The oriented homeomorphism problem for Haken 3-manifolds
The oriented homeomorphism problem for Haken 3-manifolds
Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it's often natural to consider 3-manifolds equipped with an orientation , and so my question is:
Question 1: Is there an algorithm in the literature that determines whether or not a pair of oriented Haken 3-manifolds M and N are orientedly homeomorphic?
By Haken's Algorithm, we can reduce to the case in which M and N are (unorientedly) homeomorphic, and so the question can be reduced to:
Question 2: Is there an algorithm in the literature that determines whether or not an orientable Haken 3-manifold M admits an orientation-reversing self-homeomorphism?
Notes:
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Note the words 'in the literature'. I don't really doubt that this algorithm exists, but the details seem quite complicated, particularly in the non-geometric case, so I hope that someone will be able to supply a reference. A very short argument would also be nice. Answers of the form 'I don't know, but I'm sure it can be done if you work out each geometric case individually and then think hard enough about the JSJ decomposition', while appreciated, would be less useful.
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The hyperbolic case can be deduced from Sela's solution to the homeomorphism problem, which also computes the automorphism group. I've no doubt the other non-Haken cases can be handled similarly. Hence my interest in the Haken case.
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It may be that a simple modification of Haken's Algorithm handles the oriented case. This isn't at all clear to me, but I would be very interested to hear if it's true.
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(added later) I would also be happy to confirm that this question is `open' (in the sense that an answer is not in the literature, rather than that the answer is in doubt)---I very much suspect that this is the case. An authoritative pronouncement by someone who knows the literature very well would therefore be a valid answer.
K2-Think
The oriented homeomorphism problem for Haken 3-manifolds, while not explicitly addressed in the literature with a fully detailed algorithm, can be approached by extending existing unoriented homeomorphism algorithms. However, a direct, published algorithm specifically for the oriented case or explicitly determining orientation-reversing self-homeomorphisms (Question 2) is not widely cited.
Key p...
GPT-5
Yes. For Haken 3‑manifolds there is an algorithm in the literature that decides the oriented homeomorphism problem, and in particular decides whether an orientable Haken 3‑manifold admits an orientation–reversing self‑homeomorphism.
Two complementary ways to see this (both standard in the literature):
- Via mapping class groups and Waldhausen–Johannson
- Waldhausen’s theorem (for sufficiently l...
o3 Pro
Short answer
Yes. Both questions have already been settled in the affirmative.
For an orientable Haken 3-manifold one can algorithmically produce a
finite presentation of its mapping–class group, and from this decide
(a) whether the manifold possesses an orientation–reversing self-homeomorphism, and
(b) whether two given orientable Haken manifolds are homeomorphic by an orientation–preservi...
Gemini 2.5 Pro
This is a fantastic question that gets at the heart of algorithmic 3-manifold topology. The short answer, as you suspect, is that the problem is algorithmically solvable, and all the necessary tools are in the literature, but a single, self-contained reference that explicitly details the algorithm for the general Haken case might not exist. However, the procedure is well-understood by experts.
Le...
o3
Yes – the oriented homeomorphism problem for Haken 3-manifolds is already solved in the literature.
The solution is not stated as one isolated theorem in the original papers of Haken, Hemion, Jaco–Shalen–Johannson, …, but all the ingredients were put together explicitly by
S. V. Matveev, “Algorithmic Topology and Classification of 3-Manifolds”,
Algorithms and Computation in Mathematics 9, Spr...
DeepSeek R1
First, I need to recall what Haken manifolds are. These are compact, irreducible 3-manifolds that con...
o4 mini
Here is what one needs and where it “lives” in the existing Haken‐manifold literature. In fact nothing essentially new is required—one simply carries along orientations in the usual JSJ-plus-hierarchy algorithm and in the recognition of the geometric pieces. I do not know of a single paper whose title and abstract announce “the oriented Haken homeomorphism problem,” but all of the ingredients ha...
Claude 3.7 Sonnet
The Oriented Homeomorphism Problem for Haken 3-Manifolds
This question asks about algorithms for determining whether two oriented Haken 3-manifolds are orientedly homeomorphic, or equivalently, whether a Haken 3-manifold admits an orientation-reversing self-homeomorphism.
State of the Literature
The oriented homeomorphism problem for Haken 3-manifolds is indeed solvable, but explicit algor...
Claude Opus 4
This is an excellent question about the algorithmic decidability of oriented homeomorphism for Haken 3-manifolds. Let me address what is known about this problem.
Current State of Knowledge
As far as I'm aware, there is no published algorithm in the literature that specifically addresses the oriented homeomorphism problem for Haken 3-manifolds in full generality. This appears to be an ope...
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