The Topological Classification of the Lens Spaces

Abstract

The n-dimensional lens spaces can be classified under semi-linear equivalence by the torsion invariant of Reidemeister [1]. It seems desirable to prove the topological invariance of this classification without reference to the Hauptverinutung. In this paper we carry out such a proof for n = 3, using a method which, intuitively speaking, distinguishes manifolds by the differences in their knot theories. This method was proposed by Fox [16, p. 247] at the Princeton Bicentennial Conference of 1946, and later applied [17, p. 455] to re-prove the semi-linear classification of the lens spaces. Associated with every simple closed polygon k in a complex K there is a principal ideal (A) of the 1-dimensional Betti group ring of K k; A is known as the Alexander polynomial of k. [2, 3, 4, 5]. Let M be an orientable triangulable 3-dimensional manifold, y an element of H1(M), k a simple closed curve which represents y and which is polygonal in some triangulation of M. Let A be the Alexander polynomial of k, T1(M k) and B1(M k) the torsion subgroup and Betti group of H1(M -k), i: H1(M k) H1(M) the injectionhomomorphism, and *: B1(M k) H1(M)/iT1(M k) the homomorphism induced by i. Then we prove: (i) H1(M k) and iT1(M k) depend only upon e; (ii) A* depends only upon e. Unlike the Reidemeister torsion, the invariant A* is applicable to all orientable 3-dimensional manifolds. However, if one is interested only in the lens spaces, one may restrict oneself to the case H1(M k) infinite cyclic, which allows considerable simplification of the proof (cf., ?4.). As an additional example we give the classification of the topological sums of two 3-dimensional lens spaces. In what follows, manifold will mean compact orientable 3-dimensional manifold in the sense of [6]; similarly for bounded manifold. The term polygonal curve will mean polygonal in some triangulation of the given space; it should be emphasized that the assumption that two or more curves are polygonal does not necessarily postulate their polygonality in the same triangulation unless specifically stated. The reader is also cautioned that all groups, including homology groups, will be written multiplicatively to avoid confusion with group ring notation. Also, homomorphisms will frequently be written as superscripts, that is, f (x) as xf and f(x1) as x-. 163