Constraining Mapping Class Group Homomorphisms Using Finite Subgroups

The paper’s Theorem 3 proves exactly the target statements: for g ≥ 3, every homomorphism Mod(S_g) → Homeo+(S2), Homeo+(R3), or Homeo+(S3) is trivial, and for g ≥ 5, every homomorphism Mod(S_g) → Diff+(S4) is trivial . Its method is to force a finite noncyclic subgroup of Mod(S_g) into the kernel, then invoke the Lanier–Margalit normal-generator theorem (Theorem 5) to kill the entire map , with case-by-case finite-group classifications for S2, R3, S3, and S4 . The candidate solution mirrors this strategy: it cites the same Chen–Lanier result for S2, S3, and S4, uses the normal-generator input, and handles R3 either by classification as in the paper or (as they propose) by the one-point compactification embedding Homeo+(R3) → Homeo+(S3), which also yields triviality once S3 is settled. Minor differences are present (e.g., their compactification argument for R3 versus the paper’s finite-subgroup classification), but the substance and conclusions agree.