Kirby Problem 5.1

Definitions and mathematical statement

A subset $C\subset\mathbb R^2$ is cellular if it can be written as a nested intersection \[ C=\bigcap_{n=1}^{\infty}D_n,\qquad D_{n+1}\subset\operatorname{int}(D_n), \] where each $D_n$ is an embedded closed topological $2$-disk in the plane. An embedded disk is the image of the standard closed disk under a topological embedding; no smoothness of its boundary is required. In the plane, this describes the nonempty compact connected sets whose complement is connected.

A space has the fixed point property if every continuous map $f:C\to C$ has a point $x\in C$ with $f(x)=x$. Does every planar cellular set have this property? The map is an arbitrary continuous self-map, with no injectivity, surjectivity, or extension-to-the-plane hypothesis.

The set $C$ need not be a disk, a manifold, locally connected, or locally contractible. It may have complicated limiting geometry even though each approximating $D_n$ is a disk. Compactness ensures the nested intersection is nonempty, but the requested fixed point must lie in $C$ and be fixed by the given map on $C$ itself.

The assertion is universal in both the cellular set and the continuous self-map. A counterexample would therefore consist of one planar cellular continuum with a fixed-point-free continuous self-map. The question is specifically two-dimensional; a cellular set embedded in a higher-dimensional Euclidean space belongs to a different problem.

Short English statement

Does every continuous self-map of every cellular subset of the plane have a fixed point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3008 (KP-5.1), 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 5.1. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.