#### by Joan S. Birman

#### Introduction

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:

The pleasure in truly understanding new ideas in mathematics;

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

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.

#### Background

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
__\[ \pi_0(\operatorname{Diff} S_{g,n}) .\]__
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 __\( \Sigma \)__ can be defined to be the fundamental group of the
space of __\( n \)__ distinct points on __\( \Sigma \)__, the key special case being
Artin’s braid group, which occurs when one chooses __\( \Sigma \)__ 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
__\[ \pi_0 (\operatorname{Diff} S_{g,n})
\to \pi_0(\operatorname{Diff} S_{g,0}) \]__
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 __\( \Sigma \)__, a subgroup of
__\( \operatorname{Diff}(\Sigma) \)__ that would be isomorphic to __\( \pi_1(\Sigma) \)__.

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 favorite paper

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 __\( \pi_0(\operatorname{Diff}S_{g,0}) \)__, 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 __\( \pi_0(\operatorname{Diff} S_{g,0}) \)__ we began to
understand the special nature of the case __\( g=2 \)__: unlike the cases
__\( g > 2 \)__, __\( \pi_0(\operatorname{Diff} S_{2,0}) \)__ 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*
__\[ \mathfrak h^*:S_{2,0}\to S_{0,6} .\]__
We asked: when does the isotopy class of a
map __\( f\in \operatorname{Diff}(S_{2,0}) \)__ project to the isotopy class
of a map in __\( \operatorname{Diff}(S_{0,6}) \)__? 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 __\( f^{\prime}_t \)__ 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.