Kirby Problem 2.3

Definitions and mathematical statement

Let $S$ be a connected orientable surface of finite type. The mapping class group $\operatorname{Mod}(S)$ is the group of isotopy classes of orientation-preserving homeomorphisms of $S$, fixing its boundary pointwise and preserving its finite puncture set. Composition induces the group operation. A complex linear group is a group admitting an injective homomorphism into $\operatorname{GL}_N(\mathbb C)$ for some finite positive integer $N$.

Is $\operatorname{Mod}(S)$ linear for every finite-type surface $S$? Thus, for each fixed $S$, does there exist a finite $N=N(S)$ and a representation \[ \rho_S:\operatorname{Mod}(S)\longrightarrow\operatorname{GL}_{N(S)}(\mathbb C) \quad\text{with}\quad\ker\rho_S=\{1\}? \] There is no requirement that $N$ be independent of genus, punctures, or boundary components. Faithfulness means that every nontrivial mapping class acts by a nonidentity matrix. Actions on ordinary homology are examples of linear representations, but their kernels must be checked and cannot simply be discarded. The target is an ordinary finite-dimensional representation over $\mathbb C$, as specified by the source's convention for linearity.

Short English statement

Does each finite-type surface's mapping class group have a faithful finite-dimensional complex matrix representation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2751 (KP-2.3), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.3. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.