Kirby Problem 1.45

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.

The smooth concordance group $\mathcal C$ consists of oriented knots modulo smooth annuli in $S^3\times[0,1]$, with connected sum as addition and unknot $U$ as zero. A knot is smoothly slice if it bounds a smooth disk in $B^4$, and topologically slice if it bounds a locally flat disk. For a pattern $P\subset S^1\times D^2$, the satellite $P(K)$ uses the preferred longitude; its winding number is the signed homology class in the solid torus. Satellite operations induce functions on $\mathcal C$, not necessarily homomorphisms. For a group $\pi$, put $\pi^{(0)}=\pi$ and $\pi^{(r+1)}=[\pi^{(r)},\pi^{(r)}]$. If $W$ is a compact spin $4$-manifold, let $W^{(r)}$ be the cover corresponding to $\pi_1(W)^{(r)}$. On $H_2(W^{(r)};\mathbb Z)$ the equivariant intersection and self-intersection forms, denoted $\lambda_r,\mu_r$, record intersections with their deck-transformation labels in $\mathbb Z[\pi/\pi^{(r)}]$ (with the usual self-intersection quotient). An $r$-Lagrangian is a submodule on which both forms vanish and whose image in $H_2(W;\mathbb Z)$ is a half-rank direct-summand Lagrangian for the ordinary intersection form. It has $r$-duals if it has generators represented by $r$-surfaces $\ell_i$ and there are $r$-surfaces $d_j$ with $\lambda_r(\ell_i,d_j)=\delta_{ij}$ and $\operatorname{rank}H_2(W)=2\#\{\ell_i\}$. Here an $r$-surface has fundamental group mapping into $\pi^{(r)}$, and its chosen lift represents the specified class. A knot is $r$-solvable if its zero-surgery bounds such a spin $W$ inducing an isomorphism on $H_1$ and admitting an $r$-Lagrangian with $r$-duals. It is $(r+\tfrac12)$-solvable if $W$ admits an $(r+1)$-Lagrangian whose projection has $r$-duals. Write $\mathcal F_n$ for these concordance classes, $n\in\tfrac12\mathbb N_0$. For an integer $r\geq0$, a knot is $r$-positive if it bounds a null-homologous smooth disk $\Delta$ in a compact oriented simply connected smooth $V$ with $\partial V=S^3$, positive definite intersection form, and a basis of $H_2(V)$ represented by disjoint closed oriented surfaces $S_i\subset V\setminus\Delta$ with $\pi_1(S_i)$ mapping into $\pi_1(V\setminus\Delta)^{(r)}$. Define $r$-negative by a negative definite form. Let $\mathcal T_r$ be the subgroup of topologically slice classes that are both $r$-positive and $r$-negative, allowing different $V,\Delta$ in the two conditions. These are the solvable and bipolar filtrations, with definitions from the original construction papers. The topological (respectively smooth) slice genus $g_4^{\mathrm{top}}(K)$ (respectively $g_4(K)$) is the least genus of a connected oriented locally flat (respectively smooth) properly embedded surface in $B^4$ bounding $K$. An $L^2$-signature defect compares the von Neumann signature of a cover specified by a homomorphism to a group with the ordinary signature; its boundary version is the Cheeger--Gromov $\rho$ invariant. The higher-order signature invariants in the filtration obstructions are of this type.

Statement recorded in the supplied source.

(a) For each $n\in\tfrac12\mathbb N_0$ with $n\geq\tfrac52$ and every integer $g>1$, does there exist $[K]\in\mathcal F_n$ with $g_4^{\mathrm{top}}(K)\geq g$? (b) For every $r\in\mathbb N_0$ and every integer $g>1$, does there exist $[K]\in\mathcal T_r$ with $g_4(K)\geq g$? (c) Can the $L^2$-signature invariants obstructing deeper membership in these filtrations yield lower bounds for slice genus? [S1][S2][S3][S4]

Short English statement

Look for arbitrarily large slice genus at every specified depth of the filtrations and for slice-genus bounds from the higher-order signature obstructions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2704 (KP-1.45). 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, author preliminary edition (2026), Problem 1.45 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Tim D. Cochran, Kent E. Orr and Peter Teichner, Knot concordance, Whitney towers and L2-signatures, Annals of Mathematics 157 (2003), 433--519, Definitions 1.2, 8.1, 8.3, 8.5 and 8.7; Theorem 8.8. https://arxiv.org/abs/math/9908117v2.
• S4 Tim D. Cochran, Shelly Harvey and Peter Horn, Filtering smooth concordance classes of topologically slice knots, Geometry and Topology 17 (2013), 2103--2162, Definitions 2.1, 2.2, 2.4 and 2.6. https://arxiv.org/abs/1201.6283.