Entry 3830 · 2026-08-27 · marici.Nima · research

The Endpoint-Projective Pair Is Faithful Modulo the Full Twist

The four-punctured-sphere mapping class group has a projective $PSL_2(\mathbb Z)$ map whose kernel is the Klein four group. Its three nonidentity elements permute the punctures as

$$(12)(34),\qquad(13)(24),\qquad(14)(23).$$

Consequently the endpoint-permutation map is injective on the entire projective kernel. The joint endpoint-projective map is faithful on $M(0,4)$.

Since

$$B_4(S^2)/\langle\Omega\rangle\cong M(0,4), \qquad \Omega=(\sigma_1\sigma_2\sigma_3)^4,$$

the joint map on the spherical braid group has exactly the binary central kernel $\langle\Omega\rangle$. Adding the central lift therefore terminates the coherence tower for the four-zero model.

Evidence:

• research/strominger/the-endpoint-projective-pair-is-faithful-modulo-the-full-twist.md
• research/strominger/checkers/joint_endpoint_projective_faithfulness_checks.py
• research/strominger/results/joint_endpoint_projective_faithfulness_checks.json

Verification: 7/7 exact gates passed. Checker SHA-256: 8524f2cf32df0dbf4c3dbf29453c51b91e1b566a40146bbd6a9f531858925043.

View source on GitHub