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Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In

Meyer, Werner Die Signatur von Flächenbündeln. Math. Ann. 201 (1973), 239–264.

a $2$-cocycle $\tau_g$ on $Sp(2g,\mathbb{Z})$ is constructed with the following property. Let $X$ be an arbitrary closed surface and let $f : X \rightarrow \Gamma_g$ be a homomorphism. There is an associated fiber bundle $E_f \rightarrow X$ whose fiber is $\Sigma_g$. Letting $\tau(E_f)$ be the signature of the $4$-manifold $E_f$, we then have

$$-\tau(X) = f^{\ast}(\pi^{\ast}([\tau_g]))([X]) \in \mathbb{Z}.$$

Meyer also proves that $\tau(X)$ lies in $4 \mathbb{Z}$ and that any such integer can be achieved.

I have seen the following two claims in several places with references to Meyer's paper:

  1. The homology class $[\tau_g] \in H_2(Sp(2g,\mathbb{Z}))$ is divisible by $4$.
  2. The element $\frac{1}{4} [\tau_g] \in H_2(Sp(2g,\mathbb{Z}))$ generates $H_2(Sp(2g,\mathbb{Z}))$.

But these don't appear to actually be proven in Meyer's paper! Can anyone either tell me how to prove these or point me to a paper which contains the proofs?

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At least the first statement is proved in Meyer's thesis (Die signatur von lokal...), and a newer proof can be found in:

MR1823605 (2002a:57038) Reviewed Hoster, Matthias(D-MNCH) A new proof of the signature formula for surface bundles. (English summary) Topology Appl. 112 (2001), no. 2, 205–213. 57R20 (55R10 55R40 57T10)

I am guessing that since both publications begin with "Die Signatur", and have the same author, people accidentally cite the wrong one.

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  • $\begingroup$ Thanks, this is very helpful! I'll hold off on accepting it for the moment in case someone manages to answer my second question. $\endgroup$ – Patrick Apr 13 '14 at 23:44

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