Subject: A question for the surgery theorists From: Johannes Ebert Date: Wed, 6 Jun 2007 14:53:10 +0200 (CEST) Hello, I have the following question about the classification of high-dimensional manifolds. Let M be a smooth closed manifold. Then the spherical fibration associated to the stable normal bundle of M only depends on the homotopy type of M (it is the Spivak normal fibration). The first step in the surgery program to classify all smooth manifolds homotopy equivalent to M is to understand the stable vector bundles whose associated spherical fibration are equivalent to the Spivak normal fibration. The second step concerns the surgery obstruction, but I an only interested in the first step for the moment. I understand that the isomorphism class of the stable normal vector bundles does not depend on the homotopy type of M alone, but for some homotopy types, like spheres, it does (see Kervaire-Milnor). I want to see concrete counterexamples, i.e. homotopy equivalences between smooth (!) manifolds M_0 \to M_1 which are not covered by an isomorphism of the stable normal vector bundles. Because I want to do calculations with them, I like to have "simple" examples, i.e.: highly connected, small dimension, small Betti numbers, known cohomology and computable rational homotopy type. Are there examples where the normal bundles of M_0 and M_1 are distinguished by their rational Pontryagin classes? It would certainly be very nice if one of the manifolds is stably parallelizable. Is that possible? Where shall I look in the literature? Best regards, Johannes Ebert