  This talk is devoted to the problem how to compute relative nucleation
probabilities of configurations with different topology and dimension in
quantum cosmology. Assuming the semiclassical approximation, the usual formula
for the nucleation probability induced by the no-boundary wave function is
$P_{NB}\approx\exp(-I)$, where $I$ is the Euclidean action, evaluated at a
solution of the Euclidean Einstein equations with effective cosmological
constant $\Lambda$. Relative probabilities of different configurations are
usually compared at equal values of $\Lambda$. When configurations with
different dimensions are admitted (the $n$-dimensional gravitational constant
being subject to a rather mild restriction), as e.g. ${\bf S}^n$ for any $n$,
this procedure leads to the prediction that the space-time dimension tends to
be as large as possible. In this contribution, I would like to propose an
alternative scheme, namely to compare the probabilities $P_{NB}\approx\exp(-I)$
at equal values of the {\it energy} $E$, instead of the {\it energy density}
$\Lambda$. As a result, the space-time dimension settles at $n=4$. Attempts to
predict the topology of the spacelike slices lead to the candidates ${\bf S}^3$
and ${\bf S}^1\times {\bf S}^2$. Since the ''process'' of nucleation (possibly
connected with decoherence) is not well known in detail, we expect that either
{\it both} configurations may be realized with roughly equal probability, or
the {\it latter} one is favoured. Finally, we comment on the analogous
situation based on the tunneling wave function.
