The Homology Cobordism Group

Abstract

Let Σ 0 and Σ 1 be integral homology 3-spheres. They are said to be homology cobordant, or H-cobordant, if there exists a smooth compact oriented 4-manifold W with boundary ∂W= -Σ 0 ∪ Σ 1 such that the inclusions Σ i → W induce isomorphisms H *(Σ i , ℤ) → H *(W, ℤ), i = 0, 1. Equivalently, Σ 0 and Σ 1 are homology cobordant provided (-Σ 0) # Σ 1 bounds a smooth acyclic 4-manifold, that is a smooth compact oriented 4-manifold W with H *(W, ℤ) = H *(D 4, ℤ). Note that we do not require that π;1(W) = 0, only that H 1(W, ℤ) = 0. A homology sphere Σ is said to be homology cobordant to zero if it is homology cobordant to S 3 or, equivalently, bounds a smooth acyclic 4-manifold. Note that the requirement that the homology cobordism be smooth is essential — it is known [98] that any integral homology 3-sphere bounds a topological compact oriented 4-manifold which is acyclic.