# 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:

1. 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.

2. 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.

3. 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.

4. (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.

• If you connect sum with a manifold that doesn't admit an orientation reversing isometry, then check homeomorphism, then this checks if the manifolds are orientation preserving homeomorphic. However, I'm not sure if the homeo. problem for reducible (or even connect sums of Haken) manifolds is in the literature explicitly. – Ian Agol Sep 19 '12 at 15:42
• Actually, Ian, this is precisely my motivation for asking the question. As far as I can tell, the treatments of the homeomorphism problem for reducible manifolds in the literature miss this point. – HJRW Sep 19 '12 at 16:23
• Certain special cases can be deduced from the literature. If you look at Theorem 6.1.6 in Matveev's book, it says that there is an algorithm to tell if two Haken manifolds with boundary pattern are homeomorphic taking boundary pattern to boundary pattern. If you have two oriented knot complements, then you can tell if they are orientation preserving homeomorphic by choosing a boundary pattern which is not preserved under mirror image (such as a single closed (1,1) curve). However, this is also appealing to the knot complement problem. springerlink.com/content/978-3-540-45898-2 – Ian Agol Sep 20 '12 at 21:53
• That's a very nice observation, Ian! – HJRW Sep 21 '12 at 9:25