In this project we extend the famous Aharonov–Casher result on the number of zero modes of the Pauli (or Dirac) operator on R2 with a compactly supported smooth magnetic field to the case of a planar connected, but not simply connected, region M. More specifically we consider the Dirac operator on M with a smooth magnetic field compactly supported in the bulk of Mand arbitrary magnetic field supported inside the “holes” of M. The domain is given by the famous Atyiah–Patodi–Singer boundary condition. First we prove that the problem is unitarily equivalent to the case when each of the fluxes inside the holes is normalized to a value inside the interval [−π, π) by adding an integer multiple of 2π to the original flux. Denoting Φ the sum of the flux in the bulk and the normalized fluxes we then show that if M is a disc with holes the number of zero modes is given by |[Φ/2π + 1/2]| . For M being R2 with circular holes the number is [|Φ|/2π] provided that |Φ| > 1. By means of stereographic projection we show a similar result for domain on a sphere with holes. The index of the Dirac operator is the difference of the number of its zero modes with spin up and spin down and can be expressed by the famous index formula by Atiyah, Patodi and Singer. The Aharonov–Casher theorem extends this result (in a very particular setting) telling us that all the zero modes have the same spin, that depends on the sign of the total magnetic flux. Our result in this sense agrees with and extends the index theorem, respective its generalization by Grubb to manifolds that do not have a product structure close to the boundary.