Conditions for Absence of Global SU(2) Anomaly for Four-dimensional Gauge Groups

Proofs are given group theoretically for the absence of the global SU(2) (Witten) anomaly for any compact Lie group GcontainsSU(2) satisfying Pi/sub 4/(G) = 0 (Pi/sub 4/ is the four-dimensional homotopy group). It is shown that the number of fermion zero modes is even for G = SO(N) (Ngreater than or equal to7) and the five exceptional groups G/sub 2/, F/sub 4/, E/sub 6/, E/sub 7/, and E/sub 8/. The absence of the Witten anomaly for the SU(N) (Ngreater than or equal to3) groups requires freedom from the perturbative (triangle) anomaly.