# Embedding homology 3-spheres in the 4-sphere

From Manifold Atlas

## [edit] 1 Problem

Let be an integral homology -sphere, which is not . Is there a locally flat embedding 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 into .

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