On special generic maps of rational homology spheres into Euclidean spaces

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.