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 E 7(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) → H 2(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 fo..