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Classification of knotted tori

A. Skopenkov

Abstract

For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to\mathbb R^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge 2(p+q)+\max\{p,q\}+4$. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For $m\ge2p+q+3$ we introduce an abelian group structure on $E^m(S^p\times S^q)$ and describe this group `up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe $E^m(S^p\times S^q)$. The proof is based on relations between sets $E^m(N)$ for different $N$ and $m$, in particular, on a recent exact sequence of M. Skopenkov.

Classification of knotted tori

Abstract

For a smooth manifold denote by the set of smooth isotopy classes of smooth embeddings . A description of the set was known only for or for , or for . (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For we introduce an abelian group structure on and describe this group `up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe . The proof is based on relations between sets for different and , in particular, on a recent exact sequence of M. Skopenkov.

Paper Structure

This paper contains 13 sections, 14 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Some general motivations
  3. Statements of main results
  4. Corollaries
  5. Calculations, proofs of corollaries, remarks
  6. Proofs
  7. Standardization and group structure
  8. Proof of Theorem \ref{['t:main']} using Lemmas \ref{['l:stand']} and \ref{['t:grst']}
  9. Proof of the Standardization Lemma \ref{['l:stand']}
  10. Proof of the Group Structure Lemma \ref{['t:grst']}
  11. Proof of the Smoothing Lemma \ref{['t:smo']}
  12. Appendix: new direct proof of Corollary \ref{['t:corlam']}.a
  13. Appendix: new direct proof of Lemma \ref{['l:exact']}.a

Key Result

Lemma 1.1

For $m\ge2p+q+3$ we have $E^m(S^p\times S^q)\cong E^m_\#(S^p\times S^q)\oplus E^m(S^{p+q})$ .

Figures (3)

  • Figure 1: To the proof of the Standardization Lemma \ref{['l:stand']}.a for $X=S^p$
  • Figure 2: To the proof that $\overline\sigma$ is a homomorphism. This picture illustrates the proof by the case $p=0$, $q=1$ and $m=3$ (these values are not within the dimension range $m\ge2p+q+3$). The part above plane $ABCD$ stands for $\widehat{D^m_+}$. The part below plane $A'B'C'D'$ stands for $\widehat{D^m_-}$. The part between the planes stands for $S^{m-1}\times D^1$. The upper curved lines stand for $f_+(S^p\times S^{q-1})=u(S^p\times S^{q-1})$. The bottom curved lines stand for $f_-(S^p\times S^{q-1})=u(S^p\times S^{q-1})$. The union of segments $A'A,B'B,C'C$ and $D'D$ stands for $u(S^p\times S^{q-1}\times D^1)$. The union of segments $A'A$ and $B'B$ stands for $\mathop{\fam0 i}(S^p\times 1_{q-1})\times D^1$. The quadrilateral $A'ABB'$ stands for the 'surgery disk' $\mathop{\fam0 i}(D^{p+1}\times D^{q-1}_+)\times D^1$. The union of the upper curved lines and the segment $AB$ stands for the $(p+q)$-disk $\Delta_+$. Analogously for $\Delta_-$. The union of $\Delta_+,\Delta_-$ and the segments $C'C$ and $D'D$ stands for the $(p+q)$-sphere that is the image of a representative of $\overline\sigma[u]$. The union of $\Delta_+$ and $CD$ stands for $\Sigma_+$. Analogously for $\Sigma_-$. The quadrilateral $C'CDD'$ stands for the tube $\mathop{\fam0 i}(D^{p+1}\times D^{q-1}_-)\times D^1$.
  • Figure 3: To the proofs that $\ker\sigma'\subset\mathop{\fam0 im}\lambda'$ and $\sigma'\lambda'=0$. The cube stands for $S^m\times I$; its horizontal faces stand for $S^m\times k$, $k=0,1$. The ellipses stand for $\mathop{\fam0 i}(T^{p,q})\times k$, $k=0,1$. The curved line $P'Q'$ stands for $g(D^p_-\times S^q)\times1$. The curved lines $PP'$ and $QQ'$ stand for the image of $\partial D^p_-\times S^q\times I$ under the isotopy between standard embeddings. The points $M,N,M',N'$ stand for $\mathop{\fam0 i}(\pm1_p\times S^q)\times k$, $k=0,1$.

Theorems & Definitions (27)

  • Lemma 1.1: Smoothing; proved in §\ref{['0prosmho']}
  • Theorem 1.2: proved in §\ref{['0modulo']}
  • Conjecture 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof : Proof of Corollary \ref{['t:cornum']}.d
  • proof : Proof of Corollaries \ref{['t:corlam']}.b,c,d
  • proof : Proof of Corollary \ref{['t:corlam']}.b'
  • ...and 17 more