Kirby Problem 3.62

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 closed oriented $3$-manifolds, a continuous map $f:Y\to Z$ has degree $d$ if $f_*[Y]=d[Z]$ in $H_3(Z;\mathbb Z)$. Write $\widehat{\operatorname{HF}}(Y)$ for integral hat Heegaard Floer homology with the canonical coherent orientation system and \[ \operatorname{HF}_{\rm red}(Y)=\operatorname{coker}\left(\operatorname{HF}^{\infty}(Y)\to\operatorname{HF}^{+}(Y)\right). \] Here $\operatorname{rk}A=\dim_{\mathbb Q}(A\otimes_{\mathbb Z}\mathbb Q)$, summed over all spin$^c$ structures, as in the integral convention of the source's section introduction. For knots in $S^3$, $\widehat{\operatorname{HFK}}(S^3,K;i)$ denotes the summand in Alexander grading $i\in\mathbb Z$. For a nullhomotopic knot $K\subset Z$, use its meridian and Seifert longitude to define $p/q$ Dehn surgery $Z_{p/q}(K)$, where $\gcd(p,q)=1$ and $q>0$. A proper degree-one map of knot exteriors maps boundary to boundary and sends their relative fundamental classes to one another.

Statement recorded in the supplied source.

Determine how Floer invariants behave under maps of nonzero degree. In particular, for every degree-one map of closed oriented manifolds $f:Y\to Z$, are both inequalities \[ \operatorname{rk}\widehat{\operatorname{HF}}(Y)\ge\operatorname{rk}\widehat{\operatorname{HF}}(Z),\qquad \operatorname{rk}\operatorname{HF}_{\rm red}(Y)\ge\operatorname{rk}\operatorname{HF}_{\rm red}(Z) \] valid? The source also asks the refinements: if $Y=Z_{p/q}(K)$ for a nullhomotopic knot, must $\operatorname{rk}\widehat{\operatorname{HF}}(Y)\ge p\operatorname{rk}\widehat{\operatorname{HF}}(Z)$? If there is a proper degree-one map from the exterior of $K\subset S^3$ to that of $L\subset S^3$, must \[ \operatorname{rk}\widehat{\operatorname{HFK}}(S^3,K)\ge\operatorname{rk}\widehat{\operatorname{HFK}}(S^3,L), \] and must this inequality hold in every Alexander grading? [S2]

Short English statement

Find rank inequalities under nonzero-degree maps, especially degree-one monotonicity for hat and reduced Heegaard Floer homology, the stronger surgery bound, and total and Alexander-graded knot Floer bounds under degree-one maps of exteriors.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2860 (KP-3.62). 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 3.62, p. 175 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.