Cobordism classification of knotted homology 3-spheres in S^5

Abstract

In [8], odd dimensional knot cobordism groups C2n^ (n'2>2) were studied and given a purely algebraic description. C2 Λ_j is the set of all cobordism classes of knotted (2n— l)-spheres in the (2n+l)-sphere, and it is an abelian group under connected sum. In this paper we extend this to knotted homology 3-spheres in the 5-sphere and consider their homology cobordism classes. The set of all such homology cobordism classes, denoted by Cy is also an abelian group under connected sum. We shall describe the group Cf using the usual homology cobordism group of homology 3-spheres and Levine's cobordism group of certain matrices (Theorem 1.1). Our argument heavily depends on that of J. Levine. In §3 we introduce the bounding genus and the Murasugi number of a knotted homology 3-sphere in S*. Using our techniques we can estimate these numerical invariants from above. In §4 we consider the case of algebraic 3knots which appear around isolated singularities of complex hypersurfaces in C (see [11]). For example, we shall show that two algebraic 3-knots which are homology 3-spheres and defined by weighted homogeneous polynomials are homology cobordant if and only if they are isotopic. Throughout the paper, we shall work in the C°° category. Homology groups will always be with integral coefficients. The author would like to thank Professor S. Fukuhara for suggesting the problem. He is also grateful to Professor F. Michel who was kind enough to send us a copy of S. Akbulut's note.