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Organizers |
Decision Problems for Manifolds
by
Alexey V. Chernavsky
Institute of the Information Transmission Problems RAS
Coauthors: V. P. Lexin
A.A. Markov in his well-known article [1] (see also [2]) had constructed unrecognizable pl-manifolds of dimensions n ³ 4. He based himself on a series of finite presentated groups given by S.I. Adian [3]. According to [4] for such manifold in dimension n one may take the connected sum of thirteen products S2×Sn-2, n ³ 4. These manifolds we will call the Markov manifolds Mn.
There was presented in [5] the S.P. Novikov's result which states that the spheres of dimensions n ³ 5 are unrecognizable. In [6] we have shown that the unrecognizability of any compact pl-manifold of dimension n ³ 5 follows from this result. We used the Rabin argument based on the Groushko theorem completed with a homological consideration.
We prove by similar argument that any four-dimensional pl-manifold becomes topologically unrecognizable after some stabilization. We also give an example of unsolvability of some decision problem in the class of 3-manifold groups.
We call the stabilization of a four-dimensional pl-manifold N its connected sum N# M4 with the Markov manifold M4=13#(S2×S2).
Let a compact four-dimensional manifold N be given. In the class of compact four-dimensional pl-manifolds the decision problem for the topological type of the stabilization of N is unsolvable.
According to Stallings [7], the property of the finitely presented group to belong to the class of the fundamental groups of the three-dimensional compact manifolds is markovian. It follows that the question, if a finitely presented group is fundamental group of a three-dimensional compact manifold, is algorithmically undecidable.
If we are inside the class of fundamental groups of the 3-manifolds the following takes place.
Theorem. Let us consider the class of all couples (G, H), where G=p(Q) is fundamental group of some compact 3-manifold Q and H is its finitely generated free subgroup. In this class the question if the normalizer of H coincides with the whole group G is algorithmically undecidable.
The work of the second author was supported by RFBR grant no. 02-01-00721.
[1] A.A.Markov.) Doklady, 121 (1958), 218-220.
[2] A.A.Markov.) Doklady, 123 (1958), 978-980.
[3] S.I.Adian, I.G.Durnev.) Uspekhi, 2000, 55, 3-94.
[4] M.A.Shtan'ko. Izvestia RAS, (2003), (In press)
[5] I.A.Volodin, V.E.Kuznetsov, A.T.Fomenko.) Uspekhi, 32, (1977), 71-168.
[6] V.P.Lexin, A.V.Chernavsky.) Doklady, (2003) (in press).
[7] J.R. Stallings On the recursiveness of presentations of 3-manifold groups. Fund.Math., 51 (1982), 191-194.
Date received: April 3, 2003
Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cajy-25.