By Jonathan A. Hillman
To assault definite difficulties in four-dimensional knot concept the writer attracts on numerous suggestions, targeting knots in S^T4, whose basic teams comprise abelian common subgroups. Their category comprises the main geometrically beautiful and top understood examples. in addition, it really is attainable to use fresh paintings in algebraic easy methods 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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Extra resources for 2-knots and their groups
0 Tr has a nontrivial torsion free abelian normal subgroup A then the rank of A is at most 4. Proof This is an immediate consequence of Theorems 3 and 4. 0 In Chapter 2 we saw that every finitely generated abelian group is the centre of some 3-knot group. The rank 1 case is somewhat more delicate. In the next theorem we shall let F(a ,tY' denote the set of all words in the second commutator subgroup of the free group on (a ,0. As in Chapter 2, we shall Ie t zA denote the Z -torsion subgroup of an abelian group A.
Ency I-m [Le 1978]. There are in fact irreduci- ble 2-knots whose groups have deficiency I-m, for each m Farber gave presentation the following . Then "' is cyclic of order 5, t-l acts as the identity and H 2("';Z) = 0, so " is a 3-knot group. ,p in rr'. Thus t cannot be an isometry and so rr is not a 2-knot group. Centres Since the commutator subgroup of a classical knot group rr can contain no nontrivial abelian normal subgroup [N: Chapters IV,VI, any nontrivial abelian normal subgroup of rr must map injectively to rr/rr' = Z and so be central.
The lemma now follows on applying the LHS spectral sequence HP(C;Hq(B;F» => HP+q(A;F). 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.