Characterization of knot complements in the n-sphere

• Autorzy: Vo-Thanh Liem , Gerard A. Venema
• Języki publikacji: EN

Abstrakty

EN

Knot complements in the n-sphere are characterized. A connected open subset W of $S^n$ is homeomorphic with the complement of a locally flat (n-2)-sphere in $S^n$, n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of $S^1$ in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.

Słowa kluczowe

EN

knot; n-sphere; complement; homotopy groups of end

Kategorie tematyczne

• 57N35: Embeddings and immersions
• 57N15: Topology of E n , n -manifolds ( 4 n ∞ )
• 57Q45: Knots and links (in high dimensions)
• 57N45: Flatness and tameness

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1995
• Tom: 147
• Numer: 2
• Strony: 189-196

Daty

wydano

1995

otrzymano

1994-10-12

Twórcy

autor

Vo-Thanh Liem

• epartment of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350, U.S.A.

autor

Gerard A. Venema

• Department of Mathematics and Computer Science, Calvin College, Grand Rapids, Michigan 49546, U.S.A.
• Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003, U.S.A.

Bibliografia

• [1] H. Bass, Projective modules over free groups are free, J. Algebra 1 (1964), 367-373.
• [2] W. Browder, J. Levine and G. R. Livesay, Finding a boundary for an open manifold, Amer. J. Math. 87 (1965), 1017-1028.
• [3] M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76.
• [4] R. J. Daverman, Homotopy classification of locally flat codimension two spheres, Amer. J. Math. 98 (1976), 367-374.
• [5] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453.
• [6] V. T. Liem, Homotopy characterization of weakly flat knots, Fund. Math. 102 (1979), 61-72.
• [7] V. T. Liem and G. A. Venema, Characterization of knot complements in the 4-sphere, Topology Appl. 42 (1991), 231-245.
• [8] V. T. Liem and G. A. Venema, On the asphericity of knot complements, Canad. J. Math. 45 (1993), 340-356.
• [9] F. Quinn, Ends of maps, III: dimensions 4 and 5, J. Differential Geom. 17 (1982), 503-521.
• [10] L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. dissertation, Princeton Univ., Princeton, N.J., 1965.
• [11] S. Smale, Generalized Poincaré's conjecture in dimensions >4, Ann. of Math. 74 (1961), 391-466.
• [12] P. F. Smith, A note on idempotent ideals in group rings, Arch. Math. (Basel) 27 (1976), 22-27.
• [13] G. A. Venema, Duality on noncompact manifolds and complements of topological knots, Proc. Amer. Math. Soc., to appear.
• [14] G. A. Venema, Local homotopy properties of topological embeddings in codimension two, in: Proc. 1993 Georgia Internat. Topology Conf., to appear.
• [15] C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), 56-69.
• [16] J. H. C. Whitehead, Combinatorial homotopy, I, Bull. Amer. Math. Soc. 55 (1949), 213-245.