Inertia group II (Ex)

Let $M$$<wikitex>; Let $M$ be a closed smooth oriented $n$-manifold. The ''homotopy inertia group of $M$'', $I_H(M), is the subgroup of homotopy $n$-spheres $\Theta_n$ defined by $$ I_H(M) : = \{ \Sigma | \exists f \colon \Sigma \sharp M \cong M, f \simeq {\rm Id}_M \},$$ where for the statement $f \simeq {\rm Id}_M$ we regard $M$ and $\Sigma \sharp M$ as the same topological space (with different smooth structures). {{beginthm|Exercise}} # Show that $\Sigma \in I_H(M)$ if and only if ${\rm Id}_M \colon \Sigma \sharp M \to M$ is equivalent to ${\rm Id}_M \colon M \to M$ in $\mathcal{S}(M)$. # Determine $I(\Hh P^2)$ and $I_H(\Hh P^2)$: they are subgroups of $\Theta_8 = \Zz/2$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]M$ be a closed smooth oriented $n$$n$-manifold. The homotopy inertia group of $M$$M$, $I_H(M), is the subgroup of homotopy$$I_H(M), is the subgroup of homotopy$n$-spheres$$-spheres$\Theta_n$ defined by

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$$\displaystyle I_H(M) : = \{ \Sigma | \exists f \colon \Sigma \sharp M \cong M, f \simeq {\rm Id}_M \},$$

where for the statement

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$f \simeq {\rm Id}_M$ we regard $M$$M$ and $\Sigma \sharp M$$\Sigma \sharp M$ as the same topological space (with different smooth structures).

Exercise 0.1.

1. Show that $\Sigma \in I_H(M)$$\Sigma \in I_H(M)$ if and only if

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   ${\rm Id}_M \colon \Sigma \sharp M \to M$ is equivalent to

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   ${\rm Id}_M \colon M \to M$ in $\mathcal{S}(M)$$\mathcal{S}(M)$.
2. Determine $I(\Hh P^2)$$I(\Hh P^2)$ and $I_H(\Hh P^2)$$I_H(\Hh P^2)$: they are subgroups of $\Theta_8 = \Zz/2$$\Theta_8 = \Zz/2$.