I am studying a course on differential geometry.

I saw the formula for the euler characteristic of a surface with $g$ holes and $b$ boundaries components and $n$ punctures is $2-2g -b +n$.

In addition, if all punctures are on boundaries the formula is $2-2g -b + \frac{n}{2}$.

But I could not find a reference where this fact is proved rigorously.

Is there any reference?

Thank you.

  • $\begingroup$ I was looking for the same thing, I haven't read it yet but may the book of Colin Adams : The knot Theory (math.harvard.edu/~ctm/home/text/books/adams/knot_book/…) can help you . In chapter 4 talks about surfaces with boundary $\endgroup$ – Astrid A. Olave H. Oct 14 '17 at 16:52
  • $\begingroup$ Where can I find the formula $2-2g-b +n$? I'm searching and I can't find any reference. Thanks $\endgroup$ – jon jones May 19 '18 at 21:59
  • $\begingroup$ I just found in $\textit{A Primer on Mapping Class Group - Farb,Marguli} \! $, pag. 18, the following formula $\chi = 2-2g-(b+n)$. It appears strange to me because if one take out two points of the sphere you get the infinity cylinder that is an Euclidean surface. $\endgroup$ – jon jones May 19 '18 at 22:11

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