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The following is a folklore result : Let $X$ be a compact Riemann surface of genus at least $2$ and let $f : X \rightarrow X$ be a biholomorphism. Then $f$ acts nontrivially on $H_1(X;\mathbb{Z})$.

I have two questions about this.

  1. Who proved this first?
  2. One proof I have been told derives this from the Lefschetz fixed point theorem. Namely, all the fixed points of $f$ have degree $1$, so the sum of the degrees of the fixed points is nonnegative. However, if $f$ acts as the identity on $H_1(X;\mathbb{Z})$, then the Lefschetz fixed point theorem says that the sum of the degrees of the fixed points is $1-2g+1$, which is negative. Where is the original source for this proof?

I found a couple of sources that give the reference

J.-P. Serre. Rigidit´e de foncteur d’Jacobi d’´echelon n ≥ 3. Sem. H. Cartan, 1960/1961, Appendix to Exp. 17, 1961.

for both questions 1 and 2. However, I see nothing in this reference about the Lefschetz fixed point theorem, and also no reference to Riemann surfaces (it is all about self-maps of abelian varieties).

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One can also deduce the statement from the Hurwitz formula. If $Y$ is the quotient of $X$ by the subgroup generated by $f$, then by Hurwitz $g(Y)<g(X)$. Since $g(Y)$ is the dimension of the subspace of $H^0(\omega^1_X)$ on which $f$ acts as the identity, the claim follows. –  rita Dec 27 '12 at 8:01

1 Answer 1

up vote 2 down vote accepted

The result itself seems to be due to A. Hurwitz: "Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893.

At least, this is what Babai refer to on page 42 here, as well as Macbeath on page 106 here.

If you read German, you should take a look at the paper and check if this is a correct reference and if Hurwitz used arguments similar to the ones appearing in the proof of Lefschetz fixed-point theorem. My guess, however, is that he used an argument similar to the one in Rita's comment.

Note also that Lefschetz in his original paper on the fixed point theorem, see page 48 here, already knew how to prove the result on conformal self-maps using his theorem. He also mentions earlier work by Kerekjarto and Birkhoff (in 2-dimensional case), so if you are very determined, you could try to dig out the earlier references.

Sorry, in my earlier answer I misunderstood your question.

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@Misha : You attribute it in your book to Hurwitz, but with no reference. Can you give me the reference where he proves it? Also, was his original proof the one using the Lefschetz fixed point theorem, or did it come later (and, if it came later, do you know who came up with it)? –  Walter Dec 27 '12 at 3:58
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Hurwitz's paper is freely available from the Göttinger Digitalisierungszentrum here: resolver.sub.uni-goettingen.de/purl?GDZPPN002253941 –  Theo Buehler Dec 27 '12 at 17:30
    
@Misha : Thank you very much for this great answer! I looked at Hurwitz's paper, and your speculation that his proof is similar to Rita's is correct. –  Walter Dec 27 '12 at 19:37

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