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, Èliašberg, Saeki, and Sakuma. In this paper, we study rational homology nspheres that admit special generic maps into $\mathbb{R}^p$ for $p \lt n$. 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 $S^4$ to obtain new results on the nonexistence of special generic maps.

Keywords

special generic map, definite fold point, Stein factorization, homology sphere, linking form, lens space, sphere bundle

2010 Mathematics Subject Classification

Primary 57R45. Secondary 58K15, 58K30.

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The author has been supported by JSPS KAKENHI Grant Number JP18F18752 and JP17H06128. This work was written while the author was a JSPS International Research Fellow (Postdoctoral Fellowships for Research in Japan (Standard)).

Received 20 July 2021

Received revised 16 December 2022

Accepted 13 February 2023

Published 7 April 2023