ON h-COBORDISMS OF SPHERICAL SPACE FORMS

Given a manifold M of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold W is isomorphic to a cylinder M × [0, 1] if and only if W is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of M ; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, finite group actions on homotopy spheres (in dimensions ≥ 5) with exactly two singular points. In the topology of manifolds it is often important to recognize when a manifold W with boundary is isomorphic to a cylinder M × [0, 1]. The s-cobordism theorem gives the standard principle for recognizing products, but in some situations it is useful to have other criteria involving the boundary components of W . In a series of papers [U1]–[U3] F. Ushitaki has studied this question for equivariant h-cobordisms between two free linear G-spheres S(V ), S(V ′) of dimension 2n − 1 ≥ 5 and has proved that such h-cobordisms are equivariantly isomorphic to products S(V ) × I under assumptions of an algebraic nature (e.g., the vanishing of SK1(Z[G]) ). To be more specific, the following is the main result. Theorem (Ushitaki). Let G be a finite group, and X a free G-homotopy sphere of dimension 2n− 1 ≥ 5. Then the following are equivalent. (1) Every smooth G-h-cobordism W between X and itself must be G-diffeomorphic to X × I. (2) The homomorphism c̃ : H2n ( Z2;Wh(G) ) → L2n(G) in the Rothenberg exact sequence is a monomorphism. Here one takes the standard conjugation involution on the Whitehead group Wh(G) corresponding to the trivial homomorphism G→ Z2. Let G be a finite group which can act freely (topologically, piecewise linearly or smoothly) on a homotopy sphere Σ. We shall call the manifold Σ/G = M a fake spherical space form (cf. [KS4]). In this note we elaborate on the techniques and results of our paper [KS3] and obtain the following: Theorem. Let CAT denote one of the topological, piecewise linear, or smooth categories, let G 6= {1} be a finite group, and let M be a CAT manifold that is a fake Received by the editors June 23, 1997 and, in revised form, September 2, 1997. 1991 Mathematics Subject Classification. Primary 57R80, 57S25. The first author was partially supported by NSF Grant DMS 91-01575 and by a COR grant from Tulane University. The second author was partially supported by NSF grant DMS 91-02711. c ©1999 American Mathematical Society