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

\begin{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? \end{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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