On special generic maps of rational homology spheres into Euclidean spaces

TL;DR

This paper investigates when rational homology spheres can admit special generic maps into Euclidean spaces, providing homological conditions and examining specific examples like lens spaces and $S^3$-bundles over $S^4$.

Contribution

It establishes a necessary homological condition for the existence of special generic maps into lower-dimensional Euclidean spaces for odd-dimensional rational homology spheres.

Findings

Derived a homological criterion for special generic maps on rational homology spheres.

Applied the criterion to lens spaces and $S^3$-bundles over $S^4$, deriving new non-existence results.

Provided insights into the topology of manifolds admitting special generic maps.

Abstract

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.