Kirby Problem 4.22

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $X$ be a closed smooth four-manifold with $\pi_1(X)=\pi$ a good group; orientability is not assumed.

A group $\pi$ is good in the Freedman--Quinn disk-embedding sense. The accepted capped-grope criterion is: for every based height-$1.5$ disk-like capped grope $G$ and homomorphism $\rho:\pi_1(G)\to\pi$, there is an immersed disk in $G$ with the same framed attaching boundary whose double-point loops all map to $1$ under $\rho$. A capped grope is the framed four-dimensional thickening of a surface complex built by attaching higher-stage surfaces along symplectic-basis curves and capping its tips by disks; height $1.5$ means a first stage and a second stage on one half of a symplectic basis. A double-point loop follows the two preimage branches of a self-intersection, with paths to the basepoint. The source gives this criterion in Problem 4.46, Remarks (1)--(2).

For a closed four-manifold $X$, a marked structure is a pair $(Y,f)$ with $f:Y\to X$ a homotopy equivalence carrying the local fundamental class to that of $X$. In the smooth structure set $\mathcal S_{\mathrm{DIFF}}(X)$, two pairs are equivalent when a diffeomorphism of their source manifolds makes the maps to $X$ homotopic; in $\mathcal S_{\mathrm{TOP}}(X)$ use a homeomorphism instead. The forgetful map $F$ keeps the underlying topological marked structure. Let $w:\pi_1(X)\to\{1,-1\}$ be the orientation character. The involution on $\mathbb Z[\pi_1(X)]$ is $\overline g=w(g)g^{-1}$, extended linearly. Write $L_5^s(\mathbb Z[\pi_1(X)],w)$ for Wall's degree-five simple surgery-obstruction group, with this involution; the decoration $s$ records simple, rather than merely homotopy, surgery data. The Wall realization action constructs topological marked structures by a five-dimensional normal cobordism with the specified surgery obstruction.

For $n\ge2$ and $\gcd(n,q)=1$, let $L(n,q)$ be the lens space obtained by quotienting $S^3\subset\mathbb C^2$ by $(z_1,z_2)\mapsto(\zeta z_1,\zeta^qz_2)$, where $\zeta=e^{2\pi i/n}$. A fake $S^1\times L(n,q)$ is homotopy equivalent but not homeomorphic to that product. In the dihedral case, $\operatorname{UNil}_1(\mathbb Z;\mathbb Z^-,\mathbb Z^-)$ denotes Cappell's nil subgroup of the relevant surgery group; $\mathbb Z^-$ denotes the sign-action coefficient module. Its topological realization gives the nonsplittable structures on $\mathbb{RP}^4\#\mathbb{RP}^4$ discussed in the source.

Statement recorded in the supplied source.

Does the Wall action lift to an action on $\mathcal S_{\mathrm{DIFF}}(X)$ such that \[ F(A\cdot u)=A\cdot F(u)\qquad (A\in L_5^s(\mathbb Z[\pi],w),\ u\in\mathcal S_{\mathrm{DIFF}}(X))? \] Ask this without stabilizing $X$ by $S^2\times S^2$, in particular for $\pi=\mathbb Z$, $\mathbb Z\times\mathbb Z/n\mathbb Z$ with $n\ge2$, and $\mathbb Z/2\mathbb Z*\mathbb Z/2\mathbb Z$. The source's concrete realization questions are:

1. (i) Can the generator of $L_5(\mathbb Z[\mathbb Z])$ on $S^1\times S^3$ be realized by an exotic smooth $S^1\times S^3$, homeomorphic but not diffeomorphic to the standard product?
2. (ii) For $n\ge2$, can its $L_5^s$ action produce a smooth fake $S^1\times L(n,q)$ not diffeomorphic to any product $S^1\times L(n,q')$?
3. (iii) Are the nonsplittable topological manifolds in the above nil-group realization on $\mathbb{RP}^4\#\mathbb{RP}^4$ smoothable? Here nonsplittable means not decomposable as a connected sum of two manifolds each homotopy equivalent to $\mathbb{RP}^4$.

The nonorientable third case uses its orientation character in the surgery group. [S2]

Short English statement

Can every topological Wall surgery-group action for a good fundamental group be realized smoothly without stabilization? Include the product, lens-space-product, and nonsplittable real-projective-space cases.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2898 (KP-4.22). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.22 and Remarks, PDF page 207. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.