Inertia group II (Ex)

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Latest revision as of 18:35, 29 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 $\mathcal{S}(M)$. # Determine $I(\Hh P^2)$ and $I_H(\Hh P^2)$. {{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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$\displaystyle I_H(M) : = \{ \Sigma | \exists f \colon \Sigma \sharp M \cong M, f \simeq {\rm Id}_M \},$

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.

Show that $\Sigma \in I_H(M)$ if and only if
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```
is equivalent to
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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.

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Inertia group II (Ex)

Latest revision as of 18:35, 29 August 2013

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@@ Line 1: / Line 1: @@
 <wikitex>;
-Let $M$ be a closed smooth oriented $n$-manifold.  The ''homotopy inertia group of $M$'', $I_H(M),
+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
+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}}
-# 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 $\mathcal{S}(M)$.
+# Given that $\Theta_8 \cong \textup{Coker}(J_8) \cong \Zz/2$, determine $I_H(\Hh P^2)$.
-# Determine $I(\Hh P^2)$ and $I_H(\Hh P^2)$.
+# 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>
@@ Line 14: / Line 13: @@
 {{#RefList:}}
 [[Category:Exercises]]
-[[Category:Exercises without solution]]
+[[Category:Exercises with solution]]