Download e-book for kindle: 2-knots and their groups by Jonathan A. Hillman

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 various concepts, targeting knots in S^T4, whose primary teams comprise abelian common subgroups. Their type includes the main geometrically attractive and most sensible understood examples. additionally, it truly is attainable to use contemporary 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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Proof Since PI (lTjF) = I and P2(lTjF) = 0 for any field F, the first assertion is lTI(M) .. 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.

The 0 has = 0 49 Localization and Aspbericity Suppose Corollary tbat G is a PDt-group over Q and tbat tbe cobo- mology ring H-(N;Q) is generated by Hl(N;Q). Tben N is aspberical. Proof The classifying map N from K(G,1) to is clearly a rational (co)homology equivalence and so the theorem applies. 0 Theorem Let K be a 2-knot wbose group Tr is a PDt-group over a 8 field F sucb tbat Hl(Tr';F) O. & tbe classifying map f to K(Tr,1) induces a bomology equivalence from M(K) witb coefficients F. Proof The infinite cyclic covers M and K(Tr,lY "" K(Tr',l) each satisfy Mil- nor 3 duality f':M - of K(rr',1) f'l:Hl(M;F) formal dimension f of classifies Hl(Tr';F) tr' with coefficients f' l:Hl(Tr';F) and F and Therefore the induced Hl(M;F) are also Trl(M).

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.

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2-knots and their groups by Jonathan A. Hillman

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