On the Homology Cobordism Group of Homology 3-spheres

In this paper we present our results on the homology cobordism group Θ3Z of the oriented integral homology 3-spheres. We specially emphasize the role played in the subject by the gauge theory including Floer homology and invariants by Donaldson and Seiberg – Witten. A closed oriented 3-manifold Σ is said to be an integral homology sphere if it has the same integral homology as the 3-sphere S. Two homology spheres Σ0 and Σ1 are homology cobordant, if there is a smooth compact oriented 4-manifold W with ∂W = Σ0∪−Σ1 such that H∗(W,Σ0;Z) = H∗(W,Σ1;Z) = 0. The set of all homology cobordism classes forms an abelian group Θ3Z with the group operation defined by a connected sum. Here, the zero element is homology cobordism class of 3-sphere S, and the additive inverse is obtained by a reverse of orientation. The Rochlin invariant μ is an epimorphism μ : Θ3Z → Z2, defined by the formula μ(Σ) = 1 8 sign(W ) mod 2, where W is any smooth simply connected parallelizable compact manifold with ∂W = Σ. This invariant is well-defined due to well-known Rochlin theorem [R] which states that the signature sign(V ) of any smooth simply connected closed parallelizable manifold V is divisible by 16. We will focus our attention on the following problem concerning the group Θ3Z and the homomorphism μ : does there exist an element of order two in Θ3Z with non-trivial Rochlin invariant ? This is one of R. Kirby problems [K], Problem 4.4, and positive answer to this problem would imply [GS], in particular, that all closed topological n-manifolds are simplicially triangulable if n ≥ 5 ( this is not true in dimension 4 where a counterexample is due to A. Casson and M. Freedman, see e.g. [AK] ). Our approach to this problem is trying to lift the Rochlin homomorphism to integers. If we succeded in doing this, then all elements of finite order would have to lie in the kernel of μ, and the Kirby problem would have a negative solution. In fact, our goal is less ambitious – we define a lift of μ on a certain subgroup of Θ3Z. This implies that there are no solutions to the problem inside this subgroup. This is done in Section 1 with help of the μ̄–invariant introduced by W. Neumann and L. Siebenmann in 1978, see [N] and [Sb], for the so called plumbed homology spheres. 1