Can Dehn Twists Be Applied to Non-Orientable Manifolds Like the Moebius Strip?

[tom.stoer]

I am familiar with the so-called Dehn twists applied to tori (from a "physical" perspective like in string theory).

My question: is it possible to generalize the Dehn twist or a similar concept to non-compact and/or non-orientable manifolds?

What I have in mind is the Moebius strip: it is constructed like a "twisted" torus: cut a cylindrical strip, twist it by π, 2π, 3π, ... and glue it together again; for π, 3π, ... this does not preserve the topology of the strip; for 2π, ... it does, at least locally (what is not preserved is the topology of the embedding on 3-space).

Second question: is there a classification based on the generalized Dehn twist which allows one to distinguish between the π, 3π, ... cases which are identical locally?

 

[Bacle2]

I think that the issue is that in a non-orientable manifold you cannot define tubular neighborhoods ( the ones you use to define a Dehn twist) at every point. Instead of tubular neighborhoods, you will have a sort of mobius-strip-like object. See, e.g., http://en.wikipedia.org/wiki/2-sided

I think orientability of a manifold is equivalent to every curve defined on the manifold being two-sided, as in the Wiki definition. Maybe there is some other type of construction beyond this, but not that I know of.

EDIT: I think I may have misunderstood --or misunderestimated--your question.I think you know one can't define a tubular 'hood at every point of a non-orientable surface,and you want to know if there is a variant of it that would allow you to define a version of a Dehn twists for a non-orientable surface. Sorry if I misunderstood.

 

[tom.stoer]

EDIT: I think I may have misunderstood --or misunderestimated--your question.I think you know one can't define a tubular 'hood at every point of a non-orientable surface ...

Yes, I know that.

It was my intention to find a variant which works for non-orientable manifolds.

 

[Ben Niehoff]

It seems like you want to embed the Mobius strip inside a solid torus, such that Dehn twists of the torus correspond to extra $2\pi$ rotations of the Mobius strip. On the surface of the solid torus, Dehn twists are diffeomorphisms. Does this fact extend to the solid interior? Intuitively, I would think so. If that is the case, then this will give you no new information about the Mobius band; the bands with $\pi, 3\pi, 5\pi \ldots$ twists are all diffeomorphic (and hence homeomorphic).

What you can do instead is think of the Mobius strip as defined by its edge(s). Basically, as a $\mathbb{Z}_2$ bundle over a circle. Then you can classify the various twists with knot theory. But this does depend on the embedding in 3 dimensions.

 

[Ben Niehoff]

Incidentally, even a torus cannot "see", intrinsically, that it is Dehn twisted. That's kind of the point. If you quotient the plane by a lattice, any transformation that leaves the lattice invariant will necessarily leave the quotient invariant. So tori that differ by a Dehn twist are actually identical.