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By J. A. Hillman

ISBN-10: 0387111689

ISBN-13: 9780387111681

ISBN-10: 3540111689

ISBN-13: 9783540111689

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There is an alternative description of the Crowell exact sequence in terms of the link group which we shall give as we shall use it in the next chapter. Let G be a finitely generated group and let eG : ZZ[G] § the augmentation homomorphism of the group ring of G. be the augmentation ideal, generated by { g - l ] g the ideal generated by { w - I ]w in G' } . (g- I) + ~ 2 ~= ker eG in G } , and let ~ 2 be The map sending gG' to gives an isomorphism of G/G' with ~ / ~ 2 . module of G, A(G) = ~ / ~ 2 ~ ' Let Zg be The Alexander is in a natural way a ZZEG/G' ]-module, which is finitely generated and hence finitely presentable.

R ~:(ArM)/t(~M) Since R is torsion free there results a map + R which clearly has image the ideal generated by the (q-r) x (q-r) minors of U, which are just the elements of one row of Q(q-r). Since both domain and image of ~ are rank ! torsion free modules, ~ gives an isomorphism of (ArM)/t(Ar M) with this ideal. // The projection of M onto M/tM induces an epimorphism ArM + ~r(M/tM) and hence p(M) = o(M/tM). p(N). Thus p is a homomorphism from the semigroup of finitely generated R-modules (with respect to direct sum) to the semigroup of ideal classes (with respect to product of ideals).

There are ~ and ~ in A such that ~u = By. no common factor. Then since rank Hi(X;A) = l, We may assume that ~ and B have Since fi is factorial v = ~w for some w in A n , which must actually be in H2(X;A) by the exactness of is torsion free. (2) and the fact that A n+l Therefore every 2-generator submodule of the finitely generated rank 1 A-module H2(X;A) is cyclic. The lemma follows easily. // Cochran's result extended to embeddings of arbitrary finite graphs and was published in [ 3 0 ] . In general H2(X;A) is free if and only if the projective dimension of A(L) is at most 2.

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Alexander Ideals of Links by J. A. Hillman


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