Homology 3-spheres in codimension three

Details Homology 3-spheres in codimension three Homology 3-spheres in codimension three

2005-06-22

arxiv.org/abs/math/0506464v4

International Journal of Mathematics 17 (2006) pp.869-885

Mathematics

Geometric Topology

Scientific paper

10.1142/S0129167X06003771

For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classes of all embeddings of homology 3-spheres in the 6-sphere. As a consequence, we show that two embeddings of an oriented integral homology 3-sphere in the 6-sphere are isotopic if and only if they are homology bordant. We also relate our invariant to the Rohlin invariant and accordingly characterise those embeddings which are compressible into the 5-sphere.

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