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Handles and non-orientible 3D manifolds

  1. Mar 17, 2009 #1
    I've read that with 2D manifolds, you can create any closed 2D manifold by adding "handles" or "crosscaps" (or "crosshandles"). To add a handle, cut out two disks and add the ends of a circle x interval product (cylinder). If you glue one end "the wrong way" you get a "crosshandle", which is how you get a Klein bottle. A "crosscap" is equivalant to a Mobius strip, but it has a circular boundary (instead of a figure-8 boundary folded up double). Use a crosscap to create a projective plane.

    There are some "rules" that apply for 2D surfaces: a handle in the presence of a crosscap can be converted to a crosshandle. Two crosscaps can be combined into a crosshandle.

    I'm trying to come up with a set of rules for creating 3D manifolds using "handles". If I define a "3-handle" as a 3-sphere x interval product, I can cut out two balls from an S3 and add the two ends of the handle to produce a 3-donut. Or I can glue one end the "wrong way" (adding a 3-crosshandle) and get a "solid Klein bottle".

    I can also cut out a ball and plug in a "crossball" to get a real projective space.

    You can cut out a solid torus (disk x circle product) and glue it back in a twisted way. Or you can glue in crosscap x circle product (which has a torus boundary). I'm not sure what you'd get.

    Are there any other types of "handles" you can attach? I want to cover both orientable and non-orientable cases. Are there any rules about combining these in 3D?
  2. jcsd
  3. Mar 17, 2009 #2
    The generalization of attaching handles to higher dimensions is called "http://en.wikipedia.org/wiki/Surgery_theory" [Broken]". The operation you are describing is a special case.
    I also suggest you look at the article on the http://en.wikipedia.org/wiki/Connected_sum" [Broken].
    Last edited by a moderator: May 4, 2017
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