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 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 extra 2-handles; also see
[e11]).
Definition:
A manifold belonging to a collection of manifolds
is said to be recognizable if there exists an algorithm which can
decide whether or not any manifold in is diffeomorphic
to .
The 4-manifold
for some is not
recognizable in the collection 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 , we can construct a smooth 5-manifold
such that
and has no handles of index .
Proof.
Suppose has generators , …, and relations
, …, . We can construct a 5-manifold by adding
1-handles to (obtaining ) and then adding
2-handles according to
the relations , …, . Because the attaching circles lie in
a 4-manifold, these isotopy classes are determined by their homotopy classes
in which in turn are determined by the relations.
However a 2-handle is attached by an embedding of and given the embedding of ; there are then two
ways to extend that embedding since
For example for the unknot in , the two framings give and the nontrivial bundle over , namely . Let be defined by any one choice of framings
of the 2-handles.
Two different presentations of 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 .
The map induced by inclusion,
is an isomorphism. It is an epimorphism because any loop in can
be pushed off the 2-complex
spine of and thus into .
It is a monomorphism for the same reason; if a loop in
is homotopically trivial in , then that 2-disk can also be pushed
off the spine and into .
◻
Now define to be with trivial 2-handles (attached to the
unlink in a small 4-ball in which avoids
all other attaching maps, and with the framing on the attaching
circle that gives ). Note that
The proof of the theorem will follow from the next lemma.
is trivial
if and only if
Proof.
The if part follows from
For the only if part, note that
Then the attaching circles of the trivial 2-handles can
be homotoped and thus isotoped in , to geometrically cancel
the 1-handles. (Note that we cannot do this with the original 2-handles
because they do not lie in a simply connected 4-manifold, but rather in a
connected sum of products , whereas
the extra 2-handles may
be thought of as
being added to .) Do the cancellation, and what
remains of is an unlink of components to
which are attached 2-handles, so that is
diffeomorphic to with boundary
or the twisted case. However, going back to original 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 , construct
.
If we had an algorithm to recognize as a connected sum
of s, then we
would
have an algorithm to decide whether 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 is diffeomorphic to .
Possibly the description of 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 -sphere for .
Kervaire
[e7]
showed that if a finitely presented
group is
superperfect, that is, if
then it is the
fundamental group of a homology -sphere , . If the
-sphere can be recognized, then
if and only if (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.