#### by Charles Livingston

#### 1. Introduction

Rob Kirby’s “Calculus” result, which he announced in 1974, is
simply stated: two framed link surgery diagrams in __\( S^3 \)__
produce the same three-manifold if and only if the diagrams are
related by a sequence of simple moves: the addition or removal of a
trivial circle with framing __\( \pm 1 \)__ and so-called *handle
slides*. At first the result seemed to hold great promise, but its
consequences were slow to develop. In about 1986 Rob told me, with
perhaps some feeling of disappointment, that this main theorem of his
“Calculus” paper
[4],
in fact its only theorem, was
without implication. Beyond that, the hope that we would be able to
unravel the theory of smooth four-manifolds via that paper’s perspective
of handlebody diagrams of four-manifolds had apparently
been dashed by Donaldson’s work.

In retrospect, there was no need for disappointment: the paper is a
landmark in geometric topology, a dividing line between a time when
little was understood about four-manifolds and a stretch of almost
50 years when four-manifolds have repeatedly risen to be among the most
active areas of research. Most notably, the main theorem was key to
Reshetikhin
and
Turaev’s
work
[e31]
on quantum invariants
and is now the foundation of what has become a major field in
topology, *topological quantum field theory*.

The main three-manifold result continues to be used frequently, and in a different direction, handlebody diagrams, along with the techniques highlighted in the Calculus paper and pioneered by Akbulut and Akbulut–Kirby in the 1970s and early 1980s, continue to be ubiquitous in four-manifold theory.

#### 2. Background: manifolds in dimensions other than four

From the perspective of 1970, manifold theory was alive and well in every dimension except four. Three-manifold theory, although haunted by the Poincaré Conjecture, had seen continuing progress. The work of Schubert, among others, had laid the foundations for studying three-manifolds via the surfaces they contain. In 1957, the proof by Christos Papakyriakopoulos [e1] of Dehn’s Lemma, the Loop Theorem, and the Sphere Theorem, then broke the subject open. A few of the names of major contributors in three-manifold theory that followed are Stallings [e4], Haken [e10] and Waldhausen [e18]. The transformation work of Bill Thurston [e37] began in the mid 1970s. Contributions in the late 1970s and 1980s include those of Jaco–Shalen and Johannson [e24], [e25], Gabai [e28] and Gordon–Luecke [e29].

In 1970,
higher-dimensional
smooth manifolds were a continuing topic
of investigation, with work built upon the foundations laid by
Smale
[e9],
[e5],
[e7].
Key to that work was
handlebody theory and its nearly identical cousin, Morse theory,
which provided the viewpoint Smale used in his proof of the Poincaré
Conjecture and the __\( h \)__-Cobordism Theorem; soon after,
Mazur,
Stallings, and
Barden
proved the __\( s \)__-Cobordism Theorem. The
development of surgery theory by
Kervaire–Milnor
[e11],
Browder
[e20],
Novikov,
Sullivan
and, in the nonsimply
connected case,
Wall
[e38]
offered the essential tools
that continue to be used today.

With regards to higher-dimensional manifolds, there were also other
categories to investigate. Piecewise linear manifolds could be
studied via their simplicial structure, in much the way that
handlebodies could be used in the smooth setting. By 1970 the relationship
between the smooth and PL categories
was
understood. In the realm of topological manifolds, prior to Rob’s
work,
Brown
and
Gluck
[e13],
[e14],
[e15].
had developed
the category of *stable manifolds* which provided a means to
avoid problems related to the Annulus Conjecture. Topological
manifolds themselves remained intractable until 1968, when Rob
announced his solution, via the “torus trick,” of the Annulus
Conjecture
[1],
paving the way for his joint work with
Siebenmann
[2],
[3].

#### 3. Background: 1970, smooth manifolds in dimension four

__\( h \)__-cobordant simply connected smooth four-manifolds become diffeomorphic upon forming the connect sum with some number of copies of

__\( S^2 \times S^2 \)__and

__\( S^2 \widetilde{\times} S^2 \)__; for indefinite four-manifolds

__\( W \)__, automorphisms of

__\( H_2(W\mathbin{\#} S^2 \times S^2) \)__are induced by diffeomorphisms and, again in the case of indefinite four-manifolds, primitive ordinary (noncharacteristic) classes in

__\( H_2(W \mathbin{\#} S^2 \times S^2) \)__are represented by embedded spheres; and homotopy equivalent four-manifolds are

__\( h \)__-cobordant.

At the same time, there were hints that four-manifolds have some
complexities that do not exist in higher-dimensional manifolds.
Kervaire–Milnor
[e6]
had demonstrated that
Rochlin’s
theorem provides unexpected constraints on spherical representatives
of classes in __\( H_2(M) \)__.
Massey
[e19]
had demonstrated that
the
Atiyah–Singer
Index Theorem could be used to constrain how
nonorientable surfaces could be embedded in __\( \mathbb{R}^4 \)__. In the early
1970s,
Casson–Gordon
[e22],
[e27]
expanded on the
application of the Index Theorem, using it to show that Levine’s
classification results about higher-dimensional knot concordance did
not hold in the classical dimension.

#### 4. Algebraic manifolds

By 1970, Rob’s work on topological manifolds along with his joint
work with Siebenmann, had settled many of the long standing problems
in manifold theory, in particular clarifying the relationship between
the topological, PL, and smooth categories. There is another domain
in which manifolds appear, that of the algebraic setting. One inkling
of this is represented by the Thom Conjecture, which in it simplest
form stated that the minimal genus among smooth representatives of a
homology class in __\( \mathbb{C}\mathbb{P}^2 \)__ equals the genus of a nonsingular
algebraic curve representing that class. (The conjecture was proved
by
Kronheimer
and
Mrowka
[e35];
the topological locally flat
version of the conjecture was disproved by
Rudolph
[e26]
using Freedman’s work on topological four-manifolds.)

Rob was drawn to considering algebraic surfaces through conversations with Arnie Kas, who explained to him the theory of algebraic families of surfaces such as the Kummer surface. The Calculus Theorem followed from Rob’s effort to understand these manifolds from the perspective of handlebody structures on smooth manifolds.

The influence of Rob’s initial efforts to understand complex surfaces
did not end with the Calculus Theorem. His joint efforts, starting in
1974, with
Harer
and Kas
to understand Kummer surfaces led to their
memoir that was completed in 1980 and appeared in
[6].
It
is worthwhile to note that when
Donaldson
found counterexamples to
the __\( h \)__-Cobordism Theorem in dimension four, they were built from
what are called logarithmic transforms of elliptic surfaces. The
timing was such that
[6]
could include handlebody
diagrams of these examples. The interested reader can find an
excellent discussion of these topics in the text by
Gompf
and
Stipsicz
[e39].

#### 5. Kirby’s Calculus Theorem

It was known by the work of
Lickorish
[e8]
and
Wallace
[e12]
that every three-manifold can be described
as surgery on a framed link, __\( \mathcal{F} \)__ in __\( S^3 \)__. We write the manifold
as __\( M^3(\mathcal{F}) \)__. The only theorem in the calculus paper is the
following.

__\( \mathcal{F}_1 \)__and

__\( \mathcal{F}_2 \)__,

__\( M^3(\mathcal{F}_1) \cong M^3(\mathcal{F}_2) \)__if and only if

__\( \mathcal{F}_1 \)__and

__\( \mathcal{F}_2 \)__are related by a series of moves of two types:

__\( \theta_1 \)__(add or remove an unknotted, unlinked, component of framing__\( \pm 1 \)__);__\( \theta_2 \)__(slide one component over another, with an appropriate change of framing).

Immediately one notices something: although we have described this paper as a landmark in four-manifold theory, the theorem says nothing about four-manifolds. To understand this, and to fully appreciate the significance of the paper, we need to summarize the proof. Note that the “if” portion is elementary; our focus is solely on the “only if” part.

#### 6. Proof outline

**First steps:** The first part of the proof reduces the theorem to four-manifold theory.

**Step 1:** The framed link diagrams for the
three-manifolds determine simply connected four-manifolds
__\( W^4(\mathcal{F}_1) \)__ and __\( W^4(\mathcal{F}_2) \)__ built from __\( B^4 \)__ by attaching
2-handles along the link with the given framings. We have
__\( M^3(\mathcal{F}_i) \cong \partial W^4(\mathcal{F}_i) \)__.

**Step 2:** Let
__\[
Y^4 = W^4(\mathcal{F}_1)\bigcup_\partial W^4(\mathcal{F}_2).
\]__
We would like
to say that __\( Y^4 = \partial Z^5 \)__ for some manifold __\( Z^5 \)__. This
will be the case if the signature __\( \sigma(Y^4) = 0 \)__. If the
signature is not zero, then replace __\( \mathcal{F}_1 \)__ with a new framed
link, formed by including unlinked trivial components with
framings __\( \pm 1 \)__. This has the effect of replacing __\( W^4(\mathcal{F}_1) \)__
with __\( W^4(\mathcal{F}_1)\mathbin{\#}_k \pm \mathbb{C}\mathbb{P}^2 \)__, altering the signature of
__\( Y^5 \)__ by any desired amount.

**Step 3:** View __\( Z^5 \)__ as a relative
cobordism between __\( (W^4(\mathcal{F}_1), \partial) \)__ and
__\( (W^4(\mathcal{F}_2), \partial) \)__. Surgery can ensure that there are no
1-handles or 3-handles in the cobordism. The middle level is
then diffeomorphic to __\( W^4(\mathcal{F}_1) \mathbin{\#}_k S^2 \times S^2 \)__ or
__\( W^4(\mathcal{F}_1) \mathbin{\#}_k S^2 \widetilde{\times} S^2 \)__ for some integer
__\( k \)__. The same holds for __\( W^4(\mathcal{F}_2) \)__, with perhaps a different
value of __\( k \)__.

**Step 4:** We have that
__\[
S^2 {\times} S^2 \mathbin{\#} \mathbb{C}\mathbb{P}^2 \cong S^2 \widetilde{\times} S^2 \mathbin{\#}
\mathbb{C}\mathbb{P}^2 \cong \mathbb{C}\mathbb{P}^2 \mathbin{\#} \mathbb{C}\mathbb{P}^2 \mathbin{\#} - \mathbb{C}\mathbb{P}^2.
\]__
Adding unknotted
components to __\( \mathcal{F}_i \)__ with framing __\( \pm 1 \)__ has the affect of
forming the connected sums with __\( \pm \mathbb{C}\mathbb{P}^2 \)__. Thus, after making
the appropriate moves we can assume that
__\( W^4(\mathcal{F}_1) \cong W^4(\mathcal{F}_2) \)__. Call this manifold __\( W^4 \)__.

**Next steps:** The next part of the proof shows that
if a four-manifold __\( W \)__ has two handlebody structures built only
with 2-handles, then those structures are related by adding and
deleting cancelling pairs of 1-handles and 2-handles, adding and
deleting cancelling pairs of 2-handles and 3-handles, and by
sliding handles. In the proof, this is where Morse Theory appears.

**Step 5:** The two handlebody structures on
__\( W^4 \)__ correspond to two Morse functions, __\( f_0 \)__ and __\( f_1 \)__, on
__\( W^4 \)__. As functions to __\( \mathbb{R} \)__ they are certainly homotopic; Cerf
theory provides a generic homotopy for which at all but a finite
set of __\( t \)__, the function __\( f_t \)__ is Morse. More precisely, at those
finite set of singularities there are two possibilities: (1) two
critical value of __\( f_t \)__ are the same, or (2) at __\( t \)__, a pair of
singular points are introduced or disappear, of indices __\( i \)__ and
__\( i+1 \)__ for some __\( i \)__ in the range __\( 0\le i \le 3 \)__.

**Step 6:** The first issue that arises is
that tracking the critical values of the Morse function and the
births and deaths of critical points is not sufficient. For
instance, changes in the Morse function can correspond to handle
slides. To deal with this, Rob built upon the
diagrammatics in
Cerf’s work to specifically analyze generic paths of Morse
functions in the case of four-manifolds.

**Step 7:** More challenging is that the
path of Morse functions has to be modified to simplify the
corresponding handlebody structures. For instance, any births of
0-handles and of 4-handles have to be eliminated.

**Step 8 (final step):** The last step of the proof
consists of, in Rob’s words, pushing the births and deaths of
cancelling pairs to the boundary. The affect of this is to perform
a sequence of the move __\( \theta_1 \)__ to both of the framed links.

#### 7. The impact of “A Calculus” on four-manifold theory

The Calculus paper gave researchers the hope that working with
handlebody structures on four-manifolds would provide a route to
solving some of the challenging problems of the day. For instance,
there was the problem of finding an
even, definite four-manifold __\( W \)__
with signature 16. It was known that the Kummer surface (of signature
16 and __\( \beta_2 = 22 \)__) is split by a homology sphere __\( \Sigma^3 \)__ into
a definite rank 16 piece and a second piece __\( X \)__. An approach to
constructing __\( W \)__ was to try to build a contractible manifold bounded
by __\( \Sigma \)__ which could then replace __\( X \)__ to form __\( W \)__. Of course, as
we learned from Donaldson in 1982, the approach was bound to fail.
Yet along the way, the exploration of handlebody structures on
bounded four-manifolds yielded a series of new results. One early
example includes Akbulut and Kirby’s analysis
[5]
of the
cyclic covers of four-manifolds branched over surfaces. As another
example, Robert Gompf’s
[e33]
proof that so-called
Cappell–Shaneson homotopy four-spheres are standard is a masterful
exercise in handlebody manipulation.

The same approach became instrumental following the introduction of gauge theory. A highlight is Akbulut’s proof [e32] that the Mazur manifold, a compact contractible four-manifold, has two distinct smooth structures. The proof ends with ten pages of diagrams. One more recent highlight is Piccirillo’s proof [e47] that the Conway knot is not slice (a proof that depended on the Rasmussen invariant [e40], as opposed to gauge theory). In that paper, a long series of handlebody diagrams involving handle slides and cancelling pairs of 1-handles and 2-handles yields a desired diffeomorphism between two bounded four-manifolds. A more theoretical appearance is the central role it plays in the work of Juhasz [e42] and Zemke [e45] analyzing the functoriality of Heegaard Floer theory.

#### 8. The impact of “A Calculus” on three-manifold theory

Following Rob’s work, Fenn and Rourke [e23] demonstrated that a single move on surgery diagrams could replace the pair described by Rob. Rolfsen generalized the calculus to the case of fractional surgery, as described in his text [e21]. Following this work, the use of surgery diagrams and the calculus became a common practice in studying explicit three-manifolds. A host of basic examples can be found in such books as [e39], [e21]. More recent examples include ([e46], Figure 6), ([e43], Figures 4, 5, 6), and ([e44], Figure 3).

On the other hand, for 15 years after its discovery, the deeper part of the Calculus Theorem had little impact on three-manifold theory. People put no small effort into trying to define new three-manifold invariants via surgery diagrams, but without success.

A
hint of its significance might have been seen in Casson’s
development of what is now called the *Casson invariant* of
homology three-spheres. That invariant is defined using Heegaard
diagrams, but Casson’s proof that it is an integral lifting of the
__\( \mathbb{Z}_2 \)__-valued Arf invariant depended on a study of surgery diagrams
of three-manifolds. In subsequent years, the Casson invariant was
generalized to more general three-manifolds, in such work as that of
Boyer–Lines
[e30],
Walker
[e34]
and
Lescop
[e36].

Although Casson’s work was purely three-dimensional, he was able to apply the Casson invariant to prove that there exist four-manifolds that don’t support simplicial triangulations. This triangulation result was extended thirty years later by Manolescu [e41] to include all dimensions greater than four.

Another hint that there might be deeper consequences of the Calculus
Theorem came with the work of
Jones.
His definition of the *Jones
polynomial* is given in terms of braid diagrams for a link. To
prove that it is well-defined, he called on the Markov
Theorem, a
knot theoretic analog of the Calculus Theorem that states that two
braids close to
the same link if and only if they are related by
a sequence of some basic simple moves.

The ultimate breakthrough came with the work of Reshetikhin and Turaev [e31]. Soon after Jones developed the Jones polynomial, Witten recognized its relation to three-manifold theory and its underlying connection to quantum field theory. With this in the background, Reshetikhin and Turaev proved [e31] that a three-manifold invariant can be defined via framed link diagrams. The proof that this invariant is well-defined depends on the Calculus Theorem; the authors prove that if either of the “Kirby moves” is performed on a framed link diagram, then the value of the invariant does not change. Thus, the Calculus Theorem lies at the foundations of the entire realm of topological quantum field theory, and it continues to be its keystone.

*Charles Livingston began his college studies at UCLA in 1971. Two years later he transferred to MIT, where he received his mathematics degree in 1975. His graduate work was done at the University of California, Berkeley, during the years 1975–1980. From there he took a postdoctoral position at Rice University and then moved to Indiana University, Bloomington. Beginning in 2019 he has held the title of Professor Emeritus.*