Exotic spheres and chirality (Ex)
(Difference between revisions)
(Created page with "<wikitex>; Recall that $\Theta_n$ denotes the group of h-cobordism classes of manifolds $\Sigma^n$ which are homotopy equivalent to the $n$-sphere. For $n <> 4$ this is the s...") |
m |
||
Line 1: | Line 1: | ||
<wikitex>; | <wikitex>; | ||
− | 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.