For a spin 4-manifold WW bounded by a Seifert fibered rational homology 3-sphere (Y,s)(Y,s), Ue (2022) proved μˉ\bar\mu-based constraints on the intersection form of WW and, for base orbifolds of genus 00 with orientable base, compared them with Manolescu's κ\kappa-invariant constraints, showing 0≤κ(Y,s)+μˉ(Y,s)≤20 \le \kappa(Y,s)+\bar\mu(Y,s) \le 2 with κ=−μˉ\kappa=-\bar\mu under parity and degree hypotheses. Every Seifert fibered rational homology 3-sphere has base orbifold with underlying surface S2S^2 or RP2\mathbb{RP}^2, so the problem is to complete this constraint picture for the regimes the source explicitly leaves untreated: Seifert fibrations over non-orientable base 2-orbifolds, and the degenerate cobordism cases b2+(W)=0b_2^+(W)=0 or b2−(W)=0b_2^-(W)=0 where the known κ\kappa-inequalities fail and only weaker bounds hold. Later work (Stoffregen 2020; Dai–Sasahira–Stoffregen 2023) determines κ\kappa for almost-rational plumbings — hence for all genus-00 orientable-base Seifert rational homology spheres — but covers neither untreated regime.

Background

A compact smooth spin 4-manifold (W,sW)(W,s_W) bounded by a closed spin 3-manifold (Y,s)(Y,s) carries an intersection form on H2(W;Z)H_2(W;\mathbb Z), with signature σ(W)\sigma(W) and positive/negative dimensions b2±(W)b_2^\pm(W). Classical relative 10/8-type results constrain this form in terms of invariants of the boundary. Furuta's 10/8 theorem for closed spin 4-manifolds (b2≥54∣σ∣+2b_2 \ge \tfrac{5}{4}|\sigma|+2) has boundary versions in which the role of the Rokhlin invariant is played by rational lifts of it.

Two such lifts appear here. First, the Neumann–Siebenmann invariant μˉ(Y,s)∈Q\bar\mu(Y,s) \in \mathbb Q is defined for plumbed rational homology 3-spheres (3-manifolds obtained by plumbing disk bundles over spheres according to a weighted graph) by μˉ(Y,s)=(σ(P(Γ))−wΓ⋅wΓ)/8\bar\mu(Y,s)=(\sigma(P(\Gamma))-w_\Gamma\cdot w_\Gamma)/8, where P(Γ)P(\Gamma) is the plumbed 4-manifold of a plumbing graph Γ\Gamma for YY and wΓw_\Gamma is the characteristic element of H2(P(Γ);Z)H_2(P(\Gamma);\mathbb Z) determined by the spin structure (the value is independent of the chosen graph); it is a lift of the Rokhlin invariant μ=σ(W)/8 mod 2Z\mu = \sigma(W)/8 \bmod 2\mathbb Z. Seifert fibered rational homology 3-spheres — 3-manifolds fibered in circles over a closed 2-dimensional orbifold (a surface with cone points), with Seifert invariants {b,(a1,b1),…,(an,bn)}\{b,(a_1,b_1),\dots,(a_n,b_n)\}, singular-fiber multiplicities aia_i, and degree deg⁡Y=b+∑ibi/ai\deg Y = b+\sum_i b_i/a_i (Ue's sign convention; some authors use the opposite sign) — form a prominent plumbed class; every Seifert fibered rational homology 3-sphere has base orbifold with underlying surface S2S^2 or RP2\mathbb{RP}^2 (Alfieri–Baldwin–Dai–Sivek, reference 12), and the plumbing theory of Neumann and Neumann–Raymond covers both orientable and non-orientable base orbifolds — for Seifert manifolds over non-orientable bases, whose plumbing pieces are not all disk bundles over S2S^2, the plumbing description and the definition of the μˉ\bar\mu-invariant are due to Neumann–Raymond (reference 10). Second, for any rational homology 3-sphere with spin structure (Y,s)(Y,s), Manolescu defined κ(Y,s)∈Q\kappa(Y,s) \in \mathbb Q from the Pin(2)-equivariant Seiberg–Witten Floer stable homotopy type SWF(Y,s)\mathrm{SWF}(Y,s): here Pin(2) is the normalizer of the circle in SU(2)SU(2), the symmetry group of the Seiberg–Witten equations beyond ordinary gauge symmetry, and κ\kappa is read off the equivariant koko-grading of SWF\mathrm{SWF}. Manolescu proved cobordism inequalities: for a spin cobordism (W;s0,s1)(W;s_0,s_1) with b1(W)=0b_1(W)=0 between rational homology 3-spheres (Y0,s0)(Y_0,s_0) and (Y1,s1)(Y_1,s_1),

−b2−(W)+σ(W)8−1≤κ(Y0,s0)−κ(Y1,s1)≤b2+(W)+σ(W)8+1,-b_2^-(W)+\frac{\sigma(W)}{8}-1 \le \kappa(Y_0,s_0)-\kappa(Y_1,s_1) \le b_2^+(W)+\frac{\sigma(W)}{8}+1,

where the constant −1-1 (resp. +1+1) may be dropped when b2−(W)b_2^-(W) (resp. b2+(W)b_2^+(W)) is even, with further improvements when a boundary is Floer KGK_G-split (a splitting condition on the relevant equivariant Conley index); these imply relative 10/8-type constraints on intersection forms.

Ue (arXiv:2206.05412) proved, for (Y,s)(Y,s) a Seifert rational homology 3-sphere over an orientable base 2-orbifold, that μˉ(Y,s)≡σ(W)/8(mod2Z)\bar\mu(Y,s) \equiv \sigma(W)/8 \pmod{2\mathbb Z} and

−b2−(W)+σ(W)8≤μˉ(Y,s)≤b2+(W)+σ(W)8,-b_2^-(W)+\frac{\sigma(W)}{8} \le \bar\mu(Y,s) \le b_2^+(W)+\frac{\sigma(W)}{8},

using the Fukumoto–Furuta orbifold index invariant ww, and compared the two theories for genus-00 bases: 0≤κ(Y,s)+μˉ(Y,s)≤20 \le \kappa(Y,s)+\bar\mu(Y,s) \le 2; equality κ=−μˉ\kappa=-\bar\mu holds when some singular-fiber multiplicity is even and deg⁡Y>0\deg Y>0, and for integral homology spheres κ+μˉ∈{0,2}\kappa+\bar\mu \in \{0,2\} with both values occurring. For the all-odd-multiplicity case, Remark 5 of that paper proves κ=−μˉ\kappa=-\bar\mu only under extra conditions on the spin structure. Subsequently, Dai–Sasahira–Stoffregen gave a combinatorial description of SWF\mathrm{SWF} for almost-rational plumbed homology spheres — a class containing all Seifert rational homology spheres with base orbifold S2S^2 — which determines κ\kappa there: κ(Y,s)=−μˉ(Y,s)\kappa(Y,s)=-\bar\mu(Y,s) if −μˉ(Y,s)=δ(Y,s)-\bar\mu(Y,s)=\delta(Y,s) (the monopole Frøyshov invariant) and κ(Y,s)=−μˉ(Y,s)+2\kappa(Y,s)=-\bar\mu(Y,s)+2 otherwise; Stoffregen earlier proved Manolescu's conjecture β=−μˉ\beta=-\bar\mu (with β\beta one of Manolescu's equivariant invariants) for Seifert integral homology spheres. In parallel, Pin(2)-equivariant KO-theoretic methods of Lin, refined by Hopkins–Lin–Shi–Xu's computation of the Pin(2)-equivariant Mahowald invariant, yield intersection-form constraints that are sharper than the μˉ\bar\mu- and κ\kappa-bounds in some cases. What remains outside all of these results are precisely the regimes the source explicitly leaves untreated: non-orientable base orbifolds (necessarily with underlying surface RP2\mathbb{RP}^2; never treated, even for the μˉ\bar\mu-inequalities), and the degenerate cobordism regimes b2+(W)=0b_2^+(W)=0 or b2−(W)=0b_2^-(W)=0 in which Manolescu's inequality technique provably fails and only weak one-sided bounds are known.

Problem Statement

Let (Y,s)(Y,s) be a Seifert fibered rational homology 3-sphere with spin structure over an arbitrary compact base 2-orbifold — the underlying surface of the base orbifold is necessarily S2S^2 or RP2\mathbb{RP}^2, so this includes in particular non-orientable base orbifolds — and let (W,sW)(W,s_W) be a compact smooth spin 4-manifold with boundary (Y,s)(Y,s). Ue (2022) proved, for orientable base orbifolds, that μˉ(Y,s)≡σ(W)/8(mod2Z)\bar\mu(Y,s) \equiv \sigma(W)/8 \pmod{2\mathbb Z} with −b2−(W)+σ(W)/8≤μˉ(Y,s)≤b2+(W)+σ(W)/8-b_2^-(W)+\sigma(W)/8 \le \bar\mu(Y,s) \le b_2^+(W)+\sigma(W)/8, and, for genus-00 orientable bases, the comparison 0≤κ(Y,s)+μˉ(Y,s)≤20 \le \kappa(Y,s)+\bar\mu(Y,s) \le 2 with the case analysis above. Determine the corresponding constraint picture in the regimes left untreated, as one objective — completing the μˉ\bar\mu/κ\kappa intersection-form constraints for spin fillings of all Seifert rational homology 3-spheres:

  1. (Non-orientable bases) Do the analogues of the μˉ\bar\mu-inequalities of Theorem 1 and of the κ\kappa-versus-μˉ\bar\mu comparison of Theorem 2 hold for Seifert fibrations over non-orientable base 2-orbifolds (where μˉ\bar\mu is defined via the Neumann–Raymond plumbing theory of reference 10)? Prove the analogous statements, or exhibit explicit counterexamples.
  2. (Degenerate cobordisms) Determine sharp inequalities, in terms of μˉ\bar\mu and κ\kappa, on the intersection form in the regimes b2+(W)=0b_2^+(W)=0 or b2−(W)=0b_2^-(W)=0 (with b1(W)=0b_1(W)=0 as in Manolescu's theorems): both for spin fillings (W,sW)(W,s_W) with ∂W=(Y,s)\partial W=(Y,s), where Manolescu's Theorem 5 fails and only the weak one-sided bounds −κ(Y,s)≤b2+(W)+σ(W)/8-\kappa(Y,s)\le b_2^+(W)+\sigma(W)/8 (resp. κ(Y,s)≤b2−(W)−σ(W)/8\kappa(Y,s)\le b_2^-(W)-\sigma(W)/8) are currently available, and for spin cobordisms (W,sW)(W,s_W) between two Seifert rational homology 3-spheres with spin structures, where the improvements of Manolescu's Theorem 5 likewise fail.

The related comparative question — which of μˉ\bar\mu, κ\kappa, the involutive Heegaard Floer correction terms, or the KO-theoretic bounds of Lin and Hopkins–Lin–Shi–Xu yields the optimal obstruction for a given plumbed boundary — remains open. This task concerns only the two regimes enumerated in items 1 and 2 above.

References

  1. Masaaki Ue, “Constraints on intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres in terms of μˉ\bar\mu and κ\kappa invariants”, preprint, arXiv:2206.05412 (2022), https://arxiv.org/abs/2206.05412
  2. Masaaki Ue, “On Manolescu's κ\kappa-invariants of rational homology 3-spheres”, preprint, arXiv:2405.02597 (2024), https://arxiv.org/abs/2405.02597
  3. Irving Dai, Hirofumi Sasahira, Matthew Stoffregen, “Lattice homology and Seiberg–Witten Floer spectra”, Journal of the European Mathematical Society (2026), DOI 10.4171/jems/1809, https://doi.org/10.4171/jems/1809 (arXiv:2309.01253)
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  5. Matthew Stoffregen, “Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations”, Compositio Mathematica 156 (2020) 199–250, DOI 10.1112/s0010437x19007620, https://doi.org/10.1112/s0010437x19007620 (arXiv:1505.03234)
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  7. Michael Hopkins, Jianfeng Lin, XiaoLin Danny Shi, Zhouli Xu, “Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant”, Communications of the American Mathematical Society 2 (2022) 22–132, DOI 10.1090/cams/4, https://doi.org/10.1090/cams/4 (arXiv:1812.04052)
  8. Yoshihiro Fukumoto, Mikio Furuta, Masaaki Ue, “W-invariants and Neumann–Siebenmann invariants for Seifert homology 3-spheres”, Topology and its Applications 116 (2001) 333–369, DOI 10.1016/s0166-8641(00)00103-6, https://doi.org/10.1016/s0166-8641(00)00103-6
  9. Walter D. Neumann, “An invariant of plumbed homology spheres”, in Topology Symposium Siegen 1979, Lecture Notes in Mathematics 788, Springer, Berlin (1980) 125–144, DOI 10.1007/BFb0099243, https://doi.org/10.1007/BFb0099243
  10. Walter D. Neumann, Frank Raymond, “Seifert manifolds, plumbing, μ\mu-invariant and orientation reversing maps”, in Algebraic and Geometric Topology (Santa Barbara 1977), Lecture Notes in Mathematics 664, Springer, Berlin (1978) 163–196, DOI 10.1007/BFb0061699, https://doi.org/10.1007/BFb0061699
  11. Irving Dai, Ciprian Manolescu, “Involutive Heegaard Floer homology and plumbed three-manifolds”, Journal of the Institute of Mathematics of Jussieu 18 (2019) 1115–1155, DOI 10.1017/s1474748017000329, https://doi.org/10.1017/s1474748017000329 (arXiv:1704.02020)
  12. Antonio Alfieri, John Baldwin, Irving Dai, Steven Sivek, “Instanton Floer homology of almost-rational plumbings”, Geometry & Topology 26 (2022) 2237–2294, DOI 10.2140/gt.2022.26.2237, https://doi.org/10.2140/gt.2022.26.2237 (arXiv:2010.03800)