2013/08/21 by Johnson, Jesse
#57N10 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1308.4599
Kevin Hartshorn showed that if a three-dimensional manifold M admits a Heegaard surface Σ with Hempel distance d then every incompressible surface in M has genus at least (d)/(2). Scharlemann-Tomova generalized this, proving that in such a manifold, every other Heegaard surface for M of genus g' < (d)/(2) is a stabilization of Σ. In the present paper, we show that Hartshorn's bound is sharp and Scharlemann-Tomova's bound is very close to sharp. In particular, for every pair of integers g ≥ 2, d ≥ 2, we construct a three-manifold M with a genus g, distance d Heegaard splitting and an incompressible surface of genus (d)/(2). We also construct, for every d ≥ 4, a three-manifold with a genus g, distance d Heegaard surface Σ and a second Heegaard surface with genus g' = (1)/(2) d + g - 1 that is not a stabilization of Σ.