On Peiffer transformations, link diagrams and a question of J. H. C. Whitehead

Summary

Introduction by Ronald Brown.

Whitehead's famous question: is every subcomplex of a 2-dimensional aspherical complex aspherical? was discussed at Bangor in March, 1978, during a visit of Johannes Huebschmann. He suggested a possible approach to this question, but doubts were raised about this in May, 1978, in a letter to me from Eldon Dyer. Peter Stefan then pinpointed precisely the failure of the proposed method, by finding an example of an identity sequence which was equivalent to Peiffer transformations to the empty sequence φ but which did not collapse to φ. He sent the example to several people, and Roger Fenn in replying explained the method of pictures. Peter wrote to Roger on 17 May, 1978, and circulated this letter. Peter died in a mountaineering accident on June 18.

This note contains Peter's example, and the major part of his letter, omitting some irrelevant matters or outdated points. The article “Identities among relations” by R. Brown and J. Huebschmann, in this volume, and referred to here as [Br-Hu], is intended to give the background required for understanding this note, and so some of Peter's notations, conventions and diagrams have been changed to make the two articles consistent. Other changes are few and minor. References here other than to [Br-Hu] are to the bibliography of that article.