  It is shown that flat zero-energy solutions (vacua) of the 5d Kaluza-Klein
theory admit a non-trivial homotopy structure generated by certain Kaluza-Klein
excitations. These vacua consist of an infinite set of homotopically different
spacetimes denoted by $\mathcal{M}^{(n)}_5$, among which $\mathcal{M}^{(0)}_5$
and $\mathcal{M}^{(1)}_5$ are especially identified as $M_{4} \times S^{1}$ and
$M_5$, the ground states of the 5d Kaluza-Klein theory and the 5d general
relativity, respectively (where $M_k$ represents the $k$-dimensional Minkowski
space).
