# Contradiction in Computation of Homology Groups of the Mapping Class Group of a Surface?

One of the two main results of a paper by Nathalie Wahl on homological stability of the mapping class group of a surface is the following:

Theorem 1.2 The map $$H_*(\delta_g) : H_*(\Gamma_{g,1};\mathbb{Z}) \to H_*(\Gamma_{g,0};\mathbb{Z})$$ is surjective for $$* \leq \frac23g+1$$ and an isomorphism for $$* \leq \frac23g$$.

Here, $$\Gamma_{g,r} := \pi_0\text{Diff}(S_{g,r} \, \text{rel} \, \partial)$$ denotes the mapping class group of a surface $$S_{g,r}$$ with genus $$g$$ and $$r$$ boundary components such that all diffeomorphisms fix the boundary of the surface. Moreover, the map $$\delta_g : \Gamma_{g,1} \to \Gamma_{g,0}$$ is induced by gluing a disk to the one boundary component in $$S_{g,1}$$.

There is a much earlier paper by Harer, the main result of which (in our setting) is the following:

Theorem A Consider $$\Gamma_{g,r}$$ as above. Let $$n$$ be the number of distinguished points in $$S_{g,r}$$, i.e. points which are fixed by diffeomorphisms of the surface. Then we have $$H_2(\Gamma_{g,r}) \cong \begin{cases} \mathbb{Z}^{n+1} & g \geq 5, \; r+n > 0\\ \mathbb{Z} \oplus \mathbb{Z}/(2g-2)\mathbb{Z} & g \geq 5, \;r=n=0 \end{cases}$$

In the setting of Wahl's paper, I believe we have $$n=0$$, since the author never mentions any distinguished points. Now, if we fix some $$g \geq 5$$ then we know that the map $$H_2(\delta_g)$$ is an isomorphism $$H_2(\Gamma_{g,1}) \to H_2(\Gamma_{g,0})$$, since the inequality $$2 \leq \frac23 g$$ is true by the assumption on $$g$$. However, using Theorem A, this translates to an isomorphism $$\mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}/(2g-2)\mathbb{Z}$$, since $$n=0$$, which is a contradiction, as the torsion is non-trivial.

Question: What exactly goes wrong in this line of reasoning? Am I overlooking any details?

One idea I have is that perhaps Wahl assumes that we always have one distinguished point, i.e. $$n=1$$, or that we can assume without loss of generality that all diffeomorphisms of $$S_{g,r}$$ fix a single point (outside of the boundary $$\partial S_{g,r}$$ of course). This would at least be sufficient to resolve the contradiction.

• I glanced at both papers and don't immediately see how to resolve this. Wahl does explicitly talk about punctures (fixed points) in the intro and says that the theorem for n punctures follows from the theorem for 0 punctures, but on the face of it, there does seem to be a contradiction in the 0 puncture case. It may just be a matter of conventions, but have you considered emailing Wahl and asking her. You can report back your findings in an answer to this question. Commented Jun 28 at 19:19
• @CheerfulParsnip I contacted her and will report back as soon as the problem is (hopefully) resolved! Commented Jun 29 at 12:38

Kindly Nathalie Wahl has pointed out that the contradiction can easily be resolved as follows: It turns out that Harer's result simply contained a mistake in the torsion part, and his paper was later corrected. A correct statement of the result is given in a paper by Korkmaz & Stipsicz, namely

Theorem 1.1 If $$g \geq 4$$ and $$n,r \geq 0$$, then $$H_2(\Gamma_{g,r}^n) \cong \mathbb{Z}^{n+1}$$.

In particular this covers our above case with $$g \geq 5$$, $$r=1$$ and $$n=0$$. Then Theorem 1.2. of Wahl's paper says we have an isomorphism $$\mathbb{Z} \to \mathbb{Z}$$, so no contradiction.