  The quantum cosmological version of the multidimensional Einstein-Yang-Mills
model in a $R \times S^3 \times S^d$ topology is studied in the framework of
the Hartle-Hawking proposal. In contrast to previous work in the literature, we
consider Yang-Mills field configurations with non-vanishing time-dependent
components in both $S^3$ and $S^d$ spaces. We obtain stable compactifying
solutions that do correspond to extrema of the Hartle-Hawking wave function of
the Universe. Subsequently, we also show that the regions where 4-dimensional
metric behaves classically or quantum mechanically (i.e. regions where the
metric is Lorentzian or Euclidean) will depend on the number, $d$, of compact
space dimensions.
