Manolescu's $\kappa$-invariant of Dehn surgeries on knots: the odd-odd case and the $\pm 2$ ambiguity

For a rational homology 3-sphere $Y$ with spin structure $\mathfrak{s}$, Manolescu's $\kappa$-invariant - an integer-valued invariant extracted from the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer stable homotopy type of $(Y,\mathfrak{s})$ - gives the sharpest known restrictions on the intersection forms of spin 4-manifolds bounded by $Y$. The problem, arising from M. Ue's study of these invariants (arXiv:2405.02597), concerns computing $\kappa$ for 3-manifolds obtained by Dehn surgery on knots in $S^3$. For surgeries $K_{p/q}$ with $p+q\equiv 1\pmod 2$ and the spin structure $\mathfrak{s}_0$ extending over $S^3$, $\kappa$ is knot-independent and equals a lens-space value (up to a $+2$ ambiguity for negative surgeries), but for $p\equiv q\equiv 1\pmod 2$ the unique spin structure $\mathfrak{s}_1$ gives a genuinely knot-dependent $\kappa$ (every integer is realized by a $+1$-surgery on a torus knot), and Ue's orbifold method fails because the characteristic sublink of the canonical continued-fraction surgery link contains the knot component itself. The value of $\kappa(K_{p/q},\mathfrak{s}_1)$ remains undetermined for arbitrary knots outside the almost-rational (plumbed/Seifert fibered) class; the ubiquitous $\pm 2$ ambiguity in the orbifold estimates is unresolved there; and the relation between $\kappa$ and the Frøyshov and Ozsváth-Szabó correction terms is unknown beyond the almost-rational case. The problem of making $\kappa$ effectively computable for odd-odd Dehn surgeries along arbitrary knots is open.

Background

A basic mechanism of low-dimensional topology is to constrain a smooth 4-manifold from the topology of its boundary. Let $Y$ be a rational homology 3-sphere (a closed oriented 3-manifold with $H_*(Y;\mathbb{Q})\cong H_*(S^3;\mathbb{Q})$) equipped with a spin structure $\mathfrak{s}$ (a lift of the frame bundle to $\mathrm{Spin}(3)$, equivalently a trivialization of the tangent bundle over the 2-skeleton; it exists iff the second Stiefel-Whitney class vanishes). Manolescu associated to $(Y,\mathfrak{s})$ the Seiberg-Witten Floer stable homotopy type $\mathit{SWF}(Y,\mathfrak{s})$, a finite pointed spectrum carrying an action of the group $\mathrm{Pin}(2)$ (the normalizer of the maximal torus in $SU(2)$, a double cover of $O(2)$); the quaternionic symmetry of the Seiberg-Witten equations for a spin structure produces this symmetry beyond the usual circle of gauge transformations. From the $\mathrm{Pin}(2)$-equivariant $K$-theory of $\mathit{SWF}(Y,\mathfrak{s})$ Manolescu extracted an integer-valued invariant $\kappa(Y,\mathfrak{s})$ and proved that it controls the intersection form of any compact spin 4-manifold bounded by $(Y,\mathfrak{s})$: if $(W,\mathfrak{s}_W)$ is a spin cobordism with $b_1(W)=0$ from $(Y_0,\mathfrak{s}_0)$ to $(Y_1,\mathfrak{s}_1)$, then $-b_2^-(W)+\frac{\sigma(W)}{8}-1\;\le\;\kappa(Y_0,\mathfrak{s}_0)-\kappa(Y_1,\mathfrak{s}_1)\;\le\;b_2^+(W)+\frac{\sigma(W)}{8}+1,$ where $b_2^\pm(W)$ are the ranks of maximal positive/negative subspaces for the intersection form and $\sigma$ is the signature; the upper bound improves when $(Y_0,\mathfrak{s}_0)$ is Floer $K_G$-split (a regularity condition on the equivariant spectrum introduced by Manolescu, automatic when $Y_0$ carries a positive-scalar-curvature metric, in which case also $\kappa(Y_0,\mathfrak{s}_0)=-n(Y_0,\mathfrak{s}_0,g)$ for the Seiberg-Witten Floer correction term $n$). These are the sharpest known 10/8-type restrictions on spin 4-manifolds with boundary and the main motivation for computing $\kappa$.

Explicit computations of $\kappa$ are rare and essentially confined to plumbed manifolds. A negative-definite plumbing graph $\Gamma$ (a weighted tree; each vertex contributes a framed 2-handle) defines a 4-manifold $W_\Gamma$ with boundary a rational homology sphere $Y_\Gamma$; $\Gamma$ is called almost rational (AR) if increasing a single vertex weight suitably can make the plumbing rational, and this class contains all Seifert fibered rational homology spheres with base orbifold $S^2$ (Seifert fibered meaning that $Y$ is foliated by circles over a 2-dimensional orbifold). Dai-Sasahira-Stoffregen proved that $\mathit{SWF}(Y_\Gamma,\mathfrak{s})$ is computed, as a $\mathrm{Pin}(2)$-spectrum, by a combinatorial model built from lattice homology (the Ozsváth-Szabó-Némethi combinatorial Floer theory of plumbed 3-manifolds), and deduced a dichotomy: writing $\bar\mu(Y,\mathfrak{s})$ for the Neumann-Siebenmann invariant (an integer lift of the Rohlin invariant available for plumbed and Seifert fibered rational homology spheres with spin structure) and $\delta(Y,\mathfrak{s})$ for the monopole Frøyshov invariant (the monopole-Floer analogue of the Heegaard Floer correction term $d(Y,\mathfrak{s})=2\delta(Y,\mathfrak{s})$ of Ozsváth-Szabó), $\kappa(Y,\mathfrak{s})=-\bar\mu(Y,\mathfrak{s})\ \text{if}\ \delta(Y,\mathfrak{s})=-\bar\mu(Y,\mathfrak{s}),\qquad \kappa(Y,\mathfrak{s})=-\bar\mu(Y,\mathfrak{s})+2\ \text{if}\ \delta(Y,\mathfrak{s})>-\bar\mu(Y,\mathfrak{s}).$ This sharpened an earlier theorem of Ue that $0\le\kappa(Y,\mathfrak{s})+\bar\mu(Y,\mathfrak{s})\le 2$ for every Seifert fibered rational homology 3-sphere. Outside the plumbed world - in particular for surgeries on hyperbolic knots - no computation of $\kappa$ is known.

Dehn surgery is the basic operation producing 3-manifolds from knots: for a knot $K\subset S^3$ and coprime integers $p,q$, the $p/q$-surgery $K_{p/q}$ glues a solid torus to the knot exterior $X_K$ so that the meridian of the solid torus maps to $p\mu+q\lambda$ ($\mu,\lambda$ the meridian and preferred longitude of $K$); $K_{p/q}$ is a rational homology sphere with $|H_1(K_{p/q})|=|p|$. The exterior $X_K$ carries exactly two spin structures $\mathfrak{s}_0$ (extending over $S^3$) and $\mathfrak{s}_1$ (not extending); the spin structure corresponding to $c\in H^1(X_K;\mathbb{Z}/2)$ extends to $K_{p/q}$ if and only if $p\,c(\mu)\equiv pq\pmod 2$. Hence for $p$ even both $\mathfrak{s}_0$ and $\mathfrak{s}_1$ extend (two spin structures on $K_{p/q}$), while for $p\equiv q\equiv 1\pmod 2$ only $\mathfrak{s}_1$ extends, and since $p$ is odd one has $H^1(K_{p/q};\mathbb{Z}/2)=0$, so $K_{p/q}$ carries a unique spin structure (still denoted $\mathfrak{s}_1$); this case includes the integral homology spheres $K_{1/n}$ with $n$ odd.

Ue (2024) introduced an orbifold technique to estimate $\kappa(Y,\mathfrak{s})$: given a compact spin 4-orbifold $X$ bounded by $(Y,\mathfrak{s})$ whose singularities are cones $cS_i$ over spherical 3-manifolds $S_i$ (spherical meaning a quotient of $S^3$ by a finite group of isometries), Manolescu's inequality applied to the complement of the cones - combined with $\kappa(S_i,\mathfrak{s}_i)=-\bar\mu(S_i,\mathfrak{s}_i)$ at the cone points and the Fukumoto-Furuta orbifold Dirac-index invariant $w$ - restricts $\kappa(Y,\mathfrak{s})$ to one or two explicit values whenever the Betti numbers $b_2^\pm(X)$ are small. For surgeries, $K_{p/q}$ is described by a framed link whose framings come from an even continued-fraction expansion of $p/q$, and the trick is to collapse (replace by a cone) a plumbing sublink with lens-space boundary; the sublink that can be collapsed is the characteristic sublink of the plumbing (the components whose framing parities encode the chosen spin structure). This works for $\mathfrak{s}_0$: $\kappa(K_{p/q},\mathfrak{s}_0)=\kappa(L(p,-q),\mathfrak{s}_0)$ for every knot $K$ when $p+q\equiv 1\pmod 2$, with negative surgeries determined only up to a possible $+2$. It fails for $\mathfrak{s}_1$ in the odd-odd case precisely because the characteristic sublink of this canonical surgery link contains the knot component $K$ itself, which cannot be coned off unless $K$ is unknotted. Ue nevertheless computed the special double-twist family $K(4c,2d+1)$ (a twist region of $4c$ half-twists chained to one of $2d+1$ half-twists), obtaining explicit formulas but with a residual $\pm 2$ ambiguity in several parameter ranges. In the odd-odd case the value of $\kappa(K_{p/q},\mathfrak{s}_1)$ genuinely depends on the knot: every integer occurs as $\kappa$ of a $+1$-surgery on a suitable torus knot. What is missing is any method for general knots, any criterion deciding the $\pm 2$ ambiguity outside the almost-rational class, and any extension of the correction-term dichotomy to non-plumbed rational homology spheres.

Problem Statement

Determine the value of Manolescu's $\kappa$-invariant of the rational homology 3-sphere $K_{p/q}$ obtained by $p/q$-Dehn surgery on a knot $K\subset S^3$ in the case $p\equiv q\equiv 1\pmod 2$, where $K_{p/q}$ carries its unique spin structure $\mathfrak{s}_1$ (the one restricting to the spin structure of the knot exterior that does not extend over $S^3$; the case includes the integral homology spheres $K_{1/n}$ with $n$ odd). The problem has two linked facets sharing one objective - making $\kappa$ effectively computable for odd-odd Dehn surgeries along arbitrary knots:

(1) Find a formula, algorithm, or structural identification that yields $\kappa(K_{p/q},\mathfrak{s}_1)$ for every knot $K$. The orbifold method of the source paper cannot produce such a formula: for the canonical continued-fraction surgery link the characteristic sublink contains the knot component itself, so the collapse fails there for every nontrivial knot (special link presentations reach only particular families such as $K(4c,2d+1)$, and only up to $\pm 2$), and $\kappa$ provably depends on $K$ (every integer is realized by a $+1$-surgery on a suitable torus knot).

(2) Resolve the $\pm 2$ ambiguity within this class: in the source's orbifold estimates for odd-odd surgeries on the double-twist family $K(4c,2d+1)$ - the negative $1/n$-surgeries with $n$ odd in Proposition 3, and the odd-odd case $p\equiv q\equiv 1\pmod 2$ of Proposition 4 in the ambiguous parameter ranges $0<c<p/8q$ (respectively $c>0$ or $c<-p/8q$) - the value of $\kappa(K_{\pm p/q},\mathfrak{s}_1)$ is pinned only up to two possibilities $\kappa_0$ and $\kappa_0+2$; determine which value occurs, or give a criterion that decides it, at least for manifolds outside the almost-rational class, where the Dai-Sasahira-Stoffregen dichotomy does not apply. The two facets are not independent: a formula valid for every knot settles the $\pm 2$ ambiguity of facet (2) for the double-twist family as well.

The knot $K$ may be any knot, including hyperbolic knots. Making $\kappa$ effectively computable for these surgery manifolds would also determine the sharpest available 10/8-type constraints on spin 4-manifolds bounded by them, in a setting where currently no technique exists.

References

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