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I am trying to figure out when a closed, oriented manifold admits an orientation reversing diffeomorphism. My naive argument that the orientation cover should allow you to switch orientations is apparently wrong, since not every manifold admits such a diffeomorphism.

Can anyone give me some criteria for when such a morphism should exist, or why some of the standard counterexamples (such as $\mathbb{P}^{2n}$) fail to admit one?

Thanks

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Dude, If you turn in any answers you get off of here as your solution to a homework problem, I am totally turning you in. –  Charlie Frohman Apr 18 '10 at 22:09
    
When you say $\mathbb{P}^{2n}$, do you mean complex projective space? I think real projective spaces of even dimension are very rarely orientable. –  S. Carnahan Apr 18 '10 at 22:37
    
Yes I mean $\mathbb{C}\mathbb{P}^{2n}$. Sorry if this question is actually easy, but I am not a differential geometer so I'm unsure of how to approach this. I checked google and noticed a few theses on when manifolds admit such a morphism so I assumed it wasn't completely trivial. –  Randall Apr 18 '10 at 22:45
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Can you down-vote a comment? –  Makhalan Duff Apr 19 '10 at 17:56
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I accidentally wrote "orientational" instead of orientation in the title, so I apologize for that. I'm not sure how you deduce from that and that I used the common abbreviation $\mathbb{P}^n$ for complex projective space that I had no idea what I was talking about / that this is a homework question. As I said, I'm not a topologist and since every complex manifold is orientable it seemed natural to ask when you can reverse the orientation. I know we don't want to answer calculus questions here but it seems rather silly that I can't ask questions outside my area of specialty. –  Randall Apr 26 '10 at 16:46
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4 Answers 4

up vote 12 down vote accepted

Such an endomorphism of $M$ gives an automorphism of the cohomology ring that acts by $-1$ on top cohomology. The cohomology ring of your example $M = {\mathbb C \mathbb P}^{2n}$ doesn't have such automorphisms.

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Yes thank you. My definition of orientation was in terms of homology so I was missing out on the ring structure. –  Randall Apr 19 '10 at 2:30
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The same technique Allen mentioned also shows that $\mathbb{H}P^{2n}$ doesn't admit any orientation reversing diffeomorphisms.

However, it's also true that $\mathbb{H}P^{2n+1}$ doesn't admit any orientation reversing diffeomorphisms unless n = 0. This is because the first Pontryagin class $p_1 = 2(n-1)x$ for $x\in H^4(\mathbb{H}P^n)$ a generator. Any diffeomorphism must take $p_1$ to itself, so it must take $x$ to itself (unless $n=1$). The ring structure of $\mathbb{H}P^n$ then implies that the diffeomorphism preserves orientation.

(By contrast, the map $[z_0:...:z_n]\rightarrow [\overline{z_0}:...:\overline{z_{n+1}}]$ is an orientation reversing map for $\mathbb{C}P^{2n+1}$).

Another class of (perhaps surprising) examples is exotic spheres: many of them don't admit orientation reversing diffeomorphisms, though, or course, they admit orientation reversing homeomorphisms.

This is because the collection of oriented diffeomorphism classes of an $n$-sphere ($n\neq 4$) form an abelian group under connect sum. The inverse of an oriented diffeomorphism class of sphere is the same diffeomorphism class equipped with the opposite orientation. Thus, exotic spheres with orientation reversing diffeomorphisms correspond to elements of order 1 or 2 in this group.

Then, for example, in dimension 7, the group is isomorphic to $\mathbb{Z}/28\mathbb{Z}$, so there are precisely two spheres which admit orientation reversing diffeomorphisms.

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A large number of manifolds of dimension $4k$ can't admit an orientation-reversing diffeomorphism just because of their cobordism type. That is, if $f: M\rightarrow \overline{M}$ is an orientation preserving diffeomorphism, then the cobordism class $[M^n]$ is a 2-torsion element of the cobordism group of oriented $n$-manifolds: Since $M\sqcup M \cong M\sqcup\overline{M}$ bounds the cylinder $M\times[0,1]$, thus $2[M] = [M\sqcup \overline{M}] = 0 \in \Omega^{\rm SO}_n$. By the Thom-Pontryagin theorem, if $M$ has a nonzero Pontryagin number (which requires that the dimension of $M$ to be a multiple of 4), then $[M]$ is generates a free abelian subgroup of $\Omega^{\rm SO}_n$ and is not a 2-torsion element. Thus, $M$ will not admit an orientation-reversing diffeomorphism.

In particular, this applies if the signature of $M$ is nonzero, since by Hirzebruch's signature theorem the signature is computable in terms of Pontryagin numbers. The previously mentioned examples of $\mathbb{CP}^{2k}$ and $\mathbb{HP}^k$ are special cases of this statement, since both have nonzero signature and hence are do not represent 2-torsion elements of the oriented cobordism group.

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You might want to have a look at the paper "Orientation reversal of manifolds" by Daniel Muellner.

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