Problem 8.11 — (G.

Definitions and mathematical statement

Let $\Sigma_g$ be a closed oriented surface and let $\operatorname{Mod}(\Sigma_g)$ be its orientation-preserving mapping class group. In the usual genus-$g\geq2$ setting, the Nielsen--Thurston classification distinguishes periodic classes (finite order), nonperiodic reducible classes (preserving an essential multicurve), and pseudo-Anosov classes (admitting transverse measured foliations scaled by factors $\lambda$ and $\lambda^{-1}$, with $\lambda>1$). Low-genus cases require their corresponding conventions.

A TQFT $(V,Z)$ assigns a state space $V(\Sigma_g)$. Mapping cylinders induce an action, written \[\rho_{V,Z,g}:\operatorname{Mod}(\Sigma_g)\longrightarrow \operatorname{PGL}\left(V(\Sigma_g)\right),\] when an anomaly makes it projective, or a linear action after the required extra structures or central extension are specified. Masbaum asks whether and how the Nielsen--Thurston type of a mapping class $f$ relates to its operator $\rho_{V,Z,g}(f)$. A precise result would state which TQFTs, levels and genera are considered, what operator information is used, and what implications or recovery of mapping-class type it provides.

The excerpt proposes no specific equivalence, spectral criterion or infinite-order assertion. In particular it does not assert that a single TQFT representation distinguishes all three types. An order or spectral statement in the projective setting must be invariant under multiplication of a representative matrix by a nonzero scalar. The quantified family of theories/levels and the meaning of "relation" remain open formulation choices in this source request.

Short English statement

Relate the geometric Nielsen--Thurston type of a mapping class to the linear or projective operators supplied by TQFT. The source asks for a relationship without specifying a particular detection theorem or family of theories.

Sources

• Local UnsolvedMath registry, permanent ID $10400157$, exact source code catalogue entry.
• G. Masbaum, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 8.11, PDF page 126: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access 2026-07-29; pdf-font-structure extraction.