Inertia group II (Ex)
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{{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 to ${\rm Id}_M \colon M \to M$ in $\mathcal{S}(M)$. | # 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. | # 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}} | {{endthm}} |
Revision as of 17:35, 28 August 2013
Let be a closed smooth oriented -manifold. The homotopy inertia group of , n\Theta_n$ defined by
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Tex syntax errorwe regard and as the same topological space (with different smooth structures).
Exercise 0.1.
- Show that if and only if
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is equivalent toTex syntax error
in . - Given that , determine .
- Assuming that , deduce that admits a self-homotopy equivalence which is not homotopic to a diffeomorphism.