Inertia group II (Ex)

Revision as of 17:36, 28 August 2013

Let $<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 $\mathcal{S}(M)$. # Given that $\Theta_8 \cong \textup{Coker}(J_8) \cong \Zz/2$, determine $I_H(\Hh P^2)$. # Assuming that $I(\Hh P^2) = \Theta_8$, deduce that $\Hh P^2$ admits a self-homotopy equivalence which is not homotopic to a diffeomorphism. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]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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