Quantum Cosmological Multidimensional Einstein-Yang-Mills Model in a $R \times S^3 \times S^d$ Topology

The quantum cosmological version of the multidimensional Einstein-Yang-Mills model in a ${\bf R} \times S^3 \times S^d$ topology is studied in the framework of the Hartle-Hawking proposal. In opposition 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 find that stable compactifying solutions do correspond to extrema of the wave function of the Universe and that the regions where the wave function predicts the 4-dimensional metric behaves classically or quantum mechanically (i.e. regions where the metric is Lorentzian or Euclidean), depend on the number, $d$, of compact space dimensions.