Two questions on Heegaard diagrams of S3.

Abstract

An important open question about 3-manifolds is whether or not there exists an algorithm for recognizing S3. The author poses two questions about Heegaard diagrams of S3, appropriate answers to either of which would give such an algorithm. If a Heegaard diagram contains either a wave or a cancelling pair, then one can find an equivalent diagram of smaller complexity (in the latter case, of smaller genus). Every nontrivial genus-2 diagram of S3 contains a wave [T. Homma, M. Ochiai and M. Takahashi Osaka J. Math 17 (1980), no. 3, 625–648; MR0591141 (82i:57013)], but this is false for higher genera. The author's first question is whether there are any Heegaard diagrams of S3 without waves and without cancelling pairs. {Reviewer's remark: An example of such a diagram is contained in an article of Ochiai [ibid. 22 (1985), no. 4, 871–873; MR0815455 (87a:57020)].} Given a Heegaard diagram, there is a reduction procedure which produces a so-called pseudominimal diagram. W. Haken [in Topology of manifolds (Athens, Ga., 1969), 140–152, Markham, Chicago, Ill., 1970; MR0273624 (42 #8501)] has suggested that perhaps the only pseudominimal diagrams of S3 are the trivial ones; no counterexamples are known. The author suggests a further reduction step which might be applied to a pseudominimal diagram, yielding several partial diagrams. If any of these has a cancelling pair, then the genus of the original diagram can be reduced. An example is given to show that, in general, for manifolds different from S3, even this enhanced procedure does not always detect the reducibility of a Heegaard splitting. The author's second question, however, is whether it does for splittings of S3. Thus the author is suggesting a possible algorithm for recognizing S3 which allows for the existence of nontrivial pseudominimal diagrams of S3.