The Linking Pairings of Orientable Seifert Manifolds

We compute the p-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free Saction. Using this, we show that any nonsingular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However there are pairings on inhomogeneous 2-groups which are not realizable. In [3] we computed the linking pairings of oriented 3-manifolds which are Seifert fibred over non-orientable base orbifolds. Here we shall consider the remaining case, when the base orbifold is also orientable. Thus the Seifert fibration is induced by a fixed-point free S-action on the manifold. (We shall henceforth call such a space a “Seifert manifold”, for brevity.) We give presentations for the localization of the torsion at a prime p in §2, and use these to give explicit formulae for the localized linking pairings in §3. We then study the cases p odd and p = 2 separately, in §§3-6 and §§7-8, respectively. Every nonsingular pairing on a finite abelian group whose 2-primary subgroup is isomorphic to (Z/2Z) (for some k,m) is the linking pairing of a Seifert manifold with geometry H × E, and also of one with geometry S̃L. However if the 2-primary subgroup has exponent 2 but is inhomogeneous the restrictions of the pairing to direct summands of exponent properly dividing 2 must be odd. The final section §9 summarizes briefly the earlier work of Oh on the Witt classes of such pairings [6]. 1. Bilinear pairings A linking pairing on a finite abelian group N is a symmetric bilinear function l : N × N → Q/Z which is nonsingular in the sense that l̃ : n 7→ l(−, n) defines an isomorphism from N to Hom(N,Q/Z). If L is a subgroup of N then l̃ induces an isomorphism L = {t ∈ N | lM(t, l) = 0 ∀l ∈ L} ∼= N/L. Such a pairing splits uniquely as the orthogonal sum (over primes p) of its restrictions to the p-primary 1991 Mathematics Subject Classification. 57M27, 57N10.