Subject: Re: Question for surgeons From: Tom Goodwillie Date: Fri, 8 Jun 2007 12:43:21 -0400 I am not a surgery theorist. I don't even play one on TV. (Sorry if the pop culture reference is lost on the international readership.) About the family of examples that Larry Taylor referred to: I want to give more details and also remark that you don't need a surgery exact sequence or anything about L-theory obstructions to understand these examples (from the point of view that I think Ebert is interested in). Let us construct a manifold M as a sphere bundle over a sphere: S^a --> M --> S^b. Specifically, it is the unit sphere bundle of a vector bundle V of rank a+1 on S^b. Such vector bundles are classified by elements of pi_{b-1}O(a+1). Spherical fibrations over S^b with fiber S^a (that is, fibrations over S^b with S^a being the homotopy type of the fiber) are classified, up to fiber homotopy equivalence, by elements of pi_{b-1}G(a+1). In many cases the relevant map pi_{b-1}O(a+1) ---> pi_{b-1}G(a+1) has nontrivial kernel. For example, if b=4 and a is large then pi_{b-1}O(a+1) is infinite cyclic and pi_{b-1}G(a+1) is the quotient of order 24. A nontrivial element of the kernel provides an example that is homotopy equivalent (even fiber homotopy equivalent) to the trivial example S^a\times S^b. To verify that the stable normal bundle of this manifold is nontrivial, it is enough to see that some Pontryagin class is nonzero. The tangent bundle of M has a horizontal part that is stably trivial (pulled back from tangent of S^b) and a vertical part that is stably the pullback of the vector bundle V on S^b that led to the sphere bundle. So the Pontryagin class of M is the pullback of the Pontryagin class of V, so it is nontrivial if the latter class is nontrivial. This succeeds in all cases where b=4k and a is large enough. I mean that in those cases pi_{b-1}O(a+1) is an infinite cyclic group and its nontrivial elements are detected by p_k, and the other group pi_{b-1}G(a+1) is finite (and the kernel is known since Adams). In Taylor's lowest-dimensional case, with b=4 and a=3, then pi_{b-1}O(a+1) is ZxZ and only some of its nontrivial elements are detected by p_1, but as he says you can make examples. Side remark: If you try to make this work with b=4 and a=2, you fail: you run into the funny fact that pi_3O(3)=Z injects into pi_3G(3). In some sense, some rational Pontryagin classes are defined for some unstable spherical fibrations. More precisely, the rational cohomology of BG(2k+1) or BSG(2k+1) is a polynomial ring in one generator, in degree 4k, which maps to the Pontryagin class p_k in BO(2k+1). So the kth Pontryagin class of a rank 2k+1 vector bundle depends only on the underlying spherical fibration. In contrast, the rational cohomology of BSG(2k) is a polynomial ring in one generator, in degree 2k, which is simply the Euler class. TG