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I teach at an American research 1 university. I am planning a course on combinatorial topology for undergraduates whose background is:

  • multivariable calculus
  • linear algebra
  • at least one proof course

In particular, I can't assume they have taken a class on analysis. I would like the course to carefully prove the classification of piecewise-linear surfaces with boundary. Can anyone recommend a textbook that accomplishes this?

Edit: "Topology Now!" is close to what I am looking for, but I would prefer something that builds surfaces out of triangles, in part to avoid having to talk about what it means to be locally euclidean.

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    $\begingroup$ " I would like the course to carefully prove the classification of piecewise-linear surfaces with boundary." You just say you want this, but why? Why not emphasize an easy survey course, especially given the kids lack of analysis background. Maybe they can do the careful stuff later. Perfect is the enemy of better... $\endgroup$
    – guest
    May 24, 2018 at 20:32
  • $\begingroup$ I like to teach courses that address a gap in undergrad curriculum. The course I am looking for would prepare students to take a graduate level algebraic topology course (after they take a course on general topology, which is a very different flavour than algebraic topology). I know some people like "easy survey courses," but I do not understand how to teach them, or what students receive from them. I would rather teach a rigorous course on the simplest case of what is coming in grad school (for those going to grad school) while proving an important theorem (for everyone). $\endgroup$
    – David Steinberg
    May 25, 2018 at 23:36
  • $\begingroup$ I can see the intellectual appeal for the teacher in doing something different, better. But would ask you to consider if this is best for the student. "I don't know what people get out of easy survey courses" shows a bit of a blind spot. Just be very, very, very aware that what is best for the average student may not be WHAT INTERESTS the teacher. $\endgroup$
    – guest
    May 25, 2018 at 23:42

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Check out Kinsey's book Topology of Surfaces. Kinsey proves the classification theorem of surfaces by considering triangulations of surfaces and then going through the algorithm of cutting-and-pasting to classify them. I'm pretty sure he doesn't talk about surfaces having to be locally Euclidean before this, or at least I don't remember it featuring in his proof. It's all just cutting-pasting-triangles on polygons where the edges are associated in some manner. The classification comes in Chapter 4, after only discussing some point-set topology in the previous chapters.

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