2-knots and their groups by Jonathan A. Hillman

By Jonathan A. Hillman

To assault sure difficulties in four-dimensional knot idea the writer attracts on various options, concentrating on knots in S^T4, whose primary teams include abelian common subgroups. Their type comprises the main geometrically attractive and top understood examples. furthermore, it really is attainable to use fresh paintings in algebraic tips on 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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Lemma 2 [Ke 1965] Let A be a t -1 acts in vertibly. is a A- torsion module, and the subgroup zA is a Then A Z - torsion finite A-submodule. Proof If we tensor A as A finitely generated A-module on which with the field of fractions QCt) of A we ge t 0, and is finitely generated it must be a torsion module. It is clear that is a submodule, and that t -1 acts invertibly on zA. Moreover since A zA is noetherian zA is also finitely generated, and so has finite exponent m as an abelian group. Suppose first that m is prime.

The quotient of a 2-knot group by such a subgroup is then usually a PDt-group over Q. Rosset's Lemma The keystone of the argument of this chapter (and hence of the whole book) is the following lemma of Rosset. Localinlion and Asphericity Lemma [Ro 198.. ) I" Let G be a group witb a torsion free abcliu normal subgroup A, and let S be tbe multiplicative system Z[A]-{O} In ZIG]. Tben tbe (noncentraJ/) localization R = S-1Z[G] exists and bas tbe property tbat eacb nontrivial finitely generated stably free R -module bas well defined strictly positive rank, witb R n baving rank n.

Since e 2 zA is a finitely generated Z-torsion the sequence A-module on which t -1 acts invertiblY, it is finite by Lemma 2. 0 We have assumed that t -1 acts invertibly, as it simplifies our argument, but the corollary remains true without this assnmption. Theorem 3 [Fa 1977, Le 1977) Let K be Ii 2-knot with group 7F and '" 7F'/7F" = H 1(M(K);A). Then H 2 (M(K);A) ~ e 1 A, and there is a nondegenerate Z-bilinear pairing [ , l:zAXzA - QIZ for which t acts let A as an isometry: [ta,tpl = [a,pl for all a and p in zA.

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