By B. Aulbach
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Extra info for Continuous and Discrete Dynamics near Manifolds of Equilibria
Inverted handles and M¨ obius caps. Looking at the usual immersion of the Klein bottle into R3 , we may see that it is homeomorphic to a sphere with two holes which has had a cylinder attached, but in which the attachment has been made in different directions along the two circles. This is the notion of an inverted handle; after removing two holes from the surface M , take a patch of the surface which contains both and which can be given an orientation. Then take orientations of the two circles which are not coherent with respect to this orientation, and attach the ends of the cylinder according to these.
For example, if we begin with a sphere and attach a handle in this manner, we obtain a surface homeomorphic to a torus. Consider a neighbourhood of the two holes to which the cylinder is attached; this will be homeomorphic to a disc with two holes, the so-called ‘pair of pants’ surface. Gluing one end of C to each hole, we obtain a torus with a hole; attaching a handle in the manner described above is equivalent to cutting a single hole and gluing our torus with a hole along its boundary. So far this is rather vague and imprecise; what does “cutting a hole” mean, anyway?
With this tool in hand, we will be able to construct a complete list, and to identify any planar model with a surface on our list via the method of cutting and pasting. 4. Lecture 11: Friday, Sept. 21 a. Euler characteristic of planar models. So far we have seen planar models for four different surfaces; two of these used the 2-gon and two the 4-gon. We can list these in terms of the identifications made between various sides as we complete a circuit around the boundary, as explained last time: edge identifications surface Euler characteristic aa−1 sphere 2 aa projective plane 1 −1 −1 aba b torus 0 abab−1 or aabb Klein bottle 0 To compute the Euler characteristic χ of a planar model on a 2m-gon, we may observe that F = 1 and E = m after passing to the quotient space, so the only variable is the number of vertices after all identifications have been made.
Continuous and Discrete Dynamics near Manifolds of Equilibria by B. Aulbach