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