Exotic spheres and chirality (Ex)
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| − | Recall that $\Theta_n$ denotes the group of h-cobordism classes of manifolds $\Sigma^n$ which are homotopy equivalent to the $n$-sphere. For $n | + | Recall that $\Theta_n$ denotes the group of h-cobordism classes of manifolds $\Sigma^n$ which are homotopy equivalent to the $n$-sphere. For $n \neq 4$ this is the same as the group of oriented diffeomorphism classes of such manifolds. |
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
Revision as of 13:32, 23 March 2012
Recall that 
denotes the group of h-cobordism classes of manifolds 
which are homotopy equivalent to the 
-sphere. For 
this is the same as the group of oriented diffeomorphism classes of such manifolds.
Exercise 0.1.
Let 
a homotopy -sphere with 
. Show that admits an orientation reversing diffeomorphism if and only if defines and element of order two in .
As a consequence, show that the boundary of the 
-dimensional E_8-manifold does not admit an orientation reversing diffeomorphism.
Remark 0.2. Note that manifolds admitting no orientation reversing diffeomorphism are often called chiral.
