Unsolvability of word problem for 2-knot complements

Importance: High ✭✭✭
Author(s): Gordon
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: July 23rd, 2010
Problem   Does there exist a smooth/PL embedding of $ S^2 $ in $ S^4 $ such that the fundamental group of the complement has an unsolvable word problem?

It's known that there are smooth $ 4 $-dimensional submanifolds of $ S^4 $ whose fundamental groups have unsolvable word problems. The complements of classical knots ($ S^1 \to S^3 $) are known to have solvable word problems, as do arbitrary $ 3 $-manifold groups.


A. Dranisnikov, D. Repovs, "Embeddings up to homotopy type in Euclidean Space" Bull. Austral. Math. Soc (1993).

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