Conjecture of Lin–Ruberman–Saveliev: equality of the Furuta–Ohta and Casson–Seiberg–Witten invariants for homology $S^1\times S^3$ 4-manifolds

For a closed, oriented, homology-oriented smooth 4-manifold $X$ with the integral homology of $S^1\times S^3$ and with universal abelian cover having the rational homology of $S^3$, two gauge-theoretic Casson-type invariants are defined: the Furuta–Ohta invariant $\lambda_{FO}(X)$, a signed count of anti-self-dual $SU(2)$ instantons (a degree-zero Donaldson invariant), and the Casson–Seiberg–Witten invariant $\lambda_{SW}(X)$, a signed count of Seiberg–Witten monopoles corrected by an index-theoretic term. The conjecture of Lin–Ruberman–Saveliev (2018), a slight generalization of Conjecture B of Mrowka–Ruberman–Saveliev (2011), asserts that $\lambda_{FO}(X)=-\lambda_{SW}(X)$ for every such $X$. This is a variant of Witten's conjecture relating Donaldson and Seiberg–Witten invariants for manifolds with vanishing second Betti number, where the classical forms of both theories do not directly apply. The conjecture is verified for mapping tori of finite group actions and manifolds obtained from these by torus surgery, fiber sum, and excision, but remains open in general; a proof would, among other consequences, imply vanishing of the Rohlin invariant modulo two for homotopy $S^1\times S^3$.

Background

Classical gauge-theoretic invariants of smooth 4-manifolds — the Donaldson polynomial invariants, built from signed counts of anti-self-dual (ASD) $SU(2)$ connections (instantons), and the Seiberg–Witten invariants, built from counts of monopoles, i.e. solutions of the Seiberg–Witten equations for a $spin^c$ structure — are normally only defined when $b_2^+(X)\ge 1$, where $b_2^+$ is the dimension of a maximal positive-definite subspace for the intersection form on $H_2$. This problem concerns the opposite regime: manifolds with the integral homology of $S^1\times S^3$, i.e. closed oriented smooth 4-manifolds $X$ with $H_*(X;\mathbb{Z})\cong H_*(S^1\times S^3;\mathbb{Z})$ (so $b_1=1$, $b_2=0$), equipped with a homology orientation, meaning a choice of generator of $H^1(X;\mathbb{Z})\cong\mathbb{Z}$, which orients and signs all counts below. Write $\tilde X$ for the associated universal abelian (infinite cyclic) cover, and assume $H_*(\tilde X;\mathbb{Q})\cong H_*(S^3;\mathbb{Q})$.

On such manifolds two analogues of the Casson invariant of homology 3-spheres (which counts irreducible $SU(2)$ representations of the fundamental group) exist. Furuta and Ohta (1993) defined $\lambda_{FO}(X)$ as one quarter times the degree-zero Donaldson polynomial: a signed count of gauge-equivalence classes of irreducible ASD $SU(2)$ instantons on the trivial $SU(2)$ bundle over $X$, originally under the more restrictive hypothesis that $\tilde X$ have the integral homology of $S^3$; Lin–Ruberman–Saveliev (2018) extended the definition to all $X$ satisfying the rational-homology condition above. Mrowka, Ruberman, and Saveliev (2011) introduced the Casson–Seiberg–Witten invariant $\lambda_{SW}(X)$: a signed count of irreducible Seiberg–Witten monopoles on $X$, corrected by an index-theoretic term coming from the Dirac operator on $\tilde X$ viewed as a manifold with periodic (end-periodic) ends; they proved the two terms' metric dependence cancels, so $\lambda_{SW}$ is a genuine diffeomorphism invariant, and that $\lambda_{SW}(X)$ reduces modulo two to the Rohlin invariant $\mu(X)$ of $X$, the classical $\mathbb{Z}/2$-valued invariant given by the Rohlin invariant of a 3-dimensional submanifold Poincaré dual to the generator of $H^1(X;\mathbb{Z})$.

Witten's 1994 conjecture predicted that Donaldson invariants are determined by Seiberg–Witten invariants; its established forms (via the work of Pidstrigach–Tyurin and Feehan–Leness) require $b_2^+\ge 2$, and the case $b_2=0$, $b_1=1$ treated here is an open variant of it. Mrowka–Ruberman–Saveliev conjectured ($\lambda_{SW}=-\lambda_{FO}$ for integral-cover $X$), and Lin–Ruberman–Saveliev generalized the conjecture to the rational-cover class. The conjecture has notable classical consequences: for a homotopy $S^1\times S^3$ (fundamental group $\mathbb{Z}$, so every $SU(2)$ representation is reducible) $\lambda_{FO}=0$, and since $\lambda_{SW}\equiv\mu\pmod 2$, the conjecture implies $\mu\equiv 0\pmod 2$ for homotopy $S^1\times S^3$, a long-standing question in low-dimensional topology; $\lambda_{SW}$ itself is a gauge-theoretic integer lift of $\mu$. The conjecture has been verified for mapping tori of free finite-order diffeomorphisms of homology 3-spheres (Ruberman–Saveliev 2004), for mapping tori of involutions arising as double branched covers of $S^3$ (Lin–Ruberman–Saveliev 2018, Theorem C), for mapping tori of all semi-free $\mathbb{Z}/n$-actions (Lin–Ruberman–Saveliev 2021), for manifolds with free circle action (Han 2020), and for classes of manifolds reached from these by torus surgery, fiber sum, and excision operations, under which both invariants satisfy matching formulae (Ma 2021–2024).

Problem Statement

Prove or disprove the conjecture of Lin–Ruberman–Saveliev (2018), generalizing Conjecture B of Mrowka–Ruberman–Saveliev (2011): for every closed, oriented, smooth 4-manifold $X$ equipped with a homology orientation (a choice of generator of $H^1(X;\mathbb{Z})\cong\mathbb{Z}$) such that $H_*(X;\mathbb{Z})\cong H_*(S^1\times S^3;\mathbb{Z})\quad\text{and}\quad H_*(\tilde X;\mathbb{Q})\cong H_*(S^3;\mathbb{Q}),$ where $\tilde X$ is the universal abelian cover of $X$ associated with the chosen generator, one has $\lambda_{FO}(X)=-\lambda_{SW}(X),$ with $\lambda_{FO}$ the Furuta–Ohta invariant (one quarter of the signed count of irreducible anti-self-dual $SU(2)$ connections on the trivial $SU(2)$ bundle over $X$, as extended to this class of manifolds by Lin–Ruberman–Saveliev) and $\lambda_{SW}$ the Casson–Seiberg–Witten invariant of Mrowka–Ruberman–Saveliev (the signed count of irreducible Seiberg–Witten monopoles on $X$ plus the end-periodic Dirac index correction term), both normalized as in the cited works. The original integral-cover version of Mrowka–Ruberman–Saveliev (where $H_*(\tilde X;\mathbb{Z})\cong H_*(S^3;\mathbb{Z})$) is the special case in which the rational and integral conditions coincide, and is contained in the problem as stated.

References

1. Jianfeng Lin, Daniel Ruberman, Nikolai Saveliev, On the Frøyshov invariant and monopole Lefschetz number, Journal of Differential Geometry 123 (2023), no. 3, DOI 10.4310/jdg/1683307008, arXiv:1802.07704, https://arxiv.org/abs/1802.07704
2. Tomasz S. Mrowka, Daniel Ruberman, Nikolai Saveliev, Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant, Journal of Differential Geometry 88 (2011), no. 2, 333-377, DOI 10.4310/jdg/1320067650, arXiv:0905.4319, https://arxiv.org/abs/0905.4319
3. Mikio Furuta, Hiroshi Ohta, Differentiable structures on punctured 4-manifolds, Topology and its Applications 51 (1993), no. 3, 291-301, DOI 10.1016/0166-8641(93)90083-P, https://doi.org/10.1016/0166-8641(93)90083-P
4. Edward Witten, Monopoles and four-manifolds, Mathematical Research Letters 1 (1994), no. 6, 769-796, DOI 10.4310/mrl.1994.v1.n6.a13, https://doi.org/10.4310/mrl.1994.v1.n6.a13
5. Jianfeng Lin, Daniel Ruberman, Nikolai Saveliev, On the monopole Lefschetz number of finite order diffeomorphisms, Geometry & Topology 25 (2021), no. 7, 3591-3628, DOI 10.2140/gt.2021.25.3591, arXiv:2004.05497, https://arxiv.org/abs/2004.05497
6. Daniel Ruberman, Nikolai Saveliev, Rohlin's invariant and gauge theory II. Mapping tori, Geometry & Topology 8 (2004), no. 1, 35-76, DOI 10.2140/gt.2004.8.35, arXiv:math/0306188, https://arxiv.org/abs/math/0306188
7. Daniel Ruberman, Nikolai Saveliev, Casson-type invariants in dimension four, in Geometry and Topology of Manifolds, Fields Institute Communications 47, American Mathematical Society (2005), 281-306, DOI 10.1090/fic/047/18, arXiv:math/0501090, https://doi.org/10.1090/fic/047/18
8. Jianfeng Lin, Daniel Ruberman, Nikolai Saveliev, A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$, Geometry & Topology 22 (2018), no. 5, 2865-2942, DOI 10.2140/gt.2018.22.2865, arXiv:1702.04417, https://arxiv.org/abs/1702.04417
9. Langte Ma, A surgery formula for the Casson-Seiberg-Witten invariant of integral homology $S^1\times S^3$, Journal of Topology 14 (2021), no. 3, 913-962, DOI 10.1112/topo.12202, arXiv:1909.01533, https://arxiv.org/abs/1909.01533
10. Langte Ma, Fiber sum formulae for the Casson-Seiberg-Witten invariant of integral homology $S^1\times S^3$, Mathematical Research Letters 29 (2022), no. 4, 1197-1227, DOI 10.4310/mrl.2022.v29.n4.a11, arXiv:1910.07683, https://arxiv.org/abs/1910.07683
11. Langte Ma, Surgery and excision for Furuta-Ohta invariants on homology $S^1\times S^3$, Journal of Differential Geometry 128 (2024), no. 1, DOI 10.4310/jdg/1721075264, arXiv:2006.04197, https://arxiv.org/abs/2006.04197
12. Daoyuan Han, Seiberg-Witten-Casson invariant of homology $S^1\times S^3$ with circle action, arXiv:2005.07098 (2020), https://arxiv.org/abs/2005.07098
13. Mariano Echeverria, A generalization of the Tristram-Levine knot signatures as a singular Furuta-Ohta invariant for tori, arXiv:1908.11359 (2019), https://arxiv.org/abs/1908.11359