Download 2-knots and their groups by Jonathan A. Hillman PDF

By Jonathan A. Hillman

To assault convinced difficulties in four-dimensional knot idea the writer attracts on numerous ideas, targeting knots in S^T4, whose basic teams comprise abelian common subgroups. Their classification comprises the main geometrically beautiful and top understood examples. in addition, it really is attainable to use contemporary 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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Example text

11. (We use the hyphen to avoid confusion with the stricter notion of locally-Uinite and normal) subgroup). 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.

The final assertion follows from [CE: page 262]. 0 The above proof for this corollary is from [HW 19851. In fact the finite group "' must have cohomological period 4, as we shall show in Chapter 4. Note that for each ~ 0 D the group is ZX([ ")D a 3-knot group whose commutator subgroup has centre (Z/2Z)D, and so it can only be a 2-knot group if D Hausmann and 0 or 1. = Weinberger also used the invariant q( ) the first examples of homology 5-sphere groups which are not of homology 4-spheres. (They gave to give the groups two infinite families of examples, one consisting of finite groups and the other of torsion free groups).

Proof Let with respect be to MT of M with group T is the cellular chain complex of an equivariant cell MT structure. with coefficients Then O. ) = 0 also. ),R [G IT». ) = 0, and suffice duality to show that H· = O. Let S be the multiplicative system R [UIT]-{O} in RlOIT), and let r = R lOIT)S' nonzero homology of C. S then UIT Since H 3(Hom r (C. s ,B» nontrivial is is HS RS = 0 In degree 2. S ) = 0 the so only is any left r-module Poincare by and Kiinneth theorem. 3) HS duality and the is stably free and we may split the boundary maps of the complex C.

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