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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

flag	asked Oct 9 at 15:07 Everknight 1

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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. – Ryan Budney Oct 9 at 16:40

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. – Paolo Ghiggini Oct 10 at 2:39

I don't really see a question here, so I've voted to close. – Mark Grant Nov 30 at 12:31

Llohann · Answer 1 · 2012-11-02 10:50:43Z UTC

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.

Exotic Spheres Signature

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