Embedding homology 3-spheres in the 4-sphere

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1 Problem

Let \Sigma be an integral homology 3-sphere, which is not S^3. Is there a locally flat embedding \Sigma \hookrightarrow S^4 such that one or both complementary regions are not simply-connected?

This problem is motivated by the problem of classifying such embeddings up to isotopy. If a complement has non-trivial fundamental group, then a `satellite' construction yields infinitely many isotopy classes of embeddings of \Sigma into S^4.

This problem was posed by Jonathan Hillman, Monday January 14th at MATRIX.

2 References

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