×

Nœuds Fox-résiduellement nilpotents et rigidité virtuelle des variétés hyperboliques de dimension 3. (Fox-residually nilpotent knots and virtual rigidity of hyperbolic 3-manifolds). (French) Zbl 0899.57008

Given a knot \(k\) in a 3-manifold \(M\), such that \(k\) represents a nontrivial element of \(\pi_1 (M)\), which is not a power \((k\) is primitive), the author defines the “Fox group” of \(k\). The Fox group of \(k\), \(\text{Fox} (M,k)\) is \(\pi_1 (\widetilde M-\text{int} \widetilde {N(k)})\), where the \(\widetilde{\;}\) means universal covering and \(N(k)\) is a regular neighborhood of \(k\). \(\text{Fox}(M,k)\) is the kernel of the map induced by inclusion of \(\pi_1 (M-N(k))\) in \(\pi_1 (M)\). If \(\text{Fox} (M,k)\) is residually nilpotent (the intersection of the lower central series is 1) then \(k\) is said to be Fox-residually nilpotent.
The author gives a new proof of Gabai’s theorem: If \(M,N\) are homotopy equivalent orientable, irreducible closed 3-manifolds and \(M\) is hyperbolic, then there are coverings of \(M\) and \(N\) of the same degree so that the lift of the homotopy equivalence to these coverings is homotopic to a homeomorphism.
A more general theorem is proved; namely if \(f:M\to N\) is a homotopy equivalence between orientable, closed 3-manifolds with infinite fundamental group and there exists in \(N\) a primitive nontrivial knot \(k\) such that \(f^{-1} (k)=k'\) is a satellite knot of a knot which is Fox-residually nilpotent, then \(f\) is homotopic to a homeomorphism. The proof uses standard techniques of algebra and topology in dimension 3. That Gabai’s theorem follows from the author’s depends on a result of T. Sakai.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M10 Covering spaces and low-dimensional topology
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] [Du] , Nœuds géométriquement-libres et rigidité topologique des variétés de dimension 3, Thèse, Université Toulouse-III, 1996.
[2] [Ep] , The degree of map, Proc. London Math. Soc., (3) 16 (1966), 369-383. · Zbl 0148.43103
[3] [Ga1] , Homotopy hyperbolic 3-manifolds are virtually hyperbolic, J. Amer. Math. Soc., 7 (1994), 193-198. · Zbl 0801.57009
[4] [Ga2] , Foliations and the topology of 3-manifolds, J. Differential Geom., 18 (1983), 445-503. · Zbl 0533.57013
[5] [GMT] , , , Homotopy Hyperbolic 3-manifolds are hyperbolic, preprint. · Zbl 1052.57019
[6] [He] , 3-Manifolds, Ann. of Math. Stud., vol 86, Princeton Univ. Press, Princeton NJ, 1976. · Zbl 0345.57001
[7] [Hi] , Differential topology, Springer-Verlag, Berlin-Heidelberg-New York, 1976. · Zbl 0356.57001
[8] [HS] , , Homotopy equivalence and homeomorphism of 3-manifolds, Topology, 31 (1992), 493-517. · Zbl 0771.57007
[9] [Ja] , Lecture on 3-manifold topology, CBMS Lecture Notes, No. 43, Amer. Math. Soc., Providence, RI, 1980. · Zbl 0433.57001
[10] [Jo] , Homotopy equivalences of 3-manifolds with boundaries, Lecture Note in Math. Vol. 761 (Springer, Berlin-Heidelberg-New York, 1979). · Zbl 0412.57007
[11] [JS] , , Seifert fibred spaces in 3-manifolds, Mem. Amer. Math. Soc., 220 (1980). · Zbl 0415.57005
[12] [Kr] , A note on centrality in 3-manifold groups, School of Math. Sci., Queen Mary College, London E1 4NS (1989), 261-266. · Zbl 0709.57011
[13] [Ma] , Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann., 111 (1935). · JFM 61.0102.02
[14] [Ma1] , On isomorphic matrix representations of infinite groups, Math. Sb., 8 (50) (1940), 405-422. · Zbl 0025.00804
[15] [Sa] , Geodesic knots in hyperbolic 3-manifold, Kobe J. Math., 8 (1991), 81-87. · Zbl 0749.57003
[16] [Sc1] , The geometry of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487. · Zbl 0561.57001
[17] [2] , Nilvariétés projectives, Comment. Math. Helvetici, 69 (1994), 447-473. · Zbl 0516.57006
[18] [St] , Homology and Central Series of Groups, J. Algebra, 2 (1965), 170-181. · Zbl 0135.05201
[19] [On2] , Lie groups and Lie algebras 1. Foundations of Lie theory, Lie transformations groups, Springer Verlag, 1993
[20] [Th2] , Three dimensionnal manifolds, Kleinian groups, and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982) 357-381. · Zbl 0496.57005
[21] [Wa1] , On irreducible 3-manifolds which are sufficiently large, Ann. of Math., (2) 87 (1968), 56-88. · Zbl 0157.30603
[22] [Wa2] , On the determination of some bounded 3-manifold by their fundamental groups alone, Proc. of Int. Sym. of Topology, Hercy-Novi, Yugoslavia, 1968 : Beograd (1969), 331-332. · Zbl 0202.54702
[23] [Wr] , Monotone mappings and degree one mappings between pl manifolds, Geometry Topology (Proc. Conf., Park City, Utah, 1974), Lecture Note in Math. Vol 438, Springer, Berlin (1975) 441-459. · Zbl 0309.55006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.