Classification of smooth embeddings of 4-manifolds in 7-space, I

Abstract

We work in the smooth category. Let N be a closed connected n-manifold and assume that m>n+2. Denote by E^m(N) the set of embeddings N -> R^m up to isotopy. The group E^m(S^n) acts on E^m(N) by embedded connected sum of a manifold and a sphere. If E^m(S^n) is non-zero (which often happens for 2m<3n+4) then no results on this action and no complete description of E^m(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 R^7 for certain 4-manifolds N. The proofs are based on the Kreck modification of surgery theory and on construction of a new embedding invariant. Corollary. (a) There is a unique embedding CP^2 -> R^7 up to isoposition. (b) For each embedding f : CP^2 -> R^7 and each non-trivial knot g : S^4 -> R^7 the embedding f#g is isotopic to f.