# gt.geometric topology – A certain property for Heegaard splittings

I’ve become interested in 3-manifolds with the following property (called ‘Property A’): let $$c_{i}$$ be a set of $$g$$ curves on a genus $$g$$ surface $$Sigma$$ and let $$b_{i}$$ be the $$g$$ meridional curves of $$Sigma$$. That is, if $$Sigma$$ is the boundary of a genus $$g$$ handlebody, then the $$b_{i}$$ bound compressing disks. Then $${c_{i}}_{i=1,dots,g}$$ satisfy ‘Property A’ if, for any $$b_{j}$$, the geometric intersection numbers $$iota(c_{i}, b_{j})$$ have the same sign for all $$i=1,dots,g$$. Informally, all of the attaching curves for the 2-handles run over the 1-handles in the same direction.

So my question is, which 3-manifolds admit a Heegaard splitting with the attaching curves of the 2-handles satisfy Property A?

For example, any direct sum of lens spaces will satisfy this. As does the following Heegaard diagram for the Poincaré homology sphere (taken from Manifold atlas http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere):

A possible nonexample could be the 3-torus, for which the only Heegaard diagram I know does not satisfy this property.

Lastly, and of course this depends heavily on my first question, is it possible that all rational homology spheres satisfy this?

I would be grateful for any sort of insight.