Homotopy triangulations of pseudo lens spaces and actions on products of spheres

Abstract

In the last decade the classification of free actions of cyclic groups Zr on homotopy spheres has been studied extensively. Given a free Zr action on a homotopy sphere the orbit space is homotopy equivalent (but not necessarily simple homotopy equivalent) to a lens space [t13]. So the study of free Zr actions on homotopy spheres is equivalent to the problem of classifying manifolds of the homotopy type of lens spaces. Surgery theory gives a complete classification in the PL and topological categories. The case r = 2 was attacked by Lopez de Medrano [8] and Wall [16, § 14D]. The case r odd was settled in the work of R. Lee [7], Browder et al. [4, 5]. As a next step one can consider free Zr actions on manifolds E homotopy equivalent to the cartesian product of two spheres. One might then trie to replace the lens space by a manifold defined as follows: Let a, b complex representations of Zr of degree p + 1, q + 1 in V~, V b. Let S,, Sb the unit spheres in V,, Vb and assume that they are free Zr-manifolds. Our candidade is then M'=S,, x Sb/Z r. Such an orbit space we call a pseudo lens space. However in general it is not true that the orbit space E/Zr is homotopy equivalent to such a pseudo lens space. But we think it is worthwile to study those actions having this property. They can be obtained from actions whose orbit space is simple homotopy equivalent to M by operation of the Whiteheadgroup W(Z,). So we further restrict to so called simple actions, thus obtaining at least a lower bound for the set of all actions. The precise problem we study is the following: In the PL category we consider m-dimensional closed oriented manifolds E with free Zr-actions, an orientation preserving Zfequivariant homotopy equivalence e:E---,SoxSb inducing a simple homotopy equivalence between orbit spaces. Two such (E i, e~), i= 1, 2 are considered equivalent if there is an equivariant PLhomeomorphism c:Et~Ez with e2oc equivariantly homotopic to e t. The set 5~(M) of equivalence classes of simple actions is in 1 I correspondence to the set hT(M) of equivalence classes of homotopy triangulations [16, p. 102] on M. The sugery exact sequence [16, p. 107, 1 t l ]