Timeline for Is it true that every mapping class in $\mathrm{Mod}(\Sigma_3)$ commutes with some hyperelliptic involution?
Current License: CC BY-SA 4.0
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| May 15, 2018 at 18:53 | comment | added | Ben Wieland | Arnold's argument is to produce a natural (thus MCG-invariant) homomorphism from the (mod 2) homology of the hyperelliptic curve to the homology of the sphere minus the branch locus. I think the construction is to observe that both groups receive a map from the homology of the hyperelliptic curve minus (the preimage of) the branch locus and that kernels are the same (only with mod 2 coefficients!), so the homomorphism can be defined by taking any lift, ie, perturbing the cycle to miss the branch locus. $H_1(S;\mathbb Z/2)\leftarrow H_1(S-B;\mathbb Z/2)\to H_1(\mathbb P^1-B;\mathbb Z/2)$ | |
| May 3, 2018 at 19:49 | comment | added | Ben Wieland | I got the statement wrong for $g=1$. The map factors through $S_{2g+2}$ and the representation is the doubly reduced permutation representation, but this is not faithful for $S_4$. I think that this statement is even correct for $g=0$, where the representation is trivial. | |
| May 3, 2018 at 1:40 | history | answered | Ben Wieland | CC BY-SA 4.0 |