By J. P. Levine

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This quantity involves the lawsuits of a convention held on the college collage of North Wales (Bangor) in July of 1979. It assembles study papers which replicate various currents in low-dimensional topology. The topology of 3-manifolds, hyperbolic geometry and knot thought become significant topics.

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A0 A, (and, therefore, A 0 = Ker~: Conversely, if A ÷ A~ ~ which we must take note of. is a n-primary A-module and i-th lower n-derivative of only if K then every A Ai) the is Z-torsion free if and is is a submodule of is a non-zero Ai Z-torsion free. A, necessity Z-torsion element of A, is 21 In §27ff. we will consider the problem of deciding which polynomials in A = Z[t, t -1] are Dedekind. 1. A finitely generated Z-torsion free module I(1) satisfies = ±i. The n e c e s s i t y of the condition follows On the other hand, = (t - i)~ ± i.

L free module of type §8. K. Z - t o r s i o n - f r e e modules If (~) is the annihilator minimal polynomial of A. ideal of A, we refer to For each prime factor consider the T - p r i m a r y s e q u e n c e s of n of l l, as a we may A: 0 + Ai+ 1 ÷ A i ÷ A i + A i+l ÷ 0 c o n s t r u c t e d in §i where derivatives of A i = 0, factors of l 7, i, n. and the T-primary This sequences Therefore to consider ~ - p r i m a r y modules, ring for large enough follows is the sum of its n - p r i m a r y T-primary submodule.

Is step, recall that ' #i+l a lift of and elementary of modules isomorphic A For the inductive . We now construct We f i r s t while the second is induced by ¢i+l = ~i+l wh°se composition . §13. Homogeneous §9). ~, IB/ I+I B ÷ I A / T i + I A , multiplication T-primary by is certainly . ~i > k-i Ag (see n-primary is a free ef A §7). A-module is A / ( T d) module. if and only if every non-zero is finitely generated The degree of A and all have the is the largest d 33 Proof: Consider A /(~) = Q(R), space over dimension 0 2 i < d, the quotient Q(R).