# Embedding homology 3-spheres in the 4-sphere

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## 1 Problem

Let $\Sigma$$== Problem == ; Let \Sigma be an integral homology -sphere, which is not S^3. Is there a locally flat embedding \Sigma \hookrightarrow S^4 such that one or both complementary regions are not simply-connected? This problem is motivated by the problem of classifying such embeddings up to isotopy. If a complement has non-trivial fundamental group, then a satellite' construction yields infinitely many isotopy classes of embeddings of \Sigma into S^4. This problem was posed by Jonathan Hillman, Monday January 14th at [[:Category:MATRIX 2019 Interactions|MATRIX]]. == References == {{#RefList:}} [[Category:Problems]] [[Category:Questions]] [[Category:Research questions]]\Sigma$ be an integral homology $3$$3$-sphere, which is not $S^3$$S^3$. Is there a locally flat embedding $\Sigma \hookrightarrow S^4$$\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$$\Sigma$ into $S^4$$S^4$.

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