A Splitting Theorem for Manifolds with Involution and Two Applications

Substantial results on both questions were obtained during the nineteen sixties, and further work was done during the following decade. For example, if in the first question we assume that all manifolds under consideration are simply connected and of sufficiently high dimension, then a splitting of the prescribed type always exists (compare [24], page 133). In fact, stronger results can also be proved if the fundamental groups of A, B, C are interrelated in a suitable manner (compare Wall [24], Chapter 12A, or Cappell [5]). In the second question one cannot expect the sphere bundles to be linearly or even topologically equivalent. However, a result of Atiyah [1] shows that the tangent sphere bundles are stably fiber homotopy equivalent, and a result of R. Benlian and J. Wagoner [3] shows that these bundles are already fiber homotopy equivalent before stabilization. Although this result is basically a statement in homotopy theory, the original proof in [3] was highly geometric, relying heavily on Wall’s codimension one splitting theorem in [24]; subsequent research produced an intrinsically homotopy-theoretic proof (see Dupont [12] and Sutherland [21]). In this paper we shall show that the basic geometric argument in [3] has an equivariant analog. In particular, we shall prove a splitting theorem for Z2-homotopy equivalences under general position assumptions slightly weaker than the Gap Hypothesis of [15]. Using this result we shall show that the tangent sphere bundles of homotopy equivalent manifolds are Z2-equivariantly fiber homotopy equivalent, where Z2 acts by the fiberwise antipodal map. Previous results of S. Straus [19] and M. Crabb [6] (Proposition 1.1, pages 2–3) show that these bundles with involution are stably Z2-fiber homotopy equivalent.