Instantons and the fixed point topological charge in the two-dimensional O(3) σ model

We define a fixed point topological charge for the two-dimensional O(3) lattice σ model which is free of topological defects. We use this operator in combination with the fixed point action to measure the topological susceptibility for a wide range of correlation lengths. The results strongly suggest that it is not a physical quantity in this model. The procedure, however, can be applied to other asymptotically free theories as well.