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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.
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