Problem 2.1 — Assume, for this problem, that M is a 3-manifold with non-empty boundary.

Definitions and mathematical statement

Assume, as the source explicitly requires, that $M$ is a $3$-manifold with nonempty boundary. A conventional compression body $C$ is formed from $F\times[0,1]$ by attaching $1$-handles to $F\times\{1\}$, with negative boundary $\partial_-C=F\times\{0\}$ and positive boundary $\partial_+C$; handlebody conventions cover the empty-negative-boundary case. The source does not supply the relevant handle decomposition, surface markings or boundary-equivalence category.

The preceding double-coset setup $H\phi H$ is also missing. Conventionally $H$ is the subgroup of a Heegaard-surface mapping class group whose classes extend over a handlebody. For a marked compression body, the corresponding extension subgroup would consist of boundary mapping classes extending over $C$, with the required behavior on $\partial_-C$. Depending on the gluing convention, replacing one handlebody side gives a double coset of the schematic form \[ H_C\phi H\qquad\text{or}\qquad H\phi H_C. \] These are convention-dependent descriptions, not a recovered definition of the source's groups.

Problem 2.1 asks to make this modification precise from the compression body's handle decomposition, and then determine what happens for a knot space. It requires identifying the actual subgroup/actions and equivalence relation encoded by the modified coset, not merely naming an extension subgroup. In the usual knot-exterior interpretation, $S^3\setminus\operatorname{int}\nu K$ has torus boundary, so that boundary must be retained in the compression-body setup; whether meridian/longitude data or boundary markings must be fixed is not stated. The source does not define knot space or fix the ambient manifold. No specific double-coset classification or subgroup generators are supplied.

Short English statement

Adapt the source's handlebody double-coset description to a three-manifold with boundary by replacing one side with the correct compression-body subgroup, using its handle decomposition. Work out the knot-space case. The original setup and peripheral-marking conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID $11000158$, exact source code catalogue entry, Problem 2.1, PDF page 156. No individual proposer or proposed year is supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and calls for a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; the research summary says only "Literature status: Truncated." No separate reference for the preceding double-coset construction is supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.