A CLASSIFICATION OF SMOOTH EMBEDDINGS OF FOUR-MANIFOLDS IN SEVEN-SPACE, II

Abstract

Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0. Our main result is the following classification of the set E7(N) of smooth embeddings N → ℝ7 up to smooth isotopy. Haefliger proved that E7(S4) together with the connected sum operation is a group isomorphic to ℤ12. This group acts on E7(N) by an embedded connected sum. Boéchat and Haefliger constructed an invariant ℵ: E7(N) → H2(N;ℤ) which is injective on the orbit space of this action; they also described im (ℵ). We determine the orbits of the action: for u ∈ im (ℵ) the number of elements in ℵ-1(u) is GCD (u/2, 12) if u is divisible by 2, or is GCD(u, 3) if u is not divisible by 2. The proof is based on Kreck's modified formulation of surgery.