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2-knots and their groups by Jonathan A. Hillman

By Jonathan A. Hillman

To assault convinced difficulties in four-dimensional knot concept the writer attracts on various recommendations, targeting knots in S^T4, whose primary teams comprise abelian general subgroups. Their classification comprises the main geometrically beautiful and most sensible understood examples. additionally, it truly is attainable to use contemporary paintings in algebraic how to those difficulties. New paintings in 4-dimensional topology is utilized in later chapters to the matter of classifying 2-knots.

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2-knots and their groups

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0 44 Localization and Asphericity Theorem Let I be a finitely generated group with IIJ' ;; Z and 4 which has an abelian normal subgroup A of rank at least 2. Then HS(J;ZII]) = Proof By ° for Lemma s ~ 2. E~q the 1 terms HP(JIA;Hq(A;ZII])) => HP+q(J;ZIJ» of vanish the LHS spectral for q ~ 2, if A sequence has rank greater than 2, or if it has rank 2 and is not finitely generated. If A is finitely generated and of rank 2 then we may assume that it is free, and E~q = ° for q ~ 1. I Moreover Theorem 3), so E~2 must H0(JIA ;ZIJIA]) clude that HS(J;ZII» = ° for Corollary If a 2-knot group = be infinite ° also.

Subgroups, For instance, although there may be finitely solvable presentable infinite solvable groups in which no such subgroup is torsion free. In order to get around this problem we may factor out the maximal locally-finite normal subgroup. (This idea is due to Kropholler). The quotient of a 2-knot group by such a subgroup is then usually a PDt-group over Q. Rosset's Lemma The keystone of the argument of this chapter (and hence of the whole book) is the following lemma of Rosset. Localinlion and Asphericity Lemma [Ro 198..

It is easy to see that the a universal cover of such a complex must be contractible, and so must be torsion free, and therefore infinite cyclic. 0 This corollary does sufficient to work with the quick proof that if the group not really commutative IT need Rosset's ring I\. Note Lemma, that for this it is gives a of a nontrivial classical knot K has fin- itely generated commutator subgroup then it has one end. It then follows easily from Poincare duality that is aspherical. X(K) A cyclic branched cover of S3, branched over a knot connected sum of the cyclic branched covers of ITT: rK prime K, is the of K.

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