On the integral homology of infinite cyclic coverings of links
Akio Kawauchi · Institutional Repositories DataBase (IRDB) · 1987
We consider a polygonal oriented link £ in the Euclidean 3-space R 3 .Let E(e) = R 3 -£.We have a unique infinite cyclic covering space E( £) over E( £) associated with the epimorphism y: n 1 (E(£))---+(t) sending each meridian element*J of £ to t, where (t) is the infinite cyclic group generated by a letter t.For the integral group ring A=Z(t), the integral homology H 1 (E(£)), which we denote by H(£), forms a finitely generated A-module.Throughout this paper, A-modules will mean finitely generated A-modules, unless otherwise stated.For a A-module H, we use the following notations: DH =the (unique)the A-torsion part of H, BH =H/TH, fJH =rankA H, eH =the minimal number of elements generating H over A, and EqH =Ext 1(H, A).Note that D 0 H = (t-l)NDH for a large positive integer N and hence t-1: D 0 H;;;D 0 H.By convention, eH =0 if H =0. It is known that EqH =0 for q ~ 3 and there are natural A-isomorphisms E 2 H;;;E2DH;;;Hom z(DH, Q/Z) and H has the A-projective dimension s 1 if and only if DH =0 (cf.[Ka, §3] 4 , Levine [L]).By the identity A(t) • x = A(t-1 )x for A(t) EA and x EH, H has another A-module structure.We denote this A-module by H.When H =H(£), we denote DH, D 0 H, TH, BH, fJH and eH by D(e), D 0 (£), T(£), B(£), /3(£) and e(£), respectively.Letµ(£) be the number of components of £.It is well-known that /J( £) s µ( £)-1 and the equality holds for, e.g., a slice link £ in the strong sense (cf.[Ka] 1 ).Our first purpose is to observe that there are many links £ with D(e)#0.Actually, we characterize Do(£) for all links £.THEOREM I.For all links £ we have eE 2 D 0 (£)sfJ(£)sµ(£)-1.Conversely, given a finite A-module D with t-1: D;;;D, then for any integers r, s with eE2Dsrss we have a link£ such that µ(£)=s+l, fJ(£)=r and D(£)= D 0 (£);;;D.*) "a meridian element oft" means "an element of 11: 1 (E(t)) represented by a loop homotopic to a meridian of .e"(The orientation of a meridian of .e is uniquely spec,fie l by those of .e and R 3 ).