mapping class group of the real projective plane

Question

In most literature I've read about the mapping class group, I found that many authors have stated without any explanation that any homeomorphism of a real projective 2-space to itself is isotopic to the identity. I'm guessing it is obvious but I can't seems to come up with a sound explanation, this is what i have:

$\mathbb{R}P^2$ can be constructed from from glueing the boundary of a disk $D^2$ to the boundary of a Mobius band, since the mapping class group of D^2 and the Mobius band are both trivial then the mapping class group of $\mathbb{R}P^2$ is trivial.

the reason why this doesn't seem sound is because the Klein bottle can be constructed from glueing two Mobius band's together by their boundary but the mapping class group of the Klein bottle is not trivial.

Answer 1

The proof is quite similar to the (oriented) mapping class group of the sphere is trivial

First any homeomorphism of the projective plane fixes a point; this fact comes from the fact any homeomorphism of the sphere will have a point $x$ such that $f(x)=x$ or $f(x)=-x$. You could also just show that you can always isotope so that there is a fixed point

Now puncture that point and you have a space homemorphic to an open Mobius strip, and you can find an isotopy to the identity which extends to the full projective plane (fixing the fixed point).

I feel like there could be more direct proof, but I don't know one.

Answer 2

The free mapping class group (that is, the path components of its full group of homeomorphisms) of the Möbius band M is not trivial. It has a self-homeomorphism h : M → M whose effect h_* : H₁(M;Z) → H₁(M;Z) on the first homology group (which is infinite cyclic) is multiplication by -1.

The projective plane P² can be modeled as the 2-disk D² with each pair of antipodal points on its boundary S¹ identified. From this model it's easy to see that there is an isotopy of the identity id : P² → P² to a homeomorphism taking the generator of H₁(P²;Z) (as a simple closed curve) to itself going in the reverse direction. (Just rotate the 2-disk by 180º.) This shows that the inclusion of a Möbius band into P² can be reversed by an isotopy within P².