  Special generic maps are smooth maps between smooth manifolds with only
definite fold points as their singularities. The problem of whether a closed
$n$-manifold admits a special generic map into Euclidean $p$-space for $1 \leq
p \leq n$ was studied by several authors including Burlet, de Rham, Porto,
Furuya, \`{E}lia\v{s}berg, Saeki, and Sakuma. In this paper, we study rational
homology $n$-spheres that admit special generic maps into $\mathbb{R}^{p}$ for
$p<n$. We use the technique of Stein factorization to derive a necessary
homological condition for the existence of such maps for odd $n$. We examine
our condition for concrete rational homology spheres including lens spaces and
total spaces of linear $S^{3}$-bundles over $S^{4}$, and obtain new results on
the (non-)existence of special generic maps.
