For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\text{Homeo}(S^1);\mathbb{Z})$. I have two related questions regarding the (non)vanishing of the powers of Euler class for different subgroups of $\text{Homeo}(S^1)$.

  1. Does there exist a torsion-free, finitely generated group $G\subset\operatorname{Homeo}(S^1)$ such that $\chi^k$ for all $k$ are nonzero in $H^{2k}(G;\mathbb{Z})$?

If we let the group $G$ be the group of piecewise linear homeomorphisms of the circle $\mathbb{R}/\mathbb{Z}$ which send $\mathbb{Q}_2/\mathbb{Z}$ onto itself and have singular points only on $\mathbb{Q}_2/\mathbb{Z}$, where $\mathbb{Q}_2$ is the ring of dyadic numbers. Ghys and Sergiescu proved, in this case, $\chi^k$ for all $k$ are nonzero in $H^{2k}(G;\mathbb{Z})$ but $G$, which is one of the variants of Thompson groups, has many torsion elements.

Another example related to this question is due to Solomon Jekel. Let $\Gamma_{g}^1$ be the mapping class group of a surface of genus $g$ with a marked point. It is a subgroup of $\text{Homeo}(S^1)$. Jekel showed that in this case $\chi^{g-1}$ is nonzero in $H^{2g-2}(\Gamma_{g}^1;\mathbb{Q})$. Now, one could let $G$ be a finite index torsion-free subgroup of $\Gamma_{g}^1$. But the power $g-1$ is a threshold for the non-vanishing of powers of the Euler class.

  1. In the direction of the second example, let $\text{Tor}_g^1$ be the Torelli group of the surface of genus $g$ and a marked point. What is the threshold $k$ for which $\chi^k$ is nonzero in $H^{2k}(\text{Tor}_g^1;\mathbb{Q})$? Is it less than $g-1$?

In 1994 ("Topology, Geometry and Field Theory - Proceedings of the 31st International Taniguchi Symposium", p. 105), Morita wrote

However at present even the non-triviality of $e^2 \in H^4(Tor_g^1)$ seems to be unknown.

So I imagine not much is known in general.

  • 7
    $\begingroup$ Kupers and I recently (accidentally) answered this: $e^2$ is nontrivial, even on $Tor^1_{g, 1}$. This is Corollary 8.3 of "On the cohomology of Torelli groups". $\endgroup$ Apr 30 '20 at 17:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.