Knots, i.e. embeddings of spheres

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1 Introduction

See general introduction on embeddings, notation and conventions in [Skopenkov2016c, $\S$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\S$1, $\S$$\S$3].

2 Examples

Analogously to the Haefliger trefoil knot for $k>1$$k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$$t:S^{2k-1}\to\Rr^{3k}$. For $k$$k$ even this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz$$E_D^{3k}(S^{2k-1})\cong\Zz$; it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962]. It would be interesting to know if for $k$$k$ odd this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$$E_D^{3k}(S^{2k-1})\cong\Zz_2$. The last phrase of [Haefliger1962t] suggests that this is true for $k=3$$k=3$.

3 Readily calculable classification

For $m\ge n+3$$m\ge n+3$ the group $E^m(S^n)$$E^m(S^n)$ has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs $(SG_n,SO_n)$$(SG_n,SO_n)$ for $n = n_i+1$$n = n_i+1$ and $n_i$$n_i$ [Haefliger1966a], cf. [Levine1965], [Milgram1972], [Habegger1986]. Here $SG_n$$SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$$f \colon S^{n-1} \to S^{n-1}$ of degree $1$$1$. Restricting an element of $SO_n$$SO_n$ to $S^{n-1} \subset \Rr^n$$S^{n-1} \subset \Rr^n$ identifies $SO_n$$SO_n$ as a subspace of $G_n$$G_n$.

Some readily calculable corollaries of this classification are recalled in [Skopenkov2006, $\S$$\S$3.3].

4 Codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in $\Rr^3$$\Rr^3$), a complete readily calculable classification (in the sense of Remark 1.1 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots, most of which does not present a readily calculable classification. See e.g. interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a], [Cappell&Shaneson1974] and [Ranicki1998].

5 References

• [Cappell&Shaneson1974] S. E. Cappell and J. L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277–348. MR0339216 (49 #3978) Zbl 0279.57011
• [Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
• [Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

• [Farber1984] M. Sh. Farber, An algebraic classification of some even-dimensional spherical knots I, II, Trans. Amer. Math. Soc. 281 (1984), 507-528; 529-570.
• [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244

• [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
• [Kearton1983] C. Kearton, An algebraic classification of certain simple even-dimensional knots, Trans. Amer. Math. Soc. 176 (1983), 1–53.
• [Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
• [Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
• [Milgram1972] R. J. Milgram, On the Haefliger knot groups, Bull. of the Amer. Math. Soc., 78:5 (1972) 861--865.
• [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.