Why each sphere admits an orientation reversing diffeomorphism onto itself? (For even dimensional ones can we take conjugation map?) And why complex projective spaces do not admit? Is there a geometric way to see this without using characteristic classes?
Orientation reversing diffeomorphism
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algebraic-topology
geometric-topology
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0Fun fact: there are exotic smooth structures on spheres so that the sphere with that smooth structure does not admit an orientation-reversing diffeomorphism. – 2012-02-23