Kirby Problem 5.4

Definitions and mathematical statement

Let $M$ be a Hausdorff second-countable topological manifold. Its homeomorphism group $\operatorname{Homeo}(M)$ consists of continuous bijections $M\to M$ with continuous inverses, under composition. Give it the compact-open topology: subbasic neighborhood conditions have the form $h(K)\subset U$ with $K\subset M$ compact and $U\subset M$ open. This is the usual transformation-group topology in the question.

A metrizable space $A$ is an absolute neighborhood retract (ANR, in the metrizable category) if whenever $A$ is embedded as a closed subset of any metrizable space $Z$, there is an open neighborhood $U$ of $A$ in $Z$ and a continuous retraction $r:U\to A$, meaning $r(a)=a$ for all $a\in A$. The condition must hold for every such closed embedding, not only for a preferred ambient space.

Is $\operatorname{Homeo}(M)$ an ANR for every manifold $M$? The group is regarded as a topological space for this question; the retraction need not be a homomorphism and no group structure on $Z$ is assumed.

The desired neighborhood-retraction property is stronger than merely constructing contractions of small neighborhoods of the identity. Local contractibility alone should not be substituted for the ANR condition. All homeomorphisms are included; no orientation-preserving restriction, compact-support restriction, or boundary-fixing subgroup is silently imposed.

Short English statement

Is the full homeomorphism group of every manifold, with its compact-open topology, an absolute neighborhood retract?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3011 (KP-5.4), 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 5.4. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.