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Somewhat related to this.

I also understand this might be wishful thinking but I'm looking for a textbook/notes/video giving a simple/minimal framework of the topology of surfaces for nonspecialists more specifically graph theory/discrete optimization. Videos would be especially nice. Something like here are all surfaces up to... and a list of theorems you will likely use a kind of topological toolbox for non-topologists so to speak. Proofs are not really important here.

e.g. making sense of and proving Euler's formula for a torus or higher genus surface. Classifying the types of cycles/curves of graphs embedded on a torus or higher genus. Given 2 cycles of the same "type" on a torus that intersect at two points and letting P1,P2 and Q1,Q2 be the two paths between the two points on each cycle, either P1,Q1 divide the torus into two regions or P1,Q2 do, further if P1,Q1 do then so do P2,Q2 and generalizations of this to higher genus.

Another particular thing I would like to prove is for a circle D on a torus and 5 loops (closed curves) that intersect D at 5 distinct points non of which divide the torus into two regions some 2 of the 5 loops intersect in at least 2 points.

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    $\begingroup$ I think Basic Topology by M A Armstrong will be a good start. It starts with proving euler characteristic of sphere is 0 and proceeds to the classification of compact surfaces. $\endgroup$
    – Ivin Babu
    Apr 8 at 14:18
  • $\begingroup$ @HaoS does this answer your question? $\endgroup$
    – Son Gohan
    Apr 15 at 19:33
  • $\begingroup$ @SonGohan Still reading through the references btw which reference would be best for proving/referencing a proof of the last part of my question "a circle D on a torus and 5 loops (closed curves) that intersect D at 5 distinct points non of which divide the torus into two regions some 2 of the 5 loops intersect in at least 2 points." which seems obvious enough $\endgroup$
    – Hao S
    Apr 15 at 21:47
  • $\begingroup$ I’d go through the indexes and I’d see which one explains the topic you are interested in in the suitable way for you. Hope it helped! :) $\endgroup$
    – Son Gohan
    Apr 15 at 21:54
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These notes may provide a nice introductory beginning, with lots of figures and intuitive drawings (it may be more focussed on computational topology of graphs on surfaces, but the theory is introduced at every chapter in a very accessible way for a person not in the field): http://monge.univ-mlv.fr/~colinde/proj/16epit/eric-notes.pdf

As stated in the comment, also Basic Topology by Armstrong is a nice suggestion.


A bit more technical may be Topology of Surfaces by Kinsey.

Other standard references:

  • P. C. Kainen, Some recents results in topological graph theory, 1974
  • A. T. White, Graphs, Groups, and Surfaces, 1984., Graphs of Groups on Surfaces, 2001.
  • Gross and Tucker, Topological Graph Theory, 1987.
  • Liu, Y. P. Embeddability of Graphs, 1995.
  • D. Archdeacon, Survey of topological graph theory, online

Curiosity: A book consists of the union of a finite number of closed half-spaces, all sharing the same boundary line (the spine of the book). The half-spaces are called "pages". As a result, books can be seen as a type of pseudo-manifold: they are locally Euclidean except for certain points that have neighborhoods consisting of disks with possible singularities. One can show that every finite graph can be embedded in a book with 3 pages.

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