A Classification of Smooth Embeddings of 4-manifolds in 7-space, I

We work in the smooth category. Let N be a closed connected n-manifold and assume that m > n + 2. Denote by E(N) the set of embeddings N → R up to isotopy. The group E(S) acts on E(N) by embedded connected summation of a manifold and a sphere. If E(S) is non-zero (which often happens for 2m < 3n + 4) then until recently no results on this action and no complete description of E(N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R7 for certain 4-manifolds N. The proofs use new approach based on the Kreck modified surgery theory and the construction of a new invariant. Corollary. (a) There is a unique embedding f : CP 2 → R7 up to isoposition (i.e. for each two embeddings f, f ′ : CP 2 → R7 there is a diffeomorphism h : R7 → R7 such that f ′ = h ◦ f). (b) For each embedding f : CP 2 → R7 and each non-trivial embedding g : S4 → R7 the embedding f#g is isotopic to f .