We consider a gauge-Higgs system on a fuzzy 2-sphere and study the topological structure of gauge configurations, when the U(2) gauge symmetry is spontaneously broken to U(1) × U(1) by the vev of the Higgs field. The topology is classified by the index of the Dirac operator satisfying the Ginsparg-Wilson relation, which turns out to be a noncommutative analog of the topological charge introduced by 't Hooft. It can be rewritten as a form whose commutative limit becomes the winding number of the Higgs field. We also study conditions which assure the validity of the formulation, and give a generalization of the admissibility condition. Finally we explicitly calculate the topological charge of a oneparameter family of configurations. 1 1 The GW Dirac operator on the fuzzy 2-sphere for vanishing gauge field was given earlier in [22]. The GW relation was also implemented on the noncommutative torus by using the Neuberger's overlap Dirac operator [23]. In [24], this GW Dirac operator was obtained from the general prescription [20].

We construct a topological charge of gauge field configurations on a fuzzy S 2 ×S 2 by using a Dirac operator satisfying the Ginsparg-Wilson relation. The topological charge defined on the fuzzy S 2 ×S 2 can be interpreted as a noncommutative (or matrix) generalization of the 2nd Chern character on S 2 × S 2 . We further calculate the number of chiral zero modes of the Dirac operator in topologically nontrivial gauge configurations. Generalizations of our formulation to fuzzy (S 2 ) k are also discussed.