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Celebratio Mathematica

Michael H. Freedman

Topology; quantum computing  ·  UCSD & Microsoft

Universal manifold pairings in dimension 3

by Kevin Walker

Around 2004, Mike Freed­man, in­spired by a ques­tion of Alexei Kit­aev’s, be­came in­ter­ested in uni­ver­sal man­i­fold pair­ings. This story is mainly about Mike’s tenacity but, to put it in con­text, I first need to ex­plain what uni­ver­sal man­i­fold pair­ings are.

Our start­ing point is the no­tion of a \( d \)-di­men­sion­al to­po­lo­gic­al quantum field the­ory (TQFT). If you are in­ter­ested in TQFTs (and many people are), then you should also be in­ter­ested in uni­ver­sal man­i­fold pair­ings. Man­i­folds of di­men­sion \( d \) have a cut-and-paste struc­ture: they can be glued to­geth­er along parts of their bound­ar­ies to yield new man­i­folds. For ex­ample, when \( d=2 \), we can glue two disks to­geth­er and ob­tain a sphere. A TQFT as­signs an al­geb­ra­ic ob­ject (for ex­ample, a num­ber or a vec­tor space) to each man­i­fold, in such way that glu­ings cor­res­ponds to al­geb­ra­ic op­er­a­tions such as in­ner products and tensor products (de­pend­ing on the di­men­sion of the man­i­fold). For ex­ample, every closed (that is, with an empty bound­ary) \( d \)-di­men­sion­al man­i­fold is as­signed a com­plex num­ber, every closed \( (d-1) \)-man­i­fold is as­signed a vec­tor space, and a \( d \)-di­men­sion­al man­i­fold with bound­ary is as­signed a vec­tor in the vec­tor space as­so­ci­ated to its \( (d-1) \)-di­men­sion­al bound­ary. If we glue to­geth­er two \( d \)-man­i­folds with the same bound­ary to ob­tain a closed \( d \)-man­i­fold, the num­ber as­signed to the glued-up man­i­fold is equal to the in­ner product of the vec­tors as­signed to the two pieces. A TQFT also as­signs fan­ci­er al­geb­ra­ic ob­jects to man­i­folds of di­men­sion \( d-2 \), \( d-3,\dots, 0 \), but here we will only be con­cerned with the \( d \)- and \( (d-1) \)-di­men­sion­al parts of the TQFT. There are many dif­fer­ent \( d \)-di­men­sion­al TQFTs, each mak­ing dif­fer­ent as­sign­ments of num­bers, vec­tor spaces, and so on.

Some nat­ur­al ques­tions arise: To what ex­tent does the TQFT’s re­place­ment of to­po­lo­gic­al ob­jects with al­geb­ra­ic ones lose in­form­a­tion? Can we find two dif­fer­ent \( d \)-man­i­folds with the same bound­ary such that any pos­sible TQFT as­signs to these man­i­folds equal vec­tors? One way to ad­dress this type of ques­tion is to form a sort of uni­ver­sal TQFT. In­stead of num­bers, we use the ring \( R \) of form­al lin­ear com­bin­a­tions of closed \( d \)-man­i­folds. In­stead of a vec­tor space, we as­sign to a closed \( (d-1) \)-man­i­fold \( S \) the \( R \)-mod­ule \( V(S) \) of all form­al lin­ear com­bin­a­tions of \( d \)-man­i­folds bounded by \( S \). There is a pair­ing \( V(S) \times V(S) \to R \) in­duced by glu­ing \( d \)-man­i­folds to­geth­er along \( S \), yield­ing closed \( d \)-man­i­folds. This is the uni­ver­sal man­i­fold pair­ing we spoke of.

For a fixed di­men­sion \( d \), we can ask wheth­er there ex­ists a \( (d-1) \)-man­i­fold \( S \) for which the uni­ver­sal pair­ing on \( V(S) \) has null vec­tors, that is, vec­tors \( x \in V(S) \) such that the pair­ing of \( x \) and \( y \) is zero for all \( y \in V(S) \). Such null vec­tors \( x \) would lead to uni­ver­sal re­la­tions that will hold in any par­tic­u­lar TQFT. A weak­er thing to ask for is a norm-zero vec­tor \( z \) in \( V(S) \), that is, a \( z \) which, when paired with it­self, yields zero in \( R \). Such a norm-zero vec­tor would leads to re­la­tions that hold in any TQFT with pos­it­ive-def­in­ite in­ner products.

It is easy to show that for \( d = 1 \) or 2 there are no norm-zero vec­tors for the uni­ver­sal pair­ing. Mike’s first res­ult (along with five coau­thors — Mike is gen­er­ous in shar­ing cred­it) was that null vec­tors do ex­ist for \( d = 4 \) [1]. Build­ing on this res­ult, one can show that any two closed ho­mo­topy-equi­val­ent simply con­nec­ted smooth 4-man­i­folds have the same TQFT in­vari­ants for any pos­it­ive-def­in­ite TQFT. This neg­at­ive res­ult shows that, if you hope to use TQFTs to study exot­ic smooth struc­tures on 4-man­i­folds, you will have to try something more com­plic­ated than the simplest strategy of em­ploy­ing a pos­it­ive-def­in­ite 4-di­men­sion­al TQFT. (For ex­ample, you would need to try a non-semisimple TQFT, or a 5-di­men­sion­al TQFT.) Soon after, Kreck and Teich­ner [e2] showed that such norm-zero vec­tors ex­ist for the uni­ver­sal pair­ing in di­men­sions 5 and high­er as well.

This left uni­ver­sal pair­ings in a state roughly sim­il­ar to the Poin­caré con­jec­ture (pre-Perel­man). In the 1- and 2-di­men­sion­al cases, it was easy to show that norm-zero vec­tors did not ex­ist. In di­men­sions 4 and high­er, they did ex­ist. Di­men­sion 3 was the last re­main­ing case, and it looked to be quite dif­fi­cult.

In 2007, Mike told me he had a strategy for at­tack­ing the 3-di­men­sion­al case. A re­cent res­ult of Agol, Storm and Thur­ston [e1] (build­ing on work of Perel­man) showed that there were no norm-zero vec­tors if one re­stric­ted to hy­per­bol­ic 3-man­i­folds glued along a geodes­ic bound­ary. The geo­met­riz­a­tion the­or­em for 3-man­i­folds tells us that any 3-man­i­fold can be cut along spheres and tori, to yield pieces which are either spher­ic­al, Seifert fibered, or hy­per­bol­ic. Thus, a gen­er­ic strategy for prov­ing something about all 3-man­i­folds is to

  1. prove it for spher­ic­al 3-man­i­folds,
  2. prove it for Seifert-fibered 3-man­i­folds,
  3. prove it for hy­per­bol­ic 3-man­i­folds, and
  4. show that these three spe­cial cases can some­how be com­bined when glu­ing along spheres and tori.

Typ­ic­ally, steps (3) and (4) are the hard­est parts. In the present case, Mike saw a way to solve a spe­cial case of (3), and boldly as­sumed that the re­mainder of (3), as well as (1), (2) and (4), would fall in­to place. Mike’s plan was that he and Danny Calegari would fin­ish step (3), while Mike and I would handle the com­bin­at­or­i­al ar­gu­ments of step (4).

I was privately quite skep­tic­al of this. His ar­gu­ments that step (3) (the hy­per­bol­ic case) was nearly done didn’t con­vince me, but I’m not an ex­pert in hy­per­bol­ic 3-man­i­folds, so I was will­ing to con­cede that part. I was more con­cerned about step (4), which I thought would be a hor­rible mess. I re­mem­ber feel­ing a bit smug and su­per­i­or, think­ing “Poor Mike, he’s wast­ing his time on this in­tract­able prob­lem in­stead of work­ing on more real­ist­ic pro­jects.” But Mike had been very gen­er­ous to me over the years, so I felt I owed it to him to work on this des­pite my skep­ti­cism.

Mike would present a large num­ber of ideas, and I would re­spond with (al­most) the same num­ber of counter­examples to show that these ideas couldn’t work. I was ex­pect­ing that he would even­tu­ally run out of steam and leave me in peace but, in­stead, he kept go­ing and go­ing. Some small por­tion of his ideas seemed to in­deed work, and the found­a­tions of a com­plete proof began to slowly ac­crete.

Mean­while, Danny was hav­ing a sim­il­ar ex­per­i­ence with Mike on the hy­per­bol­ic side of the ar­gu­ment. In or­der to feed in­to step (4), we needed to ex­tend the Agol–Storm–Thur­ston res­ult to cusped hy­per­bol­ic 3-man­i­folds. In Danny’s words,

One thing that im­pressed me is how san­guine Mike was about the tech­nic­al obstacles we had to over­come. Our first strategy, to avoid hav­ing to prove the strict volume in­equal­ity for the cusped hy­per­bol­ic pieces, de­pended on us­ing even­tu­al strict­ness for a se­quence of or­bi­fold fillings con­ver­ging to the cusped guy. The prob­lem is to com­pare the or­bi­fold fillings for the double and the twis­ted double; there is no easy cut-and-paste com­par­is­on, be­cause the “sur­face” is now a com­plic­ated 2-com­plex with in­ter­est­ing sin­gu­lar­it­ies along the or­bi­fold. In the case where the or­bi­fold filling has even de­gree, one gets a smooth im­mersed sur­face in a man­i­fold cov­er, and, if one knew the sur­face sub­group was sub­group-sep­ar­able, one could try to do cut-and-paste in a suit­able fi­nite cov­er. The first step of this po­ten­tial ar­gu­ment de­pended on prov­ing some ver­sion of LERF — I re­mem­ber go­ing through dozens of vari­ations on the idea of us­ing some weakened no­tion of LERF that would do what we wanted: passing to an amen­able sep­ar­at­ing cov­er and do­ing some sort of av­er­aging, re­pla­cing the hy­per­bol­ic man­i­fold with a solen­oid­al hy­per­bol­ic lam­in­a­tion in which things were sep­ar­ated, etc. After try­ing about 20 vari­ations on this idea over 8 months, it was clear that this just wasn’t go­ing to work, and I was ba­sic­ally ready to give up. But Mike just said, “well, let’s just prove the strict volume in­equal­ity for the cusped guys,” and we were in­stantly off think­ing about Ricci flow for sin­gu­lar met­rics on non­com­pact man­i­folds with no uni­form bounds on in­jectiv­ity ra­di­us…”

Even­tu­ally, after many ups and downs, we had a proof. My ini­tial as­sess­ment that the pro­ject was a big waste of time proved to be quite wrong. I don’t think I mis­judged the dif­fi­culty of the prob­lem. Rather, I mis­judged Mike’s tenacity and re­source­ful­ness. Here’s Danny again:

The great thing about this was how Mike was able to sus­tain this op­tim­ism and just try idea after idea after idea, nev­er dis­count­ing an idea be­cause it in­volved us­ing/learn­ing/gen­er­al­iz­ing some hard tech­nic­al ma­chinery, nev­er dis­count­ing an idea be­cause it seemed to de­pend on solv­ing some known “hard” prob­lem, but be­ing pre­pared to drop an idea and move on to the next one when it didn’t work out.

Works

[1] article M. H. Freed­man, A. Kit­aev, C. Nayak, J. K. Slinger­land, K. Walk­er, and Z. Wang: “Uni­ver­sal man­i­fold pair­ings and pos­it­iv­ity,” Geom. To­pol. 9 (2005), pp. 2303–​2317. MR 2209373 Zbl 1129.​57035 ArXiv math/​0503054