Homotopy spheres II (Ex)

Revision as of 23:30, 27 August 2013

In the following, $<wikitex>; In the following, $\Sigma \in \Theta_n$ is a homotopy sphere, $n \geq 5$, and $bP_{n+1} \in \Theta_n$ is the group of homotopy spheres bounding parallelisable manifolds. {{beginthm|Exercise}} # Show that $\Sigma$ embeds into $S^{n+1}$ if and only if $\Sigma$ is diffeomorphic to $S^n$. # Show that $\Sigma$ embeds into $S^{n+2}$ if and only if $\Sigma \in bP_{n+1}$. # For $k \geq 2$, show that $\Sigma$ embeds into $\Rr^{n+k+1}$ with trivial normal bundle if and only if there is a diffeomorphism $\Sigma \times S^k \cong S^n \times S^k$. # Show that if $\Sigma \in bP_{n+1}$, then for all $k \geq 2$, there is a diffeomorphism $\Sigma \times S^k \cong S^n \times S^k$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]\Sigma \in \Theta_n$ is a homotopy sphere, $n \geq 5$, and $bP_{n+1} \in \Theta_n$ is the group of homotopy spheres bounding parallelisable manifolds.

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