A 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 summation of a manifold and a sphere. If E m ( S n ) is non-zero (which often happens for 2 m < 3 n + 4 ) then until recently no complete readily calculable description of E m ( N ) or of this action were known (as far as I know). Our main results are examples of the triviality and the effectiveness of this action, and a complete readily calculable isotopy classification of embeddings into R 7 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 : C P 2 → R 7 up to isoposition ( i.e. for each two embeddings f , f ′ : C P 2 → R 7 there is a diffeomorphism h : R 7 → R 7 such that f ′ = h ○ f ) . (b) For each embedding f : C P 2 → R 7 and each non-trivial embedding g : S 4 → R 7 the embedding f # g is isotopic to f.