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