Oriention-Reversing Diffeomorphisms of a Manifold - MathOverflow

Oriention-Reversing Diffeomorphisms of a Manifold

2010-04-18T22:06:10Z

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

Answer by Allen Knutson for Oriention-Reversing Diffeomorphisms of a Manifold

2010-04-19T01:06:42Z

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.

Answer by Jason DeVito for Oriention-Reversing Diffeomorphisms of a Manifold

2010-04-19T02:54:48Z

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.

Answer by HenrikRüping for Oriention-Reversing Diffeomorphisms of a Manifold

2010-04-19T06:23:38Z

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

Answer by John Francis for Oriention-Reversing Diffeomorphisms of a Manifold

2010-04-20T19:28:43Z

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.