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The classification of simple knots. (English. Russian original) Zbl 0546.57006

Russ. Math. Surv. 38, No. 5, 63-117 (1983); translation from Usp. Mat. Nauk 38, No. 5 (233), 59-106 (1983).
A simple n-knot is an oriented submanifold K of \(S^{n+2}\) which is homeomorphic to \(S^ n\) and which bounds an \([n-1/2]-\)connected submanifold V of \(S^{n+2}\). Simple odd-dimensional knots (\(n\geq 5)\) were classified by J. Levine by means of the ”S-equivalence class” of the Seifert pairing on the middle dimensional homology of the Seifert manifold V [Comment. Math. Helv. 45, 185-198 (1970; Zbl 0211.559)]. An equivalent classification in terms of the Blanchfield pairing of the knot was later given by C. Kearton [Trans. Am. Math. Soc. 202, 141-160 (1975; Zbl 0305.57016)]. (The latter invariant is intrinsic and direct in that it does not depend on choosing a Seifert matrix and then passing to an equivalence class.) The principal result of this paper is the classification of simple even-dimensional knots (\(n\geq 8)\) by means of ”\(\Lambda\) -quintuples”. If \(\epsilon =\pm 1,\) a \(\Lambda\) -quintuple of parity \(\epsilon\) consists of finitely generated modules A,B over the Laurent polynomial ring \(\Lambda ={\mathbb{Z}}[t,t^{-1}]\), nondegenerate \(\epsilon\) -symmetric bilinear pairings \(\ell: T(A)\otimes T(A)\to {\mathbb{Q}}/{\mathbb{Z}}\) and \(\psi\) : \(B\otimes B\to {\mathbb{Z}}/4{\mathbb{Z}}\) (where T(A) is the \({\mathbb{Z}}\)-torsion submodule of A) and a monomorphism \(\alpha\) : A/2\(A\to B\). It is assumed furthermore that \(t-1\) acts invertibly on A and B, that t acts on T(A) and B as an isometry for \(\ell\) and \(\psi\) respectively and that the sequence \[ 0\to A/2A\to^{\alpha}B\to^{\beta}Hom_{{\mathbb{Z}}}(A,{\mathbb{Z}}/2{\mathbb{Z}})\to 0 \] is exact (where \(\beta (b)(a)=\psi (b\otimes\alpha (a+2A))).\)
A 2q-knot K determines a \(\Lambda\) -quintuple \((H_ q(\tilde X),\sigma_{q+2}(\tilde X),\alpha,\ell,\psi)\) where \(H_ q(\tilde X)\) is the qth integral homology of the infinite cyclic cover \(\tilde X\) of the knot complement \(X=S^{n+2}\backslash K,\) and where \(\sigma_{q+2}(\tilde X)\) is the \((q+2)^{th}\) stable homotopy group of \(\tilde X\), considered as \(\Lambda\) -modules via the action of the covering group Aut\((\tilde X/X)\). (We shall not define the map \(\alpha\) or the pairings \(\ell\) and \(\psi\) here.) Theorem 10.15 then asserts that (1) this quintuple is a \(\Lambda\) -quintuple of parity \((-1)^{q+1};\) (2) isotopic knots determine isomorphic quintuples; (3) for \(q>3\), simple 2q-knots with isomorphic quintuples are isotopic; and (4) for any \(q>3\), every \(\Lambda\) -quintuple of parity \((-1)^{q+1}\) may be realized by some simple 2q-knot. This theorem is a nontrivial application of earlier work of the author on the classification of stable knots by means of pairings in stable homotopy theory [(*) \(=\) Trans. Am. Math. Soc. 261, 185-210 (1980; Zbl 0513.57007)]. It includes earlier partial results of C. Kearton [Topology 15, 363-373 (1976; Zbl 0335.57012) - no 2- torsion in \(H_ q(\tilde X)\), Arch. Math 35, 391-393 (1980; Zbl 0443.57016) - no torsion in \(H_ q(\tilde X)]\), S. Kojima [Comment. Math. Helv. 54, 356-367 (1979; Zbl 0417.57008) - \(H_ q(\tilde X)\) finite of odd order] and the author [Mat. Sb., Nov. Ser. 115 (157), 223- 262 (1981; Zbl 0483.57013) - \(H_ q(\tilde X)\) finitely generated over \({\mathbb{Z}}\), in other words K a fibred knot].
Although the major new results in this paper are those on simple even- dimensional knots summarized above, considerable space is devoted to a reconsideration of the odd-dimensional case in order to emphasize that greater unity of approach to both cases and simpler arguments follow from use of the concept of R-equivalence introduced by the author in (*). Two Seifert manifolds V and W for a knot K are contiguous if Int V and Int W are disjoint; R-equivalence for Seifert manifolds is the equivalence relation generated by contiguity. On the algebraic level two modules over the ring \({\mathbb{Z}}[z]\) are R-equivalent if they become isomorphic after localization with respect to powers of \(z(1-z).\) (In the knot-theoretic context the ring \({\mathbb{Z}}[z,z^{-1},(1-z)^{-1}]\) is to be identified with the ring \(\Lambda [(1-t)^{-1}]\) via the isomorphism sending z to \((1-t)^{-1}.)\) Finally the author remarks that although the paper uses the language of differential topology the results hold also for simple topologically locally flat n-knots for the same range of dimensions \((n=5\) or \(n\geq 7)\). \(\{\) Reviewers remark: In fact the Blanchfield pairing also classifies simple TOP 3-knots, while a classification for fibred 4-knots may be derived from work of D. Barden [Ann. Math., II. Ser. 82, 365-385 (1965; Zbl 0136.206)] on simply connected 5- manifold\(s\}\).
Reviewer: J.Hillman

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
55Q10 Stable homotopy groups
11E16 General binary quadratic forms
57M10 Covering spaces and low-dimensional topology
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