2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68996We show that if $V^3$ is a handlebody in $\R^3$, with curves $J_1, ..., J_g \subset \partial V$ which are the attaching curves for a Heegaard splitting of a homology sphere, then there exists a homeomorphism $h\colon V \to V$ so that each of the curves $h(J_i)$ bounds an orientable surface in $\R^3 - int(V)$. This leads to a new characterization of homology spheres and also contradicts a remark of Haken (in 1969) regarding the Poincaré homology sphere.5 figures, latex2eGeometric Topologytext