By L.P., ed. Neuwirth
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Extra info for Knots, Groups and 3-Manifolds: Papers Dedicated to the Memory of R.H. Fox.
In a manner entirely analogous to the us ual linking pairing in the Z-torsion part of the homology of a manifold. This is just the Blanchfield pairing (see [B], [Ke], [T 2]). Under the 1 canonical isomorphism Hom/\. )/A) :::::: ExtA (A, A), for any A-torsion lIIodule A, the isomorphism (4) is adjoint to the Blanchfield pairing (which vanishes on Z-torsion). )/A satisfying the Hermitian property: = (_l)qtl <,B,a> . e. fta, t,B1 = fa, ,B]. In the case of a fibered knot (see [S]) T q is the Z-torsion subgroup of Hq(F), where F is the fiber, and [,] coincides with the usual linking pairing on H*(F).
Little appears to be known about the higher dimensional homotopy groups. e. \e; Specifically, 36 S. J. LOMONACO, JR. THEOREM 1. Let (S4, k(S2)) be defined as in Theorem 0 above. ' denotes a functor defined by]. H. C. Whitehead [10, 11] and later 77 generalized by Eilenberg and MacLane [12, 13]. Hence, Z77 1 -module is determined by 2. , 3 without Z77 1 -structure) is free abelian of infinite rank. Otherwise, THEOREM 3. 77 3 = O. Let k(S2) C S4 be a 2-sphere formed by spinning an arc a about the standard 2-sphere S2 and (x 1 , ..
The polynomial 1>(z) can be written as t/J(w) ~ w 2 - llw w co z(1-z). Then ( t- 6, where is a root of (2 + ( + w where uJ is a root of t/J, and K = Q(w) is a quadratic subfield of L = Q«(). The discriminant of Kover Q is 97 and of Lover K is 1-4uJ, which has norm 53. Since both discriminants are square-free, R" zl(l is the full ring of integers in L and its ideal classes form a group. The prime 2 factors into the ideals A - [2, 1 t (], B of norm 2), and C·. [2,(2 t (+1] (of norm 4). AB /\ S = (1 + () are principal ideals, so A "'" B- 1 -= [2, (], (both (7(2 -7(+4) and = and A S "'" 1.
Knots, Groups and 3-Manifolds: Papers Dedicated to the Memory of R.H. Fox. by L.P., ed. Neuwirth