Knotted tori and the normal bordism betainvariant

1
Knotted tori and the normal bordism beta-invariant

• This talk is on the classification of embeddings
• S x S ? S
• up to an isotopy. An isotopy class is a knotted
  torus.
• Motivation. This problem is interesting because
• It provided many interesting examples (Hudson,
  Milgram, Rees, A. Skopenkov)
• It generalizes the theory of 2-component links.
• It can be considered as the next step towards the
  classification of embeddings of arbitrary
  manifolds.

2
Classical results

• The range of triviality m p2q2 there is a
  unique knotted torus Whitney-Haefliger theorem.
  The case m p2q1 Hudson.
• The metastable range m p q2 there is a
  complete Haefliger-Wu a-invariant Haefliger,
  Zeeman, A. Skopenkov. The case m p q - A.
  Skopenkov, a new ß-invariant.
• The 2-metastable range m gt p q2 for p 0
  Haefliger for arbitrary p 0 and m 2pq2
  below.

3
Main concrete result

• Theorem A. Suppose m gt p q2 and m 2pq2 ().
  Then the set of smooth embeddings S x S ? S up
  to a smooth isotopy is infinite for
• m 2q1, q odd
• m p2q1, q even (Hudson series)
• 2m 2p3q3, q 3(mod 4)
• 2m 3p3q3, pq 3(mod 4)
• and is finite otherwise (if () holds).
• (New for m lt p q2)
• Example the set of knotted tori S¹ x S ? S is
  finite.

4
Main theoretical result

• an analogue of the Koschorke exact sequence,
  which reduces the classification of embeddings
  S x S ? S to the (easier) classification of
  almost embeddings S x S ? S.
• Notation.
• KT is the group of almost smooth embeddings
  S x S ?S up to an almost smooth isotopy
• KT is the group of almost embeddings S x S ?S
  up to an almost isotopy
• O p (V ), is the
  normal bordism group, where V is the Stiefel
  manifold of j-frames in R and Ngtgt0.
• Theorem B. For m gt p q2 and m 2pq2 ()
  there is an exact sequence
• O ? KT ? KT ? O ?
• Relation to the Koschorke sequence. If D ? ? D
  in the definition of an almost embedding, then
  Theorem B for p 0 ? the Koschorke sequence.

5
Commutative group structure

• Operation S parametric connected sum.
• Well-defined for m 2pq2 and m pq3
  (A. Skopenkov).
• In addition to Theorem A we give
• The rank of the group of knotted tori
• An explicite construction of its rational
  generators.

6
Main notions

• Almost smooth embeddings. A PL map is almost
  smooth, if it is smooth outside a fixed
  codimension 0 ball. Motivation local Haefliger
  knots S ? S can be removed in almost smooth
  cathegory.
• Almost embeddings. A PL map F S x S ? S is an
  almost embedding, if
• F is a smooth embedding outside a fixed
  codimension 0 ball D
• FD n F(S x S D ) ?.

7
Further investigation

• Explicite classification results. For example,
  find the group of embeddings S x S ? S up to an
  isotopy.
• Arbitrary manifolds. Generalize the ß-invariant
  and Theorem B to embeddings and almost embeddings
  of arbitrary manifolds.
• Approximability by embeddings. A version of
  Theorem B for p 0 can be applied to construct
  maps S ? S ? R not approximable by embeddings.
• Coincidence theory.
• Contact
• skopenkov_at_rambler.ru,
• Matija.Cencelj_at_fmf.uni-lj.si,
• Dusan.Repovs_at_fmf.uni-lj.si.

8
The beta-invariant

• History
• Haefliger, Hudson, Sato-Levine, Kirk,
• Normal bordism ß-invariant of link maps
  Koschorke, Habegger, Kaiser.
• integer ß-invariant of knotted tori A.
  Skopenkov.
• Proof of completeness is based on obtaining the
  ß-invariant from the homotopy information of the
  complement cf. Habegger, Kaiser, knot groups
• Idea of the construction. Take an almost
  embedding F G S S ? S. Then
• ß(F,G) ? lk(FC ,G) (mod2).