Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that **if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.**



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I found references to some H.Kneser's articles (1928, 1929, 1930). However, I am not sure that these give a complete proof. Also they are written in German with obsolete notation. Is there a survey on maps of surface where I can find the proof in English?


[Michael Albanese](https://math.stackexchange.com/a/2312535/257176) suggests that we use the Gromov norm. This path seems complicated (to me). Is it possible that the original proof (before Gromov) is more elementary?


For example, if $\chi(M) > \chi(N)$, then one can easily prove that $d$ is zero, using the quadratic form on $H^1(N ; \mathbb{Z})$. Can we use this approach, or something similar, to get a precise estimate on $d$ in general?



Maximal degree of a map between orientable surfaces
The best, elementary, self-contained, and clarifying proof of Kneser's result, including the desired inequality, is due to Richard Skora. See


Skora, Richard, The degree of a map between surfaces, Math. Ann. 276(1987), no.3, 415–423.


He careful handles the cases when the surfaces are not necessarily orientable, using absolute degree. He also includes a self-contained, general version of my "pinch followed by branched covering" theorem mentioned above, correcting my misstatement in the non-orientable case.


