By Jonathan A. Hillman

ISBN-10: 0521371732

ISBN-13: 9780521371735

ISBN-10: 0521378125

ISBN-13: 9780521378123

To assault yes difficulties in four-dimensional knot idea the writer attracts on numerous thoughts, concentrating on knots in S^T4, whose primary teams comprise abelian general subgroups. Their category includes the main geometrically attractive and top understood examples. furthermore, it truly is attainable to use fresh paintings in algebraic the way 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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Corollary If and B is a PDn -group over 0 and B have homological dimension over 0 tion = K is a 2-knot sucb tbat M(K) is aspberical tben K is not a nontrivial satellite knot. In particular, K is irreducible. Proof Let K 1 and K 2 be two 2-knots, and let "1 be an element of TrK 1· If "1 has finite order let q be that order; otherwise let q = O. Let a me ridian in Tr K 2. CTrK l' where by van Kampen's Theorem the amalgamation is over C = Z/qZ is identified with "1 in TrK 1 [Ka 1983J. (K 2;K 1,"1) = and w 52 Chaptcr 4 THE RANK 1 CASE It is a well known consequence of the asphericity of the comple- ments of classical knots that classical knot groups are torsion free.

Clearly there are no nontrivial maps from such a group to a torsion free group such as Q. This notion is of particular interest in connection with solvable groups. 111, if 0 is a finitely genera ted infinite solvable group and T maximal locally-finite normal subgroup then OIT is nontrivial. it has a nontrivial abelian normal subgroup, which is is its Therefore necessarily torsion free. Thus we may apply the theorems of the preceeding two sections to 4-manifolds with such groups. We shall consider solvable 2-knot groups in Chapter 6.

Hausmann and Kervaire have shown that every finitely generated abelian group A centre of some n -knot group, for each n that if P is a finitely presentable ~ 3 [HK 1978'1. superperfect group and is the They observe G is a knot group then by the KUnneth theorem Hi(GXP;Z) = HlG;Z) for i " 2, and If moreover there is an c1ement P in GxP has centre ((G XP) = CG XCP. P such that the subgroup [p,PI generated by {(p,xl = pxp- 1 x- 1 :x in P} is the weight whole of P, and if g element for erally if P l' in Pi Pi) CG (i)CP 1(i).

### 2-knots and their groups by Jonathan A. Hillman

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