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A PROOF OF THE GORDON CONJECTURE RUIFENG QIU AND MARTIN SCHARLEMANN
 

Summary: A PROOF OF THE GORDON CONJECTURE
RUIFENG QIU AND MARTIN SCHARLEMANN
Abstract. A combinatorial proof of the Gordon Conjecture: The sum of two
Heegaard splittings is stabilized if and only if one of the two summands is
stabilized.
1. Introduction and basic background
In 2004 the first author [Q] presented a proof of the Gordon Conjecture, that the
sum of two Heegaard splittings is stabilized if and only if one of the two summands
is stabilized. The same year, and a bit earlier, David Bachman [Ba] presented a
proof of a somewhat weaker version, in which it is assumed that the summand
manifolds are both irreducible. (A later version dropped that assumption.)
The proofs in [Q] and [Ba] are quite different. The former is heavily combinato-
rial, essentially presenting an algorithm that will create, from a pair of stabilizing
disks for the connected sum Heegaard splitting, an explicit pair of stabilizing disks
for one of the summands. (Earlier partial results towards the conjecture, e. g. [Ed]
have been of this nature.) In contrast, the proof in [Ba] is a delicate existence proof,
based on analyzing possible sequences of weak reductions of the connected sum
splitting. Both proofs have been difficult for topologists to absorb. The present
manuscript arose from the second author's efforts, following a visit to Dalian in
2007, to simplify and clarify the ideas in [Q]. (During that visit, MingXing Zhang

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics