We have studied the torsion cosmology model in Kaluza–Klein theory. We considered two simple models in which the torsion vectors are A_μ = (α,0,0,0) and A_μ = a(t)²(0,β,β,β), respectively. For the first model, the accelerating expansion of the Universe can be not explained without dark energy which is similar to that in the standard cosmology. But for the second model, we find that without dark energy the effect of torsion can give rise to the accelerating expansion of the universe and the alleviation of the well-known age problem of the three old objects for appropriated value of the model parameter β. These outstanding features of the second torsion cosmology model have been supported by the Type Ia supernovae (SNIa) data.