# A theorem by Hopf on surfaces

I am reading a paper on Riemann surfaces and faced a problem about one of the refernces the author gave in the exlaination of one of the results.

Here is a summary of what I am reading.

Let $X_1$ and $X_2$ be two surfaces. Let $\pi_1:\widetilde{X_1}\longrightarrow X_1$, $\pi_2:\widetilde{X_2}\longrightarrow X_2$ be their universal covering surfaces. Let $G_1$ and $G_2$ be the groups of the covering transformations respectively. Let Hom($G_1, G_2$) be the set of all homomorphisms between $G_1$ and $G_2$ and define an equivalence relation by setting $j_1\sim j_2$ if there exists an inner automorphism $A: G_2\longrightarrow G_2$ such that $j_2=A \circ j_1$. Denote the equivalence class of $j$ by $[j]$

Let $f: X_1\longrightarrow X_2$ be a continuous map. We also know by a lemma that every lifting, $\tilde{f}$, of $f$ induces a homomorphism $\tilde{f_*}: G_1\longrightarrow G_2$ defined by $$\tilde{f_*}(g)\circ \tilde{f}=\tilde{f}\circ g .$$ We also have the following lemma

Lemma

Continuous mappings $f_i:X_1\longrightarrow X_2\ , \ i=1,2$, are homotopic if and only if $[\tilde{f_1}_\ast]=[\tilde{f_2}_\ast]$.

Now let $\mathbf{C}(X_1, X_2)$ denote the set of continuous mappings $f: X_1\longrightarrow X_2$, and define an equivalence relation by homotopy.

By Lemma above, we can define an injective map \begin{eqnarray} \Phi: \mathbf{C}(X_1, X_2)/\simeq &\longrightarrow& \mathbf{Hom}(G_1, G_2)/\sim \
&[f] \longrightarrow & [\tilde{f_\ast}] \end{eqnarray} Clearly this map is injective.

The question is:

The author refers to a theorem by Hopf (he does not mention the theorem) to conclude that if $X_1$ and $X_2$ are compact, then $\Phi$ is bijective.

Hopf's paper was written in German and was published 80 year ago. I do not know any German. Here is a link to Hopf's paper If any body knows what theorem it is, then their help is really appreciated. Thanks in advance.

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