0
$\begingroup$

I'm studying the groups of homotpy spheres, which yields to the study of classes of diffeomorphism over the standard sphere. I dont understand why the examples are focused to construct manifolds with signature non-zero and boundary a homotopy sphere.

That is, $W$ a $4n$-manifold with $\sigma(W)\neq 0$ and $\partial W \cong S^{4n-1}$ then $\partial W$ is exotic.

Or I am just misunderstanding the invariants. Thanks

Improve this question
$\endgroup$
3
  • 6
    $\begingroup$ Your question isn't clear. "The examples" you refer to, which ones are they? Are you reading a textbook? Keep in mind, if you want to argue that a manifold is an exotic sphere, at some point you have to present an idea. Signatures of bounding manifolds is a good idea. If you have other techniques of proof, by all means use them. $\endgroup$
    – Ryan Budney
    Oct 9, 2012 at 16:40
  • $\begingroup$ Your claim sounds bizarre: take your favourite 4n-manifold with non-zero signature and remove a ball: then you have a 4n-manifold with non-zero signature, bounding a homotopy sphere, which is however the standard sphere. $\endgroup$
    – Paolo Ghiggini
    Oct 10, 2012 at 2:39
  • $\begingroup$ I don't really see a question here, so I've voted to close. $\endgroup$
    – Mark Grant
    Nov 30, 2012 at 12:31

1 Answer 1

Reset to default
1
$\begingroup$

It is like what Ryan said. At some point you have to show that what you are constructing is exotic. This is done, for example, by the signature of a parallelizable manifold bounding your candidate. Such 'coboundaries' don't always exists, however, for homotopy spheres of dimension 4n-1, a large subgroup is of this kind, and these are the easiest spheres to construct. In general, the group of homotopy spheres, $\theta^n$, seats in an exact sequence

$0\to bP_{n+1}\to \theta^n\to \tilde\pi_n$

where $bP_{n+1}$ stands for the subgroup of spheres which bound parallelizeble manifolds and $\tilde\pi_n$ is the $n$-th stable homotopy group of spheres quotiented by the image of the $J$-homomorphism.

Where are you studying these things? Differentiable Manfiolds by Kosinski is a great book. Please, let me know what you are doing, I am always interested in learning and discussing this subject.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .