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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 conception the writer attracts on various strategies, targeting knots in S^T4, whose basic teams include abelian common subgroups. Their classification comprises the main geometrically attractive and top understood examples. additionally, it really is attainable to use fresh paintings in algebraic the right 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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We shall consider solvable 2-knot groups in Chapter 6. For the present we shall give a more general result. Theorem 6 Let Tr be the group of a 2-knot K and T be its maximal locally - finite normal subgroup, and suppose Tr has a normal subgroup U such tha t U IT is a nontrivial abelian group. If U IT has rank 1 assume also that e(TrIU) < CD ; if moreover e(TrIU) = 1 assume HS(TrIT;Z[TrIT]) = 0 for s ~ 2. Then either Tr' is TrIT is a PDt-group over Q. furthermore that finite or TrIT ::; 4> or Loca1ization and Asphericity Proof We note first that U 47 is torsion free, by the maximality of T.

G = bgb- 1 for g in is finitely generated as a A-module. Since H via is Z-torsion free AIIII(H) is principal (H: page 351, and since H we must have H if necessary) HIH ::; Z H has rank 1 ::; N(mt -II) for some m, II has a presentation in Z. Thus (after relabeling . Since we must have m-II = ±1. Now I has a subgroup of finite index which maps onto Z abelian kernel A. Therefore if I with is finitely presentable, this subgroup is a constructible solvable group by (BB 19761 and (BS 19781, and so I ally torsion free.

Proof Let P and let be a presentation for G C(P) be the with g generators and r relators, corresponding 2-complex. Then def P = l-X(C(P» PI (C(P>;Q)- P2 (C(P);Q) " PI (O;Q)- P2(O;Q), and the = necessi ty of the condi- tions is clear. G = 2, the image of Z[G]r lemma. G = 2, the in Z[G]g ZIG] - Z is projective, by Schanuel's into Z[G]r splits, and L Hattori-Stallings rank O. of L is projective. is concentrated on the conjugacy class <1> of the identity [Ec 1986], and so the Kaplansky rank of L is Q~Z[G]L the dimension = 0 and so L of = Q~Z[G)L.

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