Inertia group II (Ex)
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(Created page with "<wikitex>; Let $M$ be a closed smooth oriented $n$-manifold. The ''homotopy inertia group of $M$'', $I_H(M), is the subgroup of homotopy $n$-spheres $\Theta_n$ defined by $$...") |
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is the subgroup of homotopy $n$-spheres $\Theta_n$ defined by | is the subgroup of homotopy $n$-spheres $\Theta_n$ defined by | ||
$$ I_H(M) : = \{ \Sigma | \exists f \colon \Sigma \sharp M \cong M, f \simeq {\rm Id}_M \},$$ | $$ I_H(M) : = \{ \Sigma | \exists f \colon \Sigma \sharp M \cong M, f \simeq {\rm Id}_M \},$$ | ||
| − | where for the statement $f \simeq {\rm Id}_M$ we regard $M$ and $\Sigma \sharp M$ as the same | + | where for the statement $f \simeq {\rm Id}_M$ we regard $M$ and $\Sigma \sharp M$ as the same topological space (with different smooth structures). |
| − | topological space (with different smooth structures). | + | |
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
| − | # Show that $\Sigma \in I_H(M)$ if and only if ${\rm Id}_M \colon \Sigma \sharp M \to M$ is equivalent | + | # Show that $\Sigma \in I_H(M)$ if and only if ${\rm Id}_M \colon \Sigma \sharp M \to M$ is equivalent to ${\rm Id}_M \colon M \to M$ in $\mathcal{S}(M)$. |
| − | to ${\rm Id}_M \colon M \to M$ in $\mathcal{S}(M)$. | + | # Determine $I(\Hh P^2)$ and $I_H(\Hh P^2)$: they are subgroups of $\Theta_8 = \Zz/2$. |
| − | # Determine $I(\Hh P^2)$ and $I_H(\Hh P^2)$. | + | |
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
Revision as of 00:37, 26 August 2013
Let 
be a closed smooth oriented 
-manifold. The homotopy inertia group of , 
n
\Theta_n$ defined by
Tex syntax error
Tex syntax errorwe regard
and 
as the same topological space (with different smooth structures).
Exercise 0.1.
- Show that
if and only if Tex syntax error
is equivalent toTex syntax error
in
.
- Determine
and 
: they are subgroups of 
.
