By J. A. Hillman
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I; an_d E (L) are both principal, (G' = G") then ~ ~ 2; or if G' is perfect 46 (iii) if e = ], then G'/G" has no nontrivial pseudozero submodule and Ann(G'/G") =(AI(L)/A2(L)); (iv) for each k ~ I, ek(tG'/G") ~ = (A~+k_] (L)/Ae+k(e)) 9 Proof (i) We may clearly assume that El(L) ~ 0, so that A(L) has rank I and A ( L ) / t A ( L ) ~ I. 12 the isomorphism class of this ideal is just the Steinitz-Fox-Smythe row invariant of A(L). From the exact sequence (2) we see that the column invariant is the class of the principal ideals, and Steinitz showed that the product of the row and column invariants was the class of the first nonzero elementary ideal .
C I ---+ C O explicit, for ~ as the universal space has a natural q-cells; there is an exact A -resolution Alternatively, the complex of the corresponding All the differentials ~ 7z ---+ 0 to make (3) the maps more (C,) as the tensor product complex over 7z of t-! for S I : 0 --+ A I ---+ A I --+ 0. of the complex (ZZ ( ~ A C,) are 0, so (V) TorA(zg, ZZ) eP2Z = 97. sequences = Hq((SI)~; ZZ) . d. d. d. G'/G" >. p - I . d. A(L) The second of the spectral F(V)'/F(~)" o__r lemma. d. d.
Thus in particular the commutator subgroup of a classical knot group has trivial integral homology in degree greater than I. The Crowell exact sequence for the free group F(~) is 0 ----+F(B)'/F(~)" ---+ A ~ ---+ A ---+ ~ ---+ 0 . The right hand terms constitute a partial resolution for the augmentation 44 module ~. We may obtain a complete equivariant homology of (SI) ~. The latter of ]R~ , considered the lifts to ~ vertices all have integral cellular are the Euclidean coordinates. Since covering space with (~) q of side !
Alexander Ideals of Links by J. A. Hillman