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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

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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 '12 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 '12 at 2:39
    
I don't really see a question here, so I've voted to close. –  Mark Grant Nov 30 '12 at 12:31
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1 Answer 1

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.

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