Untwisting Heegaard diagrams in 3-space

Abstract

We show that if $V^3$ is a handlebody in ${\mathbb{R}}^3$, 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 $h:V\to V$ so that each of the curves $h(J_i)$ bounds an orientable surface in ${\mathbb{R}}^3-int(V)$. 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, latex2e