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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?

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Fun fact: there are exotic smooth structures on spheres so that the sphere with that smooth structure does not admit an orientation-reversing diffeomorphism. –  Ryan Budney Feb 23 '12 at 3:16

3 Answers 3

Complex spaces of the form $\mathbb{C}P^{2n+1}$ do admit orientation reversing diffeomorphisms. In fact, the map sending $[z_0:z_1:...:z_{2n}:z_{2n+1}]$ to $[-\overline{z_1}:\overline{z_0}:...:-\overline{z_{2n+1}}:\overline{z_{2n}}]$ is orientation reversing (and gives a free action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{C}P^{2n+1}$).

One can check that when $n=0$, this map is the usual antipodal map of $\mathbb{C}P^1 = S^2$.

Also, Jim's idea works for showing that the manifolds $\mathbb{H}P^{2n}$ and $\mathbb{O}P^2$ do not admit orientation reversing diffeomorphisms. But more is true. The manifolds $\mathbb{H}P^{2n+1}$ also do not admit orientation reversing diffeomorphisms (unless $n=0$). (The map I wrote above isn't well defined when considering $\mathbb{H}P^n$). The only argument for this I'm familiar with involves characteristic classes, but I'm vaguely aware of one using Steenrod squares.

Finally, I just wanted to point out that when one says "all sphere's admit orientation reversing diffeomorphisms", one must be careful to throw out most exotic spheres. An exotic sphere is a manifold which is homeomorphic to $S^n$ but not diffeomorphic to it (and such things are known to exist, starting in dimension $7$).

It's a fact that for each fixed dimension the set of diffeomorphism classes of smooth manifolds homeomorphic to $S^n$ forms a group under connect sum. The inverse to an exotic sphere is the same sphere with reveresed orientation. Hence, exotic spheres admitting orientation reversing diffeomorphisms correspond to order $2$ elements in this group. Then, for example, in dimension $7$, the group is known to be cyclic of order $28$, so only $1$ of the 13 unoriented types of exotic spheres admits orientation reversing diffeomorphisms (though, being homeomorphic to $S^n$, it admits orientation reversing homeomorphisms!)

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The $n$-sphere $S^n$ is $\{(x_0, x_1,\ldots ,x_n):\sum {x_i}^2 = 1\}$ The map $f:S^n \to S^n$ sending $(x_0,x_1,\ldots , x_n)$ to $(-x_0,x_1,\ldots , x_n)$ is an oreintation-reversing diffeomorphism.

Sorry I don't have an answer for complex projective spaces.

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The reason complex projective space $\mathbb{CP}^{2k}$ has no orientation-reversing homeomorphism is because the top dimensional cohomology is generated by an even power of the generator, $x$, of $H^2(\mathbb{CP}^{2k})$. So any self-homeomorphism will send $x$ to $\lambda x$ ($\lambda\neq 0$), and the top cohomology will have $x^{2k}\mapsto \lambda^{2k} x^{2k}$. Since $\lambda^{2k}>0$, this preserves orientation. As Georges pointed out, this argument doesn't work for $\mathbb{CP}^{2k+1}$.

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I don't think this is correct because $x$ is in $H^2(\mathbb C \mathbb P^n)$ and not in $H^1(\mathbb C \mathbb P^n)=0$, so that the top class is not generated by an even power of $x$ if $n$ is odd. –  Georges Elencwajg Feb 22 '12 at 23:15
    
@GeorgesElencwajg: You are right. I will edit. Thanks. –  Grumpy Parsnip Feb 22 '12 at 23:20
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Actually $\mathbb{CP}^1$ is diffeomorphic to $S^2$, which definitely has an orientation-reversing homeomorphism. So the statement seems dubious for odd-dimensional complex projective spaces. –  Grumpy Parsnip Feb 23 '12 at 2:39

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