By Jonathan A. Hillman

To assault definite difficulties in four-dimensional knot idea the writer attracts on quite a few concepts, concentrating on knots in S^T4, whose basic teams comprise abelian general subgroups. Their type comprises the main geometrically attractive and most sensible understood examples. additionally, it truly is attainable to use contemporary paintings in algebraic how 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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**Sample text**

Thus W is contractible if and only if X(W) - O. 0 The assumption that to show that W be 2-dimensional is not needed in order W aspherical implies that X(W) = 0; this is in fact Rosset's application of his lemma. Gottlieb obtained the first such result in the case of an aspherical complex whose fundamental group had nontrivial centre [Go 1965]. The lemmas of Kaplansky and Rosset have been used in related ways in connection with the Whitehead conjecture on the asphericity of subcomplexes of 2-dimensional K(G ,1)-complexes (cf.

Therefore if I with is finitely presentable, this subgroup is a constructible solvable group by (BB 19761 and (BS 19781, and so I ally torsion free. We may then assume that zA = O. Moreover H is virtuis then also finitely presentable, and so it has an equivalent presentation of the form for some sufficiently large T. By a Reidemeister-Schreier rewriting process (MKS: page 1581 we get a presentation for H. ZI(m+l)Z as a subgroup. with presentation But this group maps onto the group ** via the ·map sending a 0 to c , a r+l to b and a I' . **

Cochran has shown that if the closed fibre is #r(S1XS2) the knot is homotopy ribbon, and that conversely the closed fibre of a fib red homotopy ribbon 2-knot is a connected sum of copies of S1XS 2 with a homotopy 3-sphere [Co 1983]. 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.