Abstract
The main result of this paper is a four-dimensional stable version of Kneser’s conjecture on the splitting of three-manifolds as connected sums. Namely, let M be a topological respectively smooth compact connected four-manifold (with orientation or Spin-structure). Suppose that π1(M) splits as ∗i=1Γi such that the image of π1(C) in π1(M) is subconjugated to some Γi for each component C of ∂M . Then M is stably homeomorphic respectively diffeomorphic (preserving the orientation or Spin-structure) to a connected sum ]i=1Mi with Γi = π1(Mi). Stably means that one allows additional connected sums with some copies of S2 × S2 on both sides. We also prove a uniqueness statement. As a consequence we obtain the existence and uniqueness of the stable prime decomposition of compact connected four-manifolds (with orientation or Spin-structure). The main technical ingredients are the bordism approach to the stable classification of manifolds due to the first author and the Kurosh Subgroup Theorem.
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