Decision problems for 3-manifolds and their fundamental groups

Abstract

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (word problem, conjugacy problem, isomorphism problem), and also the homeomorphism problem for 3-manifolds and the membership problem for 3-manifold groups. 57M05 IntroductionThe classical group-theoretic decision problems were formulated by Max Dehn in his work on the topology of surfaces [23] about a century ago. He considered the following questions about finite presentations hA j Ri for a group :(1) The word problem, which asks for an algorithm to determine whether a word on the generators A represents the identity element of .(2) The conjugacy problem, which asks for an algorithm to determine whether two words on the generators A represent conjugate elements of .(3) The isomorphism problem, which asks for an algorithm to determine whether two given finite presentations represent isomorphic groups.Viewing as the fundamental group of a topological space (represented as a simplicial complex, say), these questions can be thought of as asking for algorithms to determine whether a given loop is null-homotopic, whether a given pair of loops is freely homotopic, and whether two aspherical spaces are homotopy-equivalent, respectively. We add some further questions that arise naturally: (4) The homeomorphism problem, which asks for an algorithm to determine whether two given triangulated manifolds are homeomorphic.(5) The membership problem, where the goal is to determine whether a given element of a group lies in a specified subgroup.Since the 1950s, it has been known that problems (1)- (5) , as a corollary of the unsolvability of (3).In contrast to this, in this paper we show that all these problems can now be solved for compact 3-manifolds and their fundamental groups, with the caveats that in (3) we restrict ourselves to closed, orientable 3-manifolds, and in (4) we restrict ourselves to orientable, irreducible 3-manifolds. Contrary to common perception (see [89, Section 7.1]) the homeomorphism problem for 3-manifolds is still open for reducible 3-manifolds. We discuss the status of the homeomorphism problem in detail in Section 4.5.The solutions to the first four problems, with the aforementioned caveats, have been known to the experts for a while, and as is to be expected, they all rely on the geometrization theorem. An algorithm for solving the membership problem was recently given in [28], and we will provide a summary of the main ideas in this paper. The solution to the membership problem requires not only the geometrization theorem but also the tameness theorem of Once a decision problem has been shown to be solvable, a natural next question concerns its complexity, and its implementability. The complexity of decision problems around 3-manifolds is a fascinating topic which is not touched upon in the present paper, except for a few references to the literature here and there. Practical implementations of algorithms for 3-manifolds are disc...

Computational complexity and 3–manifolds and zombies

“…To our knowledge, Theorem 1.1 is a new result for the invariant #H(Γ, G), even though we specifically construct Γ to be a 3-manifold group rather than a general finitely presented group. For comparison, both the non-triviality problem and the word problem are as difficult as the halting problem for general Γ [42], while the word problem and the isomorphism problem are both recursive for 3-manifold groups [3,27].…”

Deciding Contractibility of a Non-Simple Curve on the Boundary of a 3-Manifold

“…This is much more general than surfaces, and in general 3-dimensional topology is extremely more intricate than surface topology; for example, the classification of surfaces has been known for a century, while that of 3-manifolds requires, e.g., the proof of the Poincaré conjecture by Perelman [20,21]. The general contractibility problem for 3-manifolds is known to be decidable, but the complexity of the problem has not been made explicit; see, e.g., Aschenbrenner et al [3,Section 4.1], [2]. Most of the literature concentrates on different measures of complexity of the contractibility problem (e.g., automaticity and isoperimetric inequalities [8]).…”

“…Then the the trivial algorithm for solving the word problem, see [8, Theorem 2.2.5, Lemma 2.2.4], is at least triply exponential in the worst case. See also [3].…”

2-Stratifold groups have solvable word problem

“…For example, fundamental groups of 3-manifolds are residually finite, but there are simple 2-stratifolds with non-residually finite fundamental group. Since 3-manifold groups have solvable word problem ( [1]), the question arises whether this is true for 2-stratifold groups. The main goal of this paper is to prove that this is indeed the case.…”