Another discussion of the axial vector anomaly and the index theory

A derivation is presented of the axial anomaly in a background Riemannian manifold using zeta-function regularisation. This leads directly to the relation with the index theorem. A spin-1 index theorem is derived, giving the Hirzebruch signature theorem and the expression for the Euler number in terms of the B4 coefficients. Boundary effects are briefly mentioned. Some extensions are suggested. (author)