Kirby Problem 3.56

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 $Y$ be a rational homology $3$-sphere. For $\mathfrak s\in\operatorname{Spin}^c(Y)$, $\operatorname{SWF}(Y,\mathfrak s)$ denotes its Seiberg--Witten Floer stable homotopy type, obtained from finite-dimensional approximation and the Conley index of the Chern--Simons--Dirac flow, with the standard grading desuspensions. It has an $S^1$ action; for self-conjugate $\mathfrak s$ there is a $\operatorname{Pin}(2)$ refinement. Here $\operatorname{Pin}(2)=S^1\cup jS^1\subset\mathbb H^\times$, with $j^2=-1$ and $jz=\bar z j$. Equivalence means stable homotopy equivalence in the specified category, allowing the suspensions in the definition of $\operatorname{SWF}$. Nonequivariantly, a wedge of spheres is a finite wedge of ordinary sphere spectra. In the $S^1$ version of the source's wedge question, the permitted form is a wedge of one representation sphere $S^{n\mathbb C}$ and suspensions of free $S^1$-spheres $S^{2m+1}_+$, with scalar $S^1$ action on the unit sphere in $\mathbb C^{m+1}$ and an added disjoint basepoint. Representation spheres are one-point compactifications, interpreted stably for virtual suspensions. A monopole Floer L-space has zero reduced monopole Floer homology. It is an $S^1$ homotopy L-space if each spin$^c$ summand of $\operatorname{SWF}$ is a suspension of $S^0$; it is a $\operatorname{Pin}(2)$ homotopy L-space when this also holds in the $\operatorname{Pin}(2)$ category for each self-conjugate summand.

Statement recorded in the supplied source.

Resolve the following, keeping the equivariant interpretations separate:

1. Is $\operatorname{SWF}(Y,\mathfrak s)$ always a wedge of spheres, nonequivariantly? Is it always of the specified one-representation-sphere/free-spheres form $S^1$-equivariantly?
2. Is every monopole Floer L-space an $S^1$ homotopy L-space, and is it also a $\operatorname{Pin}(2)$ homotopy L-space?
3. Find two rational homology spheres having isomorphic monopole Floer homology but different Seiberg--Witten Floer spectra, separately in the nonequivariant, $S^1$ and $\operatorname{Pin}(2)$ settings.
4. Develop algorithms computing these spectra in each of those three settings.

[S2]

Short English statement

Determine the possible Seiberg--Witten Floer spectra, compare Floer L-spaces with homotopy L-spaces, find equal-homology manifolds with different spectra, and compute spectra algorithmically, with the ordinary, circle and Pin(2) versions kept distinct.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2854 (KP-3.56). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.56, p. 171 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.