Kirby Problem 3.58

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.

For a closed oriented $3$-manifold $Y$ and a spin$^c$ structure $\mathfrak s$, write $\operatorname{HF}^{\circ}(Y,\mathfrak s)$ for Heegaard Floer homology and $\operatorname{HM}^{\circ}(Y,\mathfrak s)$ for its corresponding monopole Floer version. For $\circ=-,+,\infty$, the monopole versions are respectively the hat, check and bar monopole theories; the hat Heegaard theory corresponds to the associated $U$-mapping-cone version. Use matching coefficients, gradings, $U$ actions and orientation conventions in both theories, with canonical coherent orientations for integral Heegaard coefficients. A spin$^c$ cobordism $(W,\mathfrak t):(Y_0,\mathfrak s_0)\to(Y_1,\mathfrak s_1)$ is a compact oriented $4$-manifold with boundary $-Y_0\sqcup Y_1$ and a spin$^c$ structure restricting to $\mathfrak s_i$. Its Floer maps are $F^{\circ}_{W,\mathfrak t}$ and $\operatorname{HM}^{\circ}_{W,\mathfrak t}$, using matching homology orientations. The mixed Ozsváth--Szabó invariant $\Phi_{X,\mathfrak t}$ of a closed $4$-manifold is the mixed minus-to-plus Floer cobordism invariant, while $\operatorname{SW}_{X,\mathfrak t}$ counts the corresponding Seiberg--Witten solutions. Their comparison uses the same homology orientation, insertions and, whenever necessary, chamber data, in every setting where these invariants are defined.

Statement recorded in the supplied source.

Construct isomorphisms $\Psi^{\circ}_{Y,\mathfrak s}$ between the corresponding Heegaard and monopole Floer groups, compatible with their structures and satisfying, for every spin$^c$ cobordism, \[ \Psi^{\circ}_{Y_1,\mathfrak s_1}\circ F^{\circ}_{W,\mathfrak t} =\operatorname{HM}^{\circ}_{W,\mathfrak t}\circ\Psi^{\circ}_{Y_0,\mathfrak s_0}, \] with the standard common sign convention over $\mathbb Z$. In particular, prove equality of $\Phi_{X,\mathfrak t}$ and $\operatorname{SW}_{X,\mathfrak t}$ for closed $4$-manifolds, using corresponding orientation and chamber conventions on their common domains of definition. [S2]

Short English statement

Can the isomorphisms between Heegaard and monopole Floer homology be chosen to respect every cobordism map, so that the closed four-manifold mixed invariant equals the Seiberg--Witten invariant?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2856 (KP-3.58). 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 3.58, p. 173 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.