Montesinos Amilibia, José María and González Acuña, Francisco Javier (1983) Quasiaspherical knots with infinitely many ends. Commentarii Mathematici Helvetici , 58 (2). pp. 257-263. ISSN 0010-2571
Restricted to Repository staff only until 31 December 2020.
Official URL: http://www.springerlink.com/content/q3x151q700053g45/
A smooth n-knot K in Sn+2 is said to be quasiaspherical if Hn+1(U)=0, where U is the universal cover of the exterior of K. Let G be the group of K and H the subgroup generated by a meridian. Then (G,H) is said to be unsplittable if G does not have a free product with amalgamation decomposition A∗FB with F finite and H contained in A. The authors prove that K is quasiaspherical if and only if (G,H) is unsplittable. If the group of K has a finite number of ends, then K is quasiaspherical and it was conjectured by the reviewer [J. Pure Appl. Algebra 20 (1981), no. 3, 317–324; MR0604323 (82j:57019)] that the converse was true. The authors give a very nice and useful method of constructing knots in Sn+2 and apply this method to produce counterexamples to the conjecture.
|Uncontrolled Keywords:||quasiaspherical n-knot; knot group; free product with amalgamation over a finite group; HNN-extension over a finite subgroup; infinitely many ends|
|Subjects:||Sciences > Mathematics > Topology|
BIERI, R.,Mayer-Vietoris sequences for HNN-groups and homological duality, Math. Z.143 (1975) 123–130.
GONZÁLEZ-ACUÑA, F., andMONTESINOS, J. M.,Ends of knot groups, Annals of Math.108 (1978) 91–96.
LOMONACO, S.,The homotopy groups of knots I; how to compute the algebraic 2-type, Pacific J. Math.95 (1981) 349–390.
RATCLIFFE, J.,On the ends of higher dimensional knot groups, J. Pure and Appl. Alg.20 (1981) 317–324
STALLINGS, J.,Group theory and three-dimensional manifolds, New Haven and London, Yale University Press (1971).
SWAN, R. G.,Groups of cohomological dimension one, Journal of Algebra12 (1969) 585–601.
SWARUP, A.,An unknotting criterion, Journal of Pure and Applied Algebra6 (1975) 291–296.
WALL, C. T. C.,Pairs of relative cohomological dimension one, Journal of Pure and Applied Algebra1 (1971) 141–154.
SERRE, J. P. Arbres, Amalgames, Sl 2, Asterisque46 (1977).
DUNWOODY, M. J.,Accessibility and groups of cohomological dimension one, Proc. London Math. Soc.38 (1979), 193–215.
|Deposited On:||23 Nov 2012 11:59|
|Last Modified:||23 Nov 2012 11:59|
Repository Staff Only: item control page