Exotic Spheres Signature - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:52:05Z http://mathoverflow.net/feeds/question/109235 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109235/exotic-spheres-signature Exotic Spheres Signature Everknight 2012-10-09T15:07:34Z 2012-11-30T11:22:00Z <p>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.</p> <p>That is, $W$ a $4n$-manifold with $\sigma(W)\neq 0$ and $\partial W \cong S^{4n-1}$ then $\partial W$ is exotic.</p> <p>Or I am just misunderstanding the invariants. Thanks</p> http://mathoverflow.net/questions/109235/exotic-spheres-signature/111267#111267 Answer by Llohann for Exotic Spheres Signature Llohann 2012-11-02T10:50:43Z 2012-11-02T10:50:43Z <p>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 <strong>parallelizable</strong> 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</p> <p>$0\to bP_{n+1}\to \theta^n\to \tilde\pi_n$</p> <p>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.</p> <p>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.</p>