Kirby Problem 5.11

Definitions and mathematical statement

A $2$-dimensional CW complex is built from vertices, edges, and $2$-cells, the latter attached by continuous maps of their boundary circles to the $1$-skeleton. A subcomplex is a union of cells containing the entire attaching boundary of every cell it includes. Neither finiteness nor local finiteness is imposed unless a particular special case is being considered.

A connected CW complex $X$ is aspherical if its universal cover is contractible, equivalently if $\pi_i(X)=0$ for every $i\ge2$. A disconnected complex is called aspherical when each connected component is aspherical.

Whitehead's question asks whether every subcomplex $Y$ of an aspherical $2$-complex $X$ is itself aspherical. Explicitly, for every such pair $Y\subseteq X$, must the universal cover of each component of $Y$ be contractible?

The inclusion is a cellular subcomplex inclusion, not an arbitrary embedded subset or a map inducing an injection of fundamental groups. No injectivity assumption on $\pi_1(Y)\to\pi_1(X)$ is present. An affirmative statement would therefore follow from the dimensional and asphericity hypotheses alone.

The question includes finite complexes, the original motivating setting, as well as the unrestricted CW formulation used by the source. In dimension two, deciding the vanishing of $\pi_2$ is enough for asphericity, by applying the Hurewicz and Whitehead theorems to the simply connected universal cover; nevertheless the stated conclusion is vanishing in all higher degrees.

Short English statement

Is every subcomplex of an aspherical two-dimensional CW complex also aspherical?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3018 (KP-5.11), 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 5.11. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.