When given embedding  N → R6  of a 3-manifold  N  is isotopic to an embedding  f : N → R6  such that  f(N) ⊂ R5 ? This is a particular case of a classical compression problem in topology of manifolds studied by Haefliger, Hirsch, Gillman, Tindell, Vrabec, Rourke–Sanderson, Cencelj–Repovs, and others. The particular case is interesting because some 3-manifolds appear in the theory of integrable Hamiltonian systems together with their embeddings into  R5 .

Embeddings of closed orientable 3-manifolds into  R6  are classified by the author in http://arxiv.org/math.GT/0603429 in terms of the Whitney invariant  W(f) ∈ H1(N)  and the Kreck invariant  ηW(f)(f) ∈ Z / d(W(f)) , where  d(W(f))  is the divisibility of the projection of  W(f)  to  H1(N) / torsion . The values of the Whitney invariants of embeddings  f : N → R6  such that  f(N) ⊂ R5  were essentially described by O.Saeki, A.Szucs, and M.Takase.

The main result of this talk is a description of the values of the Kreck invariant of embeddings  f : N → R6  such that  f(N) ⊂ R5 . This gives a complete description of such embeddings. The proof is based on a formula for the Kreck invariant in terms of Seifert surfaces (analogous to the M.Takase formula for the Haefliger invariant).

In this talk I shall present explicit examples of embeddings, give definitions of the invariants, state the main result and sketch its proof.