Kirby Problem 2.7

Definitions and mathematical statement

For $n\geq2$, the braid group has presentation \[ B_n=\left\langle\sigma_1,\ldots,\sigma_{n-1}\ \middle|\ \begin{array}{ll} \sigma_i\sigma_j=\sigma_j\sigma_i&(|i-j|\geq2),\\ \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}&(1\leq i<n-1) \end{array}\right\rangle . \] Geometrically its elements are isotopy classes of $n$ strands in a disk times an interval, each transverse to the interval direction; endpoints may be permuted. The operation stacks braids. Put $B_1=\{1\}$.

Classify all homomorphisms $\phi:B_n\to B_m$ for arbitrary positive integers $n,m$. Such a homomorphism is determined by images $b_i=\phi(\sigma_i)$ in $B_m$ satisfying the displayed braid and commuting relations. The target is a structural classification of these tuples, or equivalently of all such maps, rather than merely restating those relations.

A useful classification may be given up to simultaneous conjugation of the images, provided the equivalence is stated and all maps can be recovered from the described families. It must include maps with cyclic image, noninjective maps, and injective maps; no restriction comparing $m$ and $n$ is imposed.

Short English statement

Describe every homomorphism between two braid groups, for all numbers of source and target strands.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2755 (KP-2.7), 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 2.7. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.