INTERSECTION THEORY OF MANIFOLDS WITH OPERATORS WITH APPLICATIONS TO KNOT THEORY

Let 9N be an oriented combinatorial manifold with boundary, let 9)1, be any of its covering complexes, and let G be any free abelian group of covering transformations. The homology groups of 9N* are R-modules, where R denotes the integral group ring of the (multiplicative) group G. Each such homology module H has a well-defined torsion sub-module T, and the corresponding Betti module is B = H/T. The automorphism y -> -1 of the group G extends to a unique automorphism a -a & of the ring R = R. Following Reidemeister [4] there is defined an intersection S which is a pairing of the homology modules of dual dimension to the ring R, and also a linking V which is a pairing of dual torsion sub-modules to Ro/R, where Ro is the quotient field of R. Two duality theorems are proved: (1) S is a primitive pairing to R/rm of dual Betti modules with coefficients modulo tm and Tm respectively, where or is zero or a prime element of R and m is any positive integer. (2) V is a primitive pairing of dual torsion modules to Ro/R. These theorems are analogous to the Burger duality theorems [1]; in case R is the ring of integers, they specialize to the Poincar6-Lefschetz duality theorems for manifolds with boundary. Although dual modules are not, in general, isomorphic, it is demonstrated that if one torsion module has elementary divisors A0 c Al c ... then its dual has elementary divisors Ao C Al C * ... . (The numbering differs from that of [2] in that we begin with the first non-zero ideal.) This result is applied to the maximal abelian covering of a link in a closed 3-manifold. It is proved that the elementary divisors Ai of the 1-dimensional torsion module are symmetric in the sense that Ai = At . For the case of a knot or link in Euclidean 3-space, the ideal A0 is generated by the Alexander polynomial, and the symmetry of A0 has been proved previously by Seifert [6] for knots and Torres [7] for links. The "symmetry" of the Alexander polynomial was proved by Seifert and Torres in a slightly more precise form [8, Cors. 2 and 3]. The problem of similar extra precision in the more general case of a link in an arbitrary closed 3-manifold remains open.