N ov 2 00 0 SYMMETRY OF LINKS AND CLASSIFICATION OF LENS SPACES

We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ‘symmetric’ links in S. Let T1 and T2 be solid tori, and let mi and li be the meridian and longitude of Ti (i = 1, 2). The lens space L(p, q) is a 3-manifold that is obtained from T1 and T2 by identifying their boundaries in such a way that m2 = pl1+ qm1 and l2 = ql1+ rm1, where (p, q) = 1 and qq − pr = 1. In 1935, Reidemeister classified lens spaces up to orientation-preserving PL homeomorphisms [9]. This classification was generalized to the topological category with the proof of the Hauptvermutung by Moise in 1952 [7]. Meanwhile, Fox had outlined an aproach to classification up to homeomorphisms which would not require the Hauptvermutung; see [4, Problem 2], [5]. This was implemented later by Brody [3]. We refer the reader to [6] for history of classifications of lens spaces. In this paper, we give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. Our method is motivated by that of FoxBrody. While the chief ingredient in their proof was a study of the Alexander polynomial of knots in lens spaces, we study the Alexander polynomial of ‘symmetric’ links in S. For an oriented 3-manifold M with finite first homology group, the linking form lkM : H1(M ;Z)×H1(M ;Z) −→ Q/Z is defined as follows [1], [2]. Let x and y be 1-cycles in M that represent elements [x] and [y] of H1(M ;Z) respectively. Suppose that nx bounds a 2-chain c for some n ∈ Z. Then lkM ([x], [y]) = c · y n ∈ Q/Z, where c · y is the intersection number of c and y. Let ∆K(t) be the Conway-normalized Alexander polynomial of K, i.e., ∆K(t) = ∇K(t −1/2 − t), where ∇K(z) is the Conway polynomial. Theorem 1. Let ρ : S −→ L(p, q) be the p-fold cyclic cover and K a knot in L(p, q) that represents a generator of H1(L(p, q);Z). If ∆ρ−1(K)(t) = 1, then lkL(p,q)([K], [K]) = q/p or = q/p in Q/Z. Before proving Theorem 1, we obtain the classification of lens spaces as its corollary. 1991 Mathematics Subject Classification. Primary 57M27; Secondary 57M25.