Untwisting Heegaard diagrams in 3-space

Publication date: 27 October 2003

Abstract: We show that if

V^{3}

is a handlebody in

R^{3}

, with curves

J_{1}, . . ., J_{g} s u b s e t p a r t i a l V

which are the attaching curves for a Heegaard splitting of a homology sphere, then there exists a homeomorphism

h c o l o n V o V

so that each of the curves

h (J_{i})

bounds an orientable surface in

R^{3} - i n t (V)

. This leads to a new characterization of homology spheres and also contradicts a remark of Haken (in 1969) regarding the Poincar'e homology sphere.

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