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
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.
- \( \theta_1 \) (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.
