Homology 3-spheres in codimension three

Abstract

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.

References in corpus (1)

• The Hopf invariant of a Haefliger knot

Cited by in corpus (5)

• A classification of smooth embeddings of 3-manifolds in 6-space
• Classification of embeddings below the metastable dimension
• Classification of smooth embeddings of 4-manifolds in 7-space, I
• The Hopf invariant of a Haefliger knot
• An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number