Almost normal Heegaard surfaces

TL;DR

This paper provides a concise proof that strongly irreducible Heegaard surfaces are isotopic to almost normal surfaces and re-establishes a result on normal spheres, using a reduction technique.

Contribution

It offers a shorter, more streamlined proof of key results in 3-manifold topology, improving understanding of Heegaard surfaces and normal spheres.

Findings

Shorter proof of Stocking's theorem on Heegaard surfaces

Re-proves Jaco and Rubinstein's result on normal spheres

Utilizes a novel reduction technique for proofs

Abstract

We present a new and shorter proof of Stocking's result that any strongly irreducible Heegaard surface of a closed orientable triangulated 3-manifold is isotopic to an almost normal surface. We also re-prove a result of Jaco and Rubinstein on normal spheres. Both proofs are based on the "reduction" technique introduced by the author.