Celebratio Mathematica

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.

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].

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.

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].

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.

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.

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 $W^4({\mathcal{F}}_1){\mathrm{\#}}_k\pm \mathbb{C}{\mathbb{P}}^2$, 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 $W^4({\mathcal{F}}_1){\mathrm{\#}}_k{S}^2\times S^2$ or $W^4({\mathcal{F}}_1){\mathrm{\#}}_k{S}^2\tilde{\times }{S}^2$ 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 $$S^2\times S^2\mathrm{\#}\mathbb{C}{\mathbb{P}}^2\cong S^2\tilde{\times }{S}^2\mathrm{\#}\mathbb{C}{\mathbb{P}}^2\cong \mathbb{C}{\mathbb{P}}^2\mathrm{\#}\mathbb{C}{\mathbb{P}}^2\mathrm{\#}-\mathbb{C}{\mathbb{P}}^2.$$ 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.

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 ${\mathrm{\Sigma }}^3$ 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 $\mathrm{\Sigma }$ 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.

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.