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## The A-hat genus of 4-sphere and hypersurfaces

The Atiyah-Singer index theorem for a twisted Dirac operator is used wonderfully by Vafa and Witten to to produce uniform upper bounds for eigenvalues of their Dirac operator that are independent of the bundle and the connection defining.  Slightly more precisely, suppose $M$ is a riemannian spin manifold with spin bundle $S$ and some complex vector bundle $V$ and a connection $A$ on it.  One of the Vafa-Witten bounds is

$\lambda_1 \le C$

where $C$ depends on $M$ and its metric but not on $V$ or $A$.  One of the key points of their proof is that one can choose a $V$ for which the Atiyah-Singer index theorem forces a nonzero section in the kernel of $D^+_A$.  It is useful to consider the 4-sphere whose A-hat genus is zero (as it has no middle dimensional cohomology and no signature).  The Atiyah-Singer index theorem in this case is

$Ind D^+ = ( \hat{A} \cup ch_i(V) )[M]$

(see Gromov’s ).  There is no A-hat genus normally defined for three manifolds as they involve Pontrijagin classes defined for 4j-th cohomologies but there is a notion of an index in some K-theory.