Celebratio Mathematica

My math­em­at­ic­al col­league and friend Ro­bi­on Kirby asked me to write a few words, for the volume he is pre­par­ing for the web­site Cel­eb­ra­tio Math­em­at­ica. The top­ic he sug­ges­ted was my fa­vor­ites in the list of my re­search pa­pers that he is as­sem­bling. He asked a good ques­tion, and the ques­tion lead me to ru­min­ate a bit about the events that had sur­roun­ded the pa­per that I ul­ti­mately de­cided had been es­pe­cially re­ward­ing: my work with Mike (Hugh Mi­chael) Hilden, Map­ping class groups of closed sur­faces as cov­er­ing spaces [1]. In par­tic­u­lar, I want to say a few words about three as­pects of cre­at­ive re­search in math­em­at­ics that were present in that work, all of which have been very im­port­ant to me:

1. The pleas­ure in truly un­der­stand­ing new ideas in math­em­at­ics;

2. The rare but pre­cious cre­at­ive in­sight, the ‘aha’ mo­ment; and

3. The ex­per­i­ence of un­der­stand­ing the abil­ity and cre­ativ­ity of an­oth­er hu­man be­ing. My deep­est per­son­al and math­em­at­ic­al friend­ships have come about through col­lab­or­at­ive work, and the joy in dis­cov­ery is truly won­der­ful when it’s a shared joy.

Thir­teen years after gradu­at­ing from col­lege, and two weeks after the birth of my third and young­est child, I made my first tent­at­ive move to­ward a ca­reer in math by en­rolling in an even­ing grad course in Lin­ear Al­gebra at NYU’s Cour­ant In­sti­tute. The total break from home du­ties to study math was a very wel­come breath of fresh air and the pleas­ure I ex­per­i­enced when I un­der­stood something new was ex­hil­ar­at­ing.

One course soon lead to two, but for sev­er­al years I was work­ing alone. I even­tu­ally got to know one of my fel­low stu­dents, Orin Chein and we would meet after lec­tures to go over our notes from earli­er lec­tures, adding a new ele­ment to my stud­ies: the pleas­ure in ex­chan­ging ideas with oth­ers. One of the courses on which Orin and I worked es­pe­cially hard was the course of­fer­ing in to­po­logy, taught by Pro­fess­or Jac­ob Schwartz, who had told his stu­dents that he made it a prac­tice to learn new math­em­at­ics by teach­ing every course that Cour­ant offered at least once. He ad­ded that he would be learn­ing the tools of to­po­logy with us. His present­a­tion was bril­liant and in­spir­ing. Sur­pris­ingly, it not only was dif­fer­ent from the present­a­tions in the books we con­sul­ted, but so dif­fer­ent that when it came to gaps in our un­der­stand­ing we were on our own, and even more we had no ex­amples to ease the way. We worked hard to fill both gaps.

My thes­is ad­visor, Wil­helm Mag­nus, was a fine ment­or, and he was very sens­it­ive to my in­terests, which by then had evolved in dir­ec­tions far from the core re­search top­ics at Cour­ant. He told me about his own work from the 1930’s on the map­ping class group of a tor­us with 2 punc­tures [e1], and sug­ges­ted that I think about the tor­us with $n$ punc­tures, and high­er genus. He also men­tioned, in passing, re­cent work [e2] that he had no­ticed by Fadell, Neuwirth and oth­ers about a new way to think about Artin’s braid group that lead in a nat­ur­al way to the then-new concept of braid groups of sur­faces. With the wis­dom of hind­sight I now real­ize that his in­stinct in singling out [e2] was pres­ci­ent. The task at hand was to un­der­stand what happened to the map­ping class group when one passed from a sur­face $S_{g,n}$ of genus $g$ with $n$ marked points to the sur­face $S_{g,n-1}$ or (re­peat­ing $n$ times) $S_{g,0}$? Ho­mo­topy groups were nat­ur­al in this set­ting, be­cause the map­ping class group of $S_{g,n}$ is the group $${\pi }_0(\mathrm{Diff}{S}_{g,n}).$$ It began to look as if Schwartz’s lec­tures on To­po­logy might come in very handy! The key in­sight in [e2] was that the $n$-strand braid group of a sur­face $\mathrm{\Sigma }$ can be defined to be the fun­da­ment­al group of the space of $n$ dis­tinct points on $\mathrm{\Sigma }$, the key spe­cial case be­ing Artin’s braid group, which oc­curs when one chooses $\mathrm{\Sigma }$ to be the plane ${\mathbb{R}}^2$. Still, I do not think that Mag­nus an­ti­cip­ated the work that ul­ti­mately be­came my thes­is, be­cause of his very genu­ine sur­prise when I told him about the long ex­act se­quences of (some­times nona­beli­an) ho­mo­topy groups that Orin and I had learned about from Jac­ob Schwartz, and how I used them.

The the­or­em that I even­tu­ally proved is now known as the Birman ex­act se­quence. It iden­ti­fies the ker­nel of the ho­mo­morph­ism $${\pi }_0(\mathrm{Diff}{S}_{g,n})\to {\pi }_0(\mathrm{Diff}{S}_{g,0})$$ as the $n$-strand braid group of $S_g$ mod­ulo its cen­ter. See Sec­tion 4.2 of [e2] for a dis­cus­sion of it that was aimed at a gradu­ate stu­dent in 2012. My pleas­ure in dis­cov­er­ing the map that is now known as the point push­ing map was one of those ‘aha’ mo­ments in math­em­at­ics. I re­mem­ber to this day where I was stand­ing in our home when I sud­denly un­der­stood how to con­struct, in the map­ping class group of an ori­ent­able sur­face $\mathrm{\Sigma }$, a sub­group of $\mathrm{Diff}(\mathrm{\Sigma })$ that would be iso­morph­ic to ${\pi }_1(\mathrm{\Sigma })$.

By the time that I re­ceived my PhD in Math­em­at­ics, it was clear to me that (i) I really liked the chal­lenges of re­search, that (ii) work­ing alone was pos­sible and had its re­wards, but that it might not be not ideal for me, and that (iii) my thes­is was at best a small step to­ward un­der­stand­ing the real prob­lem. While I now knew how to handle the $n$ points, there re­mained the ques­tion of un­der­stand­ing the map­ping class group of the closed sur­face, that is $S_{g,0}$. In every piece of work I have done, one solved prob­lem al­ways sug­ges­ted an­oth­er (and pos­sibly deep­er ques­tion) that I did not know how to an­swer. That phe­nomen­on turned out to be the key to the is­sue of how to keep do­ing re­search, for a life­time.

My first job was as an As­sist­ant Pro­fess­or of Math­em­at­ics at Stevens In­sti­tute of Tech­no­logy. Its pleas­ant green cam­pus was on the banks of the Hud­son River, with in­cred­ible views of New York City across the river. Math was in a low white clap­board build­ing, and the day I ar­rived to be­gin teach­ing I was greeted by a gradu­ate stu­dent, Hugh Mi­chael Hilden, who in­tro­duced him­self as ‘Mike’. His of­fice was near mine. We had lunch to­geth­er, and he told me that he would be get­ting his de­gree the fol­low­ing May, hav­ing solved his thes­is prob­lem, but that he didn’t really like his thes­is area. He wanted to know, what was I work­ing on? I told him about my ob­sess­ive wish to un­cov­er the struc­ture of the group ${\pi }_0(\mathrm{Diff}{S}_{g,0})$, the first in­ter­est­ing case (here I was re­mem­ber­ing that Mag­nus had asked me about the map­ping class group of the 3-times punc­tured tor­us) be­ing $g=2$. I told him about its po­ten­tial cent­ral role of map­ping class groups in 3-man­i­fold to­po­logy, a top­ic that in­ter­ested him very much, even though it was far from his thes­is. I was de­lighted by this un­ex­pec­ted turn of events.

That was the first of many lunch­time dis­cus­sions, and it made the year ex­cep­tion­ally in­ter­est­ing. Mike was a fast learner, and I was very happy to be both teach­ing classes with some tal­en­ted stu­dents, and hav­ing stim­u­lat­ing lunch­time dis­cus­sions about re­search. I told Mike many things about Artin’s braid group, the map­ping class group of the plane with $n$ marked points, and its close re­l­at­ive, the map­ping class group of the sphere.

Fo­cus­ing on ${\pi }_0(\mathrm{Diff}{S}_{g,0})$ we began to un­der­stand the spe­cial nature of the case $g=2$: un­like the cases $g>2$, ${\pi }_0(\mathrm{Diff}{S}_{2,0})$ had a cen­ter, the map­ping class of a dif­feo­morph­ism $$\mathfrak{h}:S_{2,0}\to S_{2,0}$$ of or­der 2 with 6 fixed points. The map $\mathfrak{h}$ was known as the hy­per­el­lipt­ic in­vol­u­tion. It had played an im­port­ant role in al­geb­ra­ic geo­metry. We soon real­ized that $S_{2,0}/\mathfrak{h}$ was iso­morph­ic to the sphere $S_{0,6}$ with 6 marked points. Mike told me about branched cov­er­ing spaces, which sug­ges­ted to us that we study the branched cov­er­ing space pro­jec­tion $${\mathfrak{h}}^{\ast }:S_{2,0}\to S_{0,6}.$$ We asked: when does the iso­topy class of a map $f\in \mathrm{Diff}(S_{2,0})$ pro­ject to the iso­topy class of a map in $\mathrm{Diff}(S_{0,6})$? It was easy to see that, up to iso­topy, every $f$ pro­jec­ted, that’s what it meant for the map­ping class group of $S_{2,0}$ to have a cen­ter. But that was not the is­sue. There were iso­top­ies of $S_{2,0}$ that did not pro­ject to iso­top­ies of $S_{0,6}$.

We pondered that mat­ter for a long time, and then the ‘aha’ mo­ment came, but this time it was with a bo­nus: I hon­estly can­not re­mem­ber wheth­er it was Mike or me or both of us who had the key idea to fo­cus on one of the 6 points $q_i$, $i=1,2,\dots$, 6 that were fixed by the hy­per­el­lipt­ic in­vol­u­tion $\mathfrak{h}$, say $q_i$, and con­sider its or­bit $f_t(q_i)$ un­der an iso­topy $f_t$ of the sur­face from $f_0$ to the iden­tity map $f_1$. Could the loop $f_t(q_i)$ rep­res­ent a non­trivi­al ele­ment of ${\pi }_1(S,q_i)$? We proved it could not. We were on our way to writ­ing the pa­per that was ul­ti­mately pub­lished as [1], my fa­vor­ite pa­per. We then went on to the more te­di­ous but routine work of modi­fy­ing the giv­en iso­topy $f_t$ to a new one $f_t^{\prime }$ that kept $q_i$ fixed for every $t\in [0,1]$. Ul­ti­mately we con­struc­ted an iso­topy ${f_t}^{\prime \prime }$ that com­muted with $\mathfrak{h}$ at every $t$ and at every point $q$ on the sur­face.

The work that we did in [1] was soon gen­er­al­ized and strengthened in many ways, e.g., see [2], which used slick­er and less trans­par­ent ar­gu­ments, but also proved much more. Mo­tiv­a­tion is some­times con­cealed in this way, but that’s part of math­em­at­ics too. Per­haps that should be the top­ic for a dif­fer­ent dis­cus­sion, at a dif­fer­ent time.