For a spin rational homology sphere (Y,s)(Y,\mathfrak{s}), Manolescu's invariant κ(Y,s)∈Q\kappa(Y,\mathfrak{s})\in\mathbb{Q}, defined from the Pin(2)-equivariant K-theory of the Seiberg–Witten Floer (SWF) stable homotopy type, sharpens the relative 10/8-ths inequality for spin 4-manifolds with boundary YY. Montague (arXiv:2210.08565) refined κ\kappa to an equivariant invariant κ~(Y,s,ι)\widetilde{\kappa}(Y,\mathfrak{s},\iota) for involutions ι\iota whose spin lift is of odd type, computed from the G2oddG_2^{\mathrm{odd}}-equivariant complex K-theory of the SWF spectrum, where G2odd=Pin(2)×Z2Z4G_2^{\mathrm{odd}}=\mathrm{Pin}(2)\times_{\mathbb{Z}_2}\mathbb{Z}_4, and proved that κ~∈{κ,κ+2}\widetilde{\kappa}\in\{\kappa,\kappa+2\}. The paper conjectures (Conjecture 1.8 in v1, Conjecture 1.9 in v2) that every Z2\mathbb{Z}_2-equivariant spin rational homology sphere of odd type is SWF–Pin(2)-surjective, i.e. that κ~=κ\widetilde{\kappa}=\kappa always holds; the main supporting evidence is Theorem 1.10, covering negative-fibration Seifert-fibered spaces with at most four singular fibers.

Background

A rational homology sphere is a closed 3-manifold YY with the same rational homology as S3S^3; a spin structure s\mathfrak{s} is a lift of its oriented orthonormal frame bundle (an SO(3)SO(3)-bundle) to the double cover Spin(3)≅SU(2)\mathrm{Spin}(3)\cong SU(2). To such a pair, Manolescu associated the Seiberg–Witten Floer (SWF) stable homotopy type SWF(Y,s)\mathrm{SWF}(Y,\mathfrak{s}), a based spectrum built from finite-dimensional approximations of the Seiberg–Witten equations (the gauge-theoretic PDE system coupling a spinor field and a U(1)U(1) connection), well defined up to stable homotopy equivalence. Because the spinor bundle of a 3-manifold is quaternionic, the conjugation symmetry of the equations produces an action of the group Pin(2)=S1⋊Z2\mathrm{Pin}(2)=S^1\rtimes\mathbb{Z}_2 (the normalizer of a maximal torus in SU(2)SU(2)) on SWF(Y,s)\mathrm{SWF}(Y,\mathfrak{s}). From the Pin(2)\mathrm{Pin}(2)-equivariant (complex) K-theory of this spectrum Manolescu extracted a numerical invariant κ(Y,s)∈Q\kappa(Y,\mathfrak{s})\in\mathbb{Q} — twice the kk-invariant of the spectrum, the grading threshold read off from the vanishing behavior of the Pin(2)\mathrm{Pin}(2)-equivariant K-theory, in the source paper's terminology — and proved the relative 10/8-ths inequality q+κ(Y,s)≥p+1q+\kappa(Y,\mathfrak{s})\ge p+1 for any indefinite spin 4-manifold (W,t)(W,\mathfrak{t}) bounding (Y,s)(Y,\mathfrak{s}), where p=−σ(W)/8p=-\sigma(W)/8 with σ(W)\sigma(W) the signature of the intersection form of WW and q=b2+(W)q=b_2^+(W) the dimension of a maximal positive-definite subspace of H2(W;R)H^2(W;\mathbb{R}); this refines Furuta's 10/8-ths theorem for closed spin 4-manifolds and is the technical engine behind applications from the 11/8-conjecture to knot genus bounds.

Montague (arXiv:2210.08565) extended this picture equivariantly. Let ι:Y→Y\iota:Y\to Y be a diffeomorphism of order two preserving s\mathfrak{s}. The lift ι^\widehat{\iota} of ι\iota to the spinor bundle comes in two flavors: since ι\iota has order two, ι^ 2\widehat{\iota}^{\,2} is either the identity or the spin flip (the nontrivial deck transformation of the double cover Spin(3)→SO(3)\mathrm{Spin}(3)\to SO(3)), and the lift is said to be of even type in the first case and of odd type in the second. A triple (Y,s,ι^)(Y,\mathfrak{s},\widehat{\iota}) with odd-type lift is called a Z2\mathbb{Z}_2-equivariant spin rational homology sphere of odd type. For such triples the symmetry group of the SWF spectrum enlarges to the compact Lie group G2odd=Pin(2)×Z2Z4G_2^{\mathrm{odd}}=\mathrm{Pin}(2)\times_{\mathbb{Z}_2}\mathbb{Z}_4 (the fiber product modding out by the diagonal Z2\mathbb{Z}_2). Working with the representation ring R(G2odd)R(G_2^{\mathrm{odd}}) and a Q\mathbb{Q}-graded additive poset built from it (isomorphic to (Q,≤,+)(\mathbb{Q},\le,+) in the odd m=2m=2 case), Montague extracted an equivariant invariant κ~(Y,s,ι)∈Q\widetilde{\kappa}(Y,\mathfrak{s},\iota)\in\mathbb{Q} from the G2oddG_2^{\mathrm{odd}}-equivariant complex K-theory of SWF(Y,s,ι^)\mathrm{SWF}(Y,\mathfrak{s},\widehat{\iota}), and proved the dichotomy κ~(Y,s,ι)∈{κ(Y,s), κ(Y,s)+2}\widetilde{\kappa}(Y,\mathfrak{s},\iota)\in\{\kappa(Y,\mathfrak{s}),\,\kappa(Y,\mathfrak{s})+2\}. The two cases are distinguished by a purely algebraic condition (Definition 4.41 of the paper): writing I(X)⊂R(G2odd)\mathfrak{I}(X)\subset R(G_2^{\mathrm{odd}}) for the vanishing ideal of the G2oddG_2^{\mathrm{odd}}-equivariant K-theory of a spectrum XX and IPin(2)(X)⊂R(Pin(2))\mathfrak{I}_{\mathrm{Pin}(2)}(X)\subset R(\mathrm{Pin}(2)) for its Pin(2)\mathrm{Pin}(2)-counterpart, XX is called Pin(2)-surjective if the restriction map res:R(G2odd)→R(Pin(2))\mathrm{res}:R(G_2^{\mathrm{odd}})\to R(\mathrm{Pin}(2)) carries I(X)\mathfrak{I}(X) onto IPin(2)(X)\mathfrak{I}_{\mathrm{Pin}(2)}(X); for odd-type triples this is equivalent to κ~=κ\widetilde{\kappa}=\kappa. All examples computed in the paper satisfy the equality, and Theorem 1.10 proves it for spin rational homology Seifert–fibered spaces — closed 3-manifolds foliated by circles, with at most finitely many exceptional (singular) fibers — of negative fibration (in the sense of Mrowka–Ozsváth–Yu) with at most four singular fibers. The surjectivity question is not merely cosmetic: the value of κ~\widetilde{\kappa} versus κ\kappa enters directly as the boundary correction term in Montague's equivariant relative 10/8-ths inequality (Theorem 1.11) for odd-type involutions, which in turn yields obstructions to extending cyclic group actions over spin fillings, equivariant knot concordance results, and genus bounds for knots in punctured 4-manifolds. Related but distinct refinements of the 10/8 philosophy for involutions include the κKMT\kappa_{\mathrm{KMT}} invariant of Konno–Miyazawa–Taniguchi (constructed in their paper Involutions, knots, and Floer K-theory), which extends an earlier inequality of Kato to odd-type involutions and appears alongside κ~\widetilde{\kappa} in the applications; the surjectivity conjecture, however, is specific to Montague's K-theoretic framework and is not addressed by those works.

Problem Statement

Conjecture (Montague, arXiv:2210.08565, Conjecture 1.8 in v1 / Conjecture 1.9 in v2): every Z2\mathbb{Z}_2-equivariant spin rational homology sphere (Y,s,ι)(Y,\mathfrak{s},\iota) of odd type is SWF–Pin(2)-surjective. Equivalently, for every such triple the equivariant correction term satisfies

κ~(Y,s,ι)=κ(Y,s),\widetilde{\kappa}(Y,\mathfrak{s},\iota)=\kappa(Y,\mathfrak{s}),

i.e. the case κ~=κ+2\widetilde{\kappa}=\kappa+2 never occurs; equivalently again, the vanishing ideal I(SWF(Y,s,ι^))⊂R(G2odd)\mathfrak{I}(\mathrm{SWF}(Y,\mathfrak{s},\widehat{\iota}))\subset R(G_2^{\mathrm{odd}}) maps surjectively onto IPin(2)(SWF(Y,s,ι^))⊂R(Pin(2))\mathfrak{I}_{\mathrm{Pin}(2)}(\mathrm{SWF}(Y,\mathfrak{s},\widehat{\iota}))\subset R(\mathrm{Pin}(2)) under the restriction map res:R(G2odd)→R(Pin(2))\mathrm{res}:R(G_2^{\mathrm{odd}})\to R(\mathrm{Pin}(2)), in the sense of Definition 4.41 of the source paper. The conjecture is a universal statement asserted without auxiliary hypotheses: it is claimed for every odd-type triple, and in particular it is not restricted to the negative-fibration Seifert-fibered spaces of Theorem 1.10 of the source paper, nor to triples satisfying a locally SWF-spherical or locally SWF-⟨jμ⟩\langle j\mu\rangle-spherical condition in the sense of the source paper. Consequently a proof resting on any such extra hypothesis does not settle the conjecture, whereas a single odd-type triple (Y,s,ι)(Y,\mathfrak{s},\iota) with κ~(Y,s,ι)=κ(Y,s)+2\widetilde{\kappa}(Y,\mathfrak{s},\iota)=\kappa(Y,\mathfrak{s})+2 refutes it.

References

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