Kirby Problem 3.36

Definitions and mathematical statement

Let $F$ be a connected surface and $M$ a three-manifold, and let $f:F\to M$ be a smooth immersion. Its normal line bundle is $\nu_f=f^*TM/TF$. The immersion is two-sided if this line bundle is trivial, equivalently if it admits a continuous nowhere-zero normal vector field. Choose basepoints to define $f_*:\pi_1(F)\to\pi_1(M)$; injectivity and the existence of the loops below do not depend on that choice.

Assume that $f_*$ is not injective. The simple loop conjecture asks whether there exists an embedded circle $c:S^1\hookrightarrow F$ with \[ [c]\ne1\text{ in }\pi_1(F),\qquad f_*[c]=1\text{ in }\pi_1(M). \] An unbased curve determines a conjugacy class, and lying in the kernel is well-defined because the kernel is normal. The first condition says the curve is essential, or not null-homotopic in $F$; without it, tiny circles bounding disks would make the question vacuous.

The image loop $f\circ c$ need not be embedded in $M$. It is the curve on the domain surface that must be simple. Thus the question asks whether a nontrivial kernel necessarily contains a simple representative, not merely whether some possibly self-intersecting loop lies in it.

Short English statement

If a two-sided immersed surface fails to inject on fundamental groups, must an essential simple loop on that surface become null-homotopic in the ambient three-manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2834 (KP-3.36), 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 3.36. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.