2-knots and their groups by Jonathan A. Hillman

By Jonathan A. Hillman

To assault sure difficulties in four-dimensional knot conception the writer attracts on numerous recommendations, targeting knots in S^T4, whose primary teams include abelian basic subgroups. Their type comprises the main geometrically attractive and top understood examples. additionally, it really is attainable to use fresh paintings in algebraic ways 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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Example text

Equivalently, by duality in the universal cover, either 1TK ~ Z or it has one end). As X is a 3-manifold with nonempty boun- dary, it collapses to a finite 2-complex [N: Chapter IIIl. In particular, " has cohomological dimension at most 2. Moreover, " has a Wirtinger presenta tion of deficiency , in presen ta tion which each of the relation form asserts the conjugacy of two of the generators. In this section we shall examine some of the connections be tween these properties.

Admits some such pair of subgroups) if and only if it satisfies conditions (ii) and (iii) of Theorem D. 0 We shall not give proofs for these results, as we present them only for contrast group of a with nontrivial the lower dimensional cases. The commutator sub- I-knot group has a nonabelian free subgroup (the image of the fundamental group of an incompressible Seifert surface for the knot) and so is never abelian. 11. shall see then it that if is either the commutator subgroup of a 2-knot group Z3 or Z[~1, the dyadic We is abelian rationals, or is cyclic of odd order.

Clear. If = 0 q(lT) then there is a closed 4-manifold M with and X(M) = O. Surgery on a weight class then gives a simply IT connected closed 4-manifold L with X(O .. 2, which must therefore be S4, and the cocore of the surgery is a 2-knot with group clear, for if K is any 2-knot then X(M(K» Let Corollary IT prime p subgroup that IT' IT' is cyclic, and so of finite. Then every has periodic cohomology. has a non cyclic abelian subgroup. Then there is a and an abelian subgroup A H IT The coverse is IT.

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