Let M be a closed spin manifold and let N be a closed manifold. For maps
f:M→N and Riemannian metrics g on M and h on N, we consider
the Dirac operator Dg,hf of the twisted Dirac bundle ΣM⊗Rf∗TN. To this Dirac operator one can associate an index
in KO−dim(M)(pt). If M is 2-dimensional, one gets a lower bound for
the dimension of the kernel of Dg,hf out of this index. We investigate
the question whether this lower bound is obtained for generic tupels (f,g,h)