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I was reading the book "Elements of algebraic topology", by Munkres, and I am struggling to understand this example.

example

Why doesn't the diagram of figure 3.6 determine the torus?

Eduardo Longa
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  • Hint: Count the number of edges in Figure 3.6 with endpoints labelled $a$ and $d$. Or, look at this question, which pretty much answers your question completely: https://math.stackexchange.com/questions/686142/a-diagram-which-is-not-the-torus?rq=1 – Lee Mosher Oct 29 '17 at 16:54
  • The answer in the above post uses Euler characteristic, which hasn't been introduced so far in the book. – Eduardo Longa Oct 29 '17 at 16:56

2 Answers2

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The second diagram is not a Torus because notice that your picture is also telling you to glue together the two instances of the edge $be$, which we would usually not do when creating a Torus. Similarly with the edge $cf$. In the first part this is avoided by carefully creating enough edges in the interior so that this problem does not occur.

Geometrically what this is doing is taking two parallel circles along the tube of the torus and flattening (pinching) each to a flat line. The picture below demonstrates this beautifully:

enter image description here

Arkady
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This diagram defines a simplicial complex, thus triangles sharing two vertices must share edge between them. Hence you should identify all four edges $ad$.

G. Strukov
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