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

By Jonathan A. Hillman

ISBN-10: 0521371732

ISBN-13: 9780521371735

ISBN-10: 0521378125

ISBN-13: 9780521378123

To assault definite difficulties in four-dimensional knot conception the writer attracts on quite a few suggestions, concentrating on knots in S^T4, whose primary teams include abelian general subgroups. Their type includes the main geometrically beautiful and most sensible understood examples. in addition, it's 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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Extra resources for 2-knots and their groups

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0 Groups abelianization Z with are cohomological dimension 2, nontrivial centre and iterated free products of (one or more) torus knot groups, amalgamated over central subgroups [St 1976]. Acyclic covering spaees Our second application of Rosset's Lemma leads ultimately to more substantial results on 2-knot groups. Theorem 3 Let M be a closed connected orientable 4-manifold with fundamental group G. Suppose tbat tbere are normal subgroups T < U of G and a subring R of trivial torsion a such that Hom(TIT ,R) = 0, U IT is a non- free abelian group and HS(GIT;R [GIT» - 0 for s ~ 2.

For fibred ribbon knots this follows from an argument of Trace. If V an n -knot K q of is any Seifert hypersurface for then the embedding of V in X extends to an embedding of == VuDn +1 in M, which lifts to an embedding in M'. Since the image V H n +1(M;Z) in is Poincare dual to Hom(7I',Z) == 1M,S1], its image in H n +1(M';Z) fibred is homotopy M' degree 1 map from equivalent to a generator of H 1(M;Z) is a generator. If K Z the closed fibre P, so there is is a q to P and hence to any factor of P.

Theorem only if Let S IT IT be a 3-knot group. Then q(rr) ~ 0, and is 0 if and is a 2-knot group. 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.

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2-knots and their groups by Jonathan A. Hillman

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