2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/45250We construct a gauge field model based on $SU(2)_L\times SU(2)_R\times U(1)_Y \times π_4(SU(2)_L\times SU(2)_R\times U(1)_Y) $ from the principle that both the original gauge group $G_{YM}$ and the discrete group $π_4(G_{YM})$ should be taken as gauge groups in the sense of non-commutative geometry. We show that the Yukawa coupling and the Higgs mechanism appear as natural results.9 pages, LATEXHigh Energy Physics - Phenomenologytext