette Yes, the Dirac operator of a spin manifold is a KO-fundamental class (and for a spin^c manifold the corresponding Dirac type operator is a K-fundamental class). That it can not be compatible in the naiv way with the homological fundamental class of the induced orientation of the manifold can be seen by noting the following: different spin structures (and there may be many) give rise to different Poincare duality maps, but the homological Poincare duality only depends on the induced orientation (of which there are only two). $\endgroup$
    – AlexE
    Commented Feb 9, 2016 at 9:30
  • 7
    $\begingroup$ Another topological description: the $\hat{A}$ class is the genus associated to the $E_\infty$ ring homomorphism $M Spin \to KO$. This might somehow mediate between your $1$ and $2$. $\endgroup$
    – Paul Siegel
    Commented Feb 9, 2016 at 9:54
  • 3
    $\begingroup$ Also, I thought quite a bit about the relation between 1 and 3 in graduate school, but all I was left with was a question: can one prove Bott periodicity using heat kernels? I think this would have to be the first step, though I'm not sure. $\endgroup$
    – Paul Siegel
    Commented Feb 9, 2016 at 9:59