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Celebratio Mathematica

Martin Scharlemann

Markov’s theorem on the nonrecognizablility of 4-manifolds:
an exposition

by Rob Kirby

In the mid 1950s A. A. Markov proved [e6] that there could be no al­gorithm to dis­tin­guish all pairs of closed, com­pact, smooth 4-man­i­folds. His proof star­ted with the fact that there is no al­gorithm to de­cide wheth­er a fi­nitely presen­ted group G is trivi­al, a fact us­ing ideas of Gödel and proved by P. S. Novikov [e1], [e2] and W. W. Boone [e5] (also see [e8] and [e9]).

Markov’s proof is in Rus­si­an, and not too widely known in the West. Marty Schar­le­mann men­tioned to me a few years ago that, as a gradu­ate stu­dent in the early 70s, he had wanted to un­der­stand Markov’s proof but knew no Rus­si­an. So he looked up the pa­per in Math­em­at­ic­al Re­views and found the ac­count of the proof there un­con­vin­cing, which led him to work out what he thought Markov’s ar­gu­ment must have been. To­geth­er we mined his memory and sor­ted out the proof which is ex­pos­ited here. To the best of my know­ledge this simple proof is not writ­ten up else­where (but see ([e10], p. 149) where the proof is giv­en as an ex­er­cise with help from earli­er ex­er­cises but con­tains no hint as to adding g ex­tra 2-handles; also see [e11]).

Defin­i­tion: A man­i­fold X be­long­ing to a col­lec­tion C of man­i­folds is said to be re­cog­niz­able if there ex­ists an al­gorithm which can de­cide wheth­er or not any man­i­fold Y in C is dif­feo­morph­ic to X.
The 4-man­i­fold X4=#kS2×S2 for some k>0 is not re­cog­niz­able in the col­lec­tion C of all smooth, com­pact, closed 4-man­i­folds.
There does not ex­ist an al­gorithm to dis­tin­guish com­pact, smooth, closed 4-man­i­folds.
Giv­en a fi­nitely presen­ted group G, we can con­struct a smooth 5-man­i­fold YG5 such that π1(YG)=π1(YG)=G and YG has no handles of in­dex >2.

Proof.  Sup­pose G has g gen­er­at­ors g1, …, gg and r re­la­tions r1, …, rr. We can con­struct a 5-man­i­fold YG5 by adding g 1-handles to B5 (ob­tain­ing #g(S1×B4)) and then adding r 2-handles ac­cord­ing to the re­la­tions r1, …, rr. Be­cause the at­tach­ing circles lie in a 4-man­i­fold, these iso­topy classes are de­term­ined by their ho­mo­topy classes in #gS1×S3 which in turn are de­term­ined by the re­la­tions.

However a 2-handle B2×B3 is at­tached by an em­bed­ding of S1×B3 and giv­en the em­bed­ding of S1×0; there are then two ways to ex­tend that em­bed­ding since π1(SO(3))=Z/2. For ex­ample for the un­knot in B5, the two fram­ings give S2×B3 and the non­trivi­al B3 bundle over S2, namely S2×~B3. Let YG be defined by any one choice of fram­ings of the 2-handles.

Two dif­fer­ent present­a­tions of G dif­fer by Tiet­ze moves, but these moves cor­res­pond to the birth or death of a 1-2 handle pair and to handle slides, and these do not change the 5-man­i­fold YG.

The map in­duced by in­clu­sion, π1(YG)π1(YG), is an iso­morph­ism. It is an epi­morph­ism be­cause any loop in YG can be pushed off the 2-com­plex spine of YG and thus in­to YG. It is a mono­morph­ism for the same reas­on; if a loop in YG is ho­mo­top­ic­ally trivi­al in YG, then that 2-disk can also be pushed off the spine and in­to YG.  ◻

Now define Yg to be YG with g trivi­al 2-handles (at­tached to the un­link in a small 4-ball in B5 which avoids all oth­er at­tach­ing maps, and with the fram­ing on the at­tach­ing circle that gives B2×B3 ). Note that Yg=YG#gS2×B3.

The proof of the the­or­em will fol­low from the next lemma.

G is trivi­al if and only if Yg=#rS2×S2.

Proof.  The if part fol­lows from G=π1(Yg)=π1(YG)=π1(YG)=0.

For the only if part, note that π1(YG)=G=0. Then the at­tach­ing circles of the g trivi­al 2-handles can be ho­mo­toped and thus iso­toped in YG, to geo­met­ric­ally can­cel the g 1-handles. (Note that we can­not do this with the ori­gin­al r 2-handles be­cause they do not lie in a simply con­nec­ted 4-man­i­fold, but rather in a con­nec­ted sum of products S1×S3, where­as the ex­tra r 2-handles may be thought of as be­ing ad­ded to YG.) Do the can­cel­la­tion, and what re­mains of Yg is an un­link of r com­pon­ents to which are at­tached r 2-handles, so that Yg is dif­feo­morph­ic to #rS2×B3 with bound­ary #rS2×S2 or the twis­ted case. However, go­ing back to ori­gin­al r 2-handles, we could have chosen their fram­ings so that we avoid the twis­ted case.  ◻

The proof of the The­or­em now fol­lows be­cause giv­en a fi­nitely presen­ted group G, con­struct YG. If we had an al­gorithm to re­cog­nize YG as a con­nec­ted sum of S2×S2s, then we would have an al­gorithm to de­cide wheth­er G is the trivi­al group.

It is still an open ques­tion as to wheth­er the 4-sphere is re­cog­niz­able, that is, if there is an al­gorithm which in fi­nite time will de­cide wheth­er a ho­mo­logy 4-sphere H is dif­feo­morph­ic to S4. Pos­sibly the de­scrip­tion of H as a tri­sec­tion de­term­ined by a dif­feo­morph­ism of a 3-di­men­sion­al handle­body will lead to such an al­gorithm.

Note that there is no al­gorithm to re­cog­nize the n-sphere for n5. Ker­vaire [e7] showed that if a fi­nitely presen­ted group G is su­per­per­fect, that is, if H1(G)=H2(G)=0, then it is the fun­da­ment­al group of a ho­mo­logy n-sphere H, n>4. If the n-sphere can be re­cog­nized, then H=Sn if and only if π1(H)=1 (by the Poin­caré the­or­em). However the class of su­per­per­fect groups, by the Ad­jan–Ra­bin the­or­em [e3] [e4], can­not have an al­gorithm to re­cog­nize the trivi­al group.