Celebratio Mathematica

Dennis Lee Johnson

Preface to Dennis Johnson's unpublished preprints

Among Den­nis John­son’s com­plete list of works on Cel­eb­ra­tio, the read­er will find two un­pub­lished pre­prints:

  1. “Non-bor­d­ing dif­feo­morph­isms of sur­faces which act trivi­ally on ho­mo­logy” (1978–80) [◊];
  2. “A geo­met­ric ver­sion of Cas­son’s in­vari­ant and its ap­plic­a­tion to Re­idemeister tor­sion” (1988) [◊].

We high­light these here as ex­amples of 20th-cen­tury math­em­at­ic­al “rar­it­ies”: pre­prints that pred­ate the cre­ation of ArX­iv (1992) and so con­sti­tute a cat­egory of un­pub­lished re­search that will be­come dif­fi­cult for his­tor­i­ans to ac­cess as math­em­aticians old enough to have amassed per­son­al lib­rar­ies of such pa­pers re­tire and dis­pose of those ma­ter­i­als.

Neither Den­nis John­son nor Klaus Jo­hann­son pub­lished ex­tens­ively dur­ing their re­search ca­reers, so “Non-bor­d­ing dif­feo­morph­isms” is un­usu­al, moreover, as an in­stance of col­lab­or­a­tion. Ed­monds and Ewing have sug­ges­ted [e3] that the pre­print was the first time the Jaco–Shalen–Jo­han­son (JSJ) de­com­pos­i­tion was ap­plied to ques­tions about sur­face dif­feo­morph­isms. Pri­or­ity in such in­stances can be tricky to es­tab­lish, however, as Mar­tin Schar­le­mann un­der­scores in a re­cent email:1

In fact, be­fore see­ing this pre­print I had thought my pa­per [e2] was the first use of JSJ de­com­pos­i­tions in study­ing cobor­d­isms of sur­face auto­morph­isms. The \( [\dots] \) last para­graph \( [ \)of Jo­hann­son–John­son\( ] \), say­ing I poin­ted out an er­ror in their ori­gin­al pa­per, makes me think that might in­deed be true, but it’s hard to tell and really doesn’t mat­ter. In any case, Bo­nahon then wrote the defin­it­ive pa­per on the sub­ject [e1], in­vent­ing “com­pres­sion-bod­ies” in the pro­cess. Cas­son must have been in­volved too, but how ex­actly I’ve for­got­ten — and there are no pub­lic­a­tions.

We note that a con­cise state­ment of John­son and Jo­hann­son’s proof of the main res­ult is pub­lished, fully cred­ited, in Sec­tion 6 of Lein­inger and Re­id’s “The co-rank con­jec­ture for 3-man­i­fold groups” [e5]. The au­thors fur­ther point out that the res­ult was proven in­de­pend­ently by Cas­son (also un­pub­lished, with the cita­tion be­ing a lec­ture Cas­son gave at the Uni­versity of Cali­for­nia, Santa Bar­bara in 1979).2

The second pre­print (“A geo­met­ric form of Cas­son’s in­vari­ant…”) has proved use­ful to phys­i­cists. Ed­ward Wit­ten char­ac­ter­ized it as “un­pub­lished lec­ture notes” when he cited it for rel­ev­ant back­ground in­form­a­tion in 1991 [e4], and in a re­cent com­mu­nic­a­tion, he re­it­er­ated its use­ful­ness for a cal­cu­la­tion he re­cently un­der­took with Douglas Stan­ford (see [◊]). Sev­er­al math­em­aticians have at­tested that a typescript ver­sion of these notes once ex­is­ted, but our ed­it­ors have not been able to loc­ate a copy.

The copy we make avail­able here, in John­son’s hand­writ­ing, is clearly a second- or third-gen­er­a­tion pho­to­copy, and we re­gret that the legib­il­ity of the doc­u­ment is poor in places. We make it avail­able non­ethe­less as a re­source for oth­ers and in the hopes that a typescript ver­sion of the same may yet be found.

 – Ed­it­ors


[1]K. Jo­hann­son and D. John­son: Non-bor­d­ing dif­feo­morph­isms of sur­faces which act trivi­ally on ho­mo­logy. Pre­print, 1980. techreport

[2]D. John­son: A geo­met­ric form of Cas­son’s in­vari­ant and its con­nec­tion to Re­idemeister tor­sion. Pre­print, 1988. techreport