Problem 2 — Show that the inclusion of Proposition 5 is a bijection.

Definitions and mathematical statement

Let $i:A\hookrightarrow B$ denote the inclusion in the source's Proposition 5. The principal problem is to prove that this inclusion is a bijection; since it is an inclusion, the remaining condition is \[ \forall b\in B\quad\exists a\in A:\quad i(a)=b. \] Neither Proposition 5 nor the definitions of $A,B,i$ are supplied, so this logical form does not identify or replace the missing geometric assertion. The source expects additional analysis of the analytic and geometric foundations of the theory.

A further question concerns an asymptotic sequence $(f_k)$: given one element $f_{k_0}$, can one describe $f_l$ at other indices, beyond those covered by the Auroux--Katzarkov doubling formula? The extracted index is printed as "$2\ r k_0$"; repeated doubling naturally suggests $2^rk_0$, but the lost exponent must be checked. The objects $f_k$, allowed indices and equivalence/description notion are absent. The final phrase "Or perhaps better" announces an unfinished reformulation and supplies no additional target.

Short English statement

Prove surjectivity of the missing Proposition 5 inclusion, and determine how one asymptotic-sequence element controls other indices beyond the stated doubling construction.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 2 and following incomplete question, supplied PDF page 177. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Proposition 5 and the Auroux--Katzarkov doubling formula are named, but their statements and a complete formula reference are absent.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000180$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports unverified status.