Homotopy spheres II (Ex)
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# Show that $\Sigma$ embeds into $\Rr^{n+1}$ if and only if $\Sigma$ is diffeomorphic to $S^n$. | # Show that $\Sigma$ embeds into $\Rr^{n+1}$ if and only if $\Sigma$ is diffeomorphic to $S^n$. | ||
# Show that $\Sigma$ embeds into $\Rr^{n+2}$ if and only if $\Sigma \in bP_{n+1}$. | # Show that $\Sigma$ embeds into $\Rr^{n+2}$ if and only if $\Sigma \in bP_{n+1}$. | ||
− | # Show that $\Sigma$ embeds into $\Rr^{n+k+1}$ with trivial normal bundle if and only if there is a diffeomorphism $\Sigma \times S^k \cong S^n \times S^k$. | + | # 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 $\Sigma \times S^k \cong S^n \times S^k$. | # 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}} | {{endthm}} |
Revision as of 23:55, 25 August 2013
In the following, is a homotopy sphere, , and is the group of homotopy spheres bounding parallelisable manifolds.
Exercise 0.1.
- Show that embeds into if and only if is diffeomorphic to .
- Show that embeds into if and only if .
- For , Show that embeds into with trivial normal bundle if and only if there is a diffeomorphism .
- Show that if , then for all , there is a diffeomorphism .