Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Dehn twist and Moebius strip

  1. May 15, 2013 #1

    tom.stoer

    User Avatar
    Science Advisor

    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 allowes one to distinguish between the π, 3π, ... cases which are identical locally?
     
    Last edited: May 15, 2013
  2. jcsd
  3. May 19, 2013 #2

    Bacle2

    User Avatar
    Science Advisor

    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.
     
    Last edited: May 19, 2013
  4. May 30, 2013 #3

    tom.stoer

    User Avatar
    Science Advisor

    Yes, I know that.

    It was my intention to find a variant which works for non-orientable manifolds.
     
  5. May 30, 2013 #4

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  6. May 30, 2013 #5

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Dehn twist and Moebius strip
  1. Mobius strip and 4D (Replies: 7)

Loading...