Given a triangulated closed compact 3-manifold $M^3$, is there an algorithm to compute a presentation for $\pi_1(M)$? Is there an algorithm to determine if for a fixed group $G$, we have $\pi_1(M) \cong G$?
Algorithm to compute $\pi_1(M^3)$
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algebraic-topology
geometric-topology
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11) is easy, since you can just consider it as a CW complex and use a standard algorithm given in say Hatcher. 2) is harder, but still true. It's more or less equivalent to the classification of 3-manifolds. – 2017-01-29
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0The answer to 2) is no; it's already an undecidable problem to determine if a group (as given by a finite presentation) is isomorphic to the fundamental group of $S^3$, namely the trivial group. – 2017-01-29
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3@QiaochuYuan That's not quite correct the way I interpreted the question. The groups we're working with here are not general groups, but rather 3-manifold groups, which is restrictive enough that the isomorphism problem becomes solvable. So one should be careful and say that $G$ is also a 3-manifold group, rather than being described by an arbitrary finite presentation, or of course you're right. – 2017-01-29
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0If $G$ has decidable word problem, there is an algorithm to decide if $G$ is isomorphic to the fundamental group of a hyperbolic 3-manifold. This is probably even true without assuming hyperbolicity. I will eventually find a reference. – 2017-02-23