Witten's conjecture predicts that the Donaldson series of a smooth four-manifold of Seiberg–Witten simple type with b1=0b_1=0 and odd b+≥3b^+\ge3 is determined by its Seiberg–Witten invariants. The strongest general route is the Feehan–Leness SO(3)-monopole program: a cobordism in the moduli space of SO(3) monopoles yields a universal cobordism formula relating Donaldson and Seiberg–Witten invariants, from which Witten's conjecture for every such manifold was deduced — but only modulo an unproved analytic hypothesis (Hypothesis 3.1 of Feehan–Leness, the subject of this problem): that the local gluing maps for SO(3) monopoles give continuous, injective, and surjective parametrizations of neighborhoods of Seiberg–Witten strata in the Uhlenbeck compactification. As of the authors' December 2025 writings this gluing construction remains an assumed technical hypothesis, so Witten's conjecture for general smooth (non-complex-projective) four-manifolds of simple type is still open. The problem is to prove these gluing-map properties, thereby making the cobordism formula and its consequences unconditional.

Background

Gauge theory produces two families of invariants of smooth four-manifolds. Donaldson invariants, defined by integrating cohomology classes over moduli spaces of anti-self-dual SO(3)\mathrm{SO}(3) connections (connections whose curvature two-form FF satisfies F=−∗FF=-*F; the moduli space is compactified by allowing curvature to concentrate at points, the Uhlenbeck compactification), were the first to reveal the subtlety of smooth four-dimensional topology. Seiberg–Witten invariants, defined via moduli spaces of solutions of the monopole equations (a Spinc\mathrm{Spin}^c connection — a unitary connection with just enough extra structure for spinor fields to be defined on an oriented Riemannian four-manifold — coupled to a spinor field), are far easier to compute. Witten's conjecture (Monopoles and Four-Manifolds, 1994) asserts the following for a standard four-manifold — smooth, closed, connected, oriented, with first Betti number b1=0b_1=0 and odd b+≥3b^+\ge3, where b+b^+ is the dimension of a maximal positive-definite subspace for the intersection form on cohomology — of Seiberg–Witten simple type (the Seiberg–Witten invariant SWX(s)\mathrm{SW}_X(\mathfrak{s}), which counts solutions of the monopole equations for a Spinc\mathrm{Spin}^c structure s\mathfrak{s}, is nonzero only when c12(s)=c12(X)c_1^2(\mathfrak{s})=c_1^2(X); the images c1(s)c_1(\mathfrak{s}) of its support are the finitely many basic classes K∈B(X)K\in B(X)): XX then also has Kronheimer–Mrowka simple type (the analogous finiteness condition on the Donaldson-invariant side), the Kronheimer–Mrowka and Seiberg–Witten basic classes coincide, and the Donaldson series — the generating function of all Donaldson invariants, indexed by w∈H2(X;Z)w\in H^2(X;\mathbb{Z}) and h∈H2(X;R)h\in H_2(X;\mathbb{R}) — is the explicit expression

DXw(h)=22−(χh−c12)eQX(h)/2∑s∈Spinc(X)(−1)12(w2+c1(s)⋅w)SWX(s)e⟨c1(s),h⟩,\mathcal{D}_X^w(h)=2^{2-(\chi_h-c_1^2)}e^{Q_X(h)/2}\sum_{\mathfrak{s}\in\mathrm{Spin}^c(X)}(-1)^{\frac{1}{2}(w^2+c_1(\mathfrak{s})\cdot w)}\mathrm{SW}_X(\mathfrak{s})e^{\langle c_1(\mathfrak{s}),h\rangle},

where QXQ_X is the intersection form on H2(X;R)H_2(X;\mathbb{R}), and χh=(χ+σ)/4\chi_h=(\chi+\sigma)/4 and c12=2χ+3σc_1^2=2\chi+3\sigma are the characteristic numbers defined, by analogy with their values for complex surfaces, from the Euler characteristic χ\chi and signature σ\sigma. Pidstrigach and Tyurin proposed proving this via the moduli space of SO(3) monopoles, which interpolates between anti-self-dual connections and Seiberg–Witten monopoles. Feehan and Leness carried this out across a series of works: their Memoir (Mem. Amer. Math. Soc. 256 (2018), no. 1226) proves — modulo a gluing hypothesis, discussed below — an SO(3)\mathrm{SO}(3)-monopole cobordism formula — the link of the anti-self-dual stratum (a multiple of the Donaldson invariant) is cobordant to the links of strata Ms×Symℓ(X)M_\mathfrak{s}\times\mathrm{Sym}^\ell(X), where MsM_\mathfrak{s} is a Seiberg–Witten monopole moduli space and Symℓ(X)\mathrm{Sym}^\ell(X) the ℓ\ell-fold symmetric product recording curvature concentration points — yielding a universal formula with real-analytic coefficients ai,j,ka_{i,j,k}; and their JEMS paper (17 (2015) 899–923) determined enough coefficients, via Fintushel–Park–Stern example manifolds and blow-up analysis, to deduce Witten's conjecture for manifolds that are abundant (the orthogonal complement B(X)⊥B(X)^\perp of the basic classes contains a hyperbolic summand of the intersection form) or satisfy c12≥χh−3c_1^2\ge\chi_h-3. Two later Advances in Mathematics papers (356 (2019), articles 106817 and 106821) removed that restriction: the cobordism formula implies that every standard four-manifold of SW simple type has superconformal simple type (a vanishing condition on low-degree Seiberg–Witten invariants introduced by Mariño–Moore–Peradze, conjectured to always hold, yielding the Fintushel–Stern lower bound on the number of basic classes), and that the undetermined low-index coefficients then contribute zero, so the cobordism formula implies Witten's conjecture for all standard SW-simple-type four-manifolds. Independently, Göttsche–Nakajima–Yoshioka proved Witten's conjecture unconditionally for complex projective surfaces via Mochizuki's formula and Nekrasov instanton counting, and proposed (their Conjecture 4.5) that their formula extends to all smooth four-manifolds of Seiberg–Witten simple type. Every step of the Feehan–Leness route, however, is conditional: the cobordism formula is proved modulo a single analytic hypothesis, Hypothesis 3.1, asserting that the local gluing maps for SO(3) monopoles (constructed in Feehan–Leness, PU(2) monopoles III, arXiv:math/9907107) give continuous parametrizations of neighborhoods of Ms×ΣM_\mathfrak{s}\times\Sigma in the Uhlenbeck-compactified moduli space for each smooth stratum Σ⊂Symℓ(X)\Sigma\subset\mathrm{Sym}^\ell(X), with continuity with respect to Uhlenbeck limits, injectivity, and surjectivity (nearby ideal monopoles lie in the image of some local gluing map). Analogous properties for anti-self-dual connections were proved by Taubes and by Donaldson–Kronheimer in special cases, and the global assembly of local maps (the overlap problem') is solved within the Memoir; but the local analytic properties for the coupled SO(3)-monopole equations have never been established. A monograph Gluing maps for SO(3) monopoles and invariants of smooth four-manifolds' has been cited as in preparation since at least the 2018 Memoir, with a framework preprint (arXiv:1910.14580) introducing a Banach-manifolds-with-corners approach, but no proof has appeared; in the December 2025 revision of arXiv:2410.13809 the authors still describe the gluing construction as `assumed (as a technical hypothesis)'.

Problem Statement

Prove the local gluing-map properties for SO(3) monopoles, stated as Hypothesis 3.1 in Feehan–Leness (J. Eur. Math. Soc. 17 (2015) 899–923) and recorded in greater detail as Hypothesis 7.8.1 in their Memoir (Mem. Amer. Math. Soc. 256 (2018), no. 1226, Sections 7.8–7.9): for a standard four-manifold XX (closed, connected, oriented, smooth, with b1=0b_1=0 and odd b+≥3b^+\ge3), the local gluing maps for SO(3) monopoles constructed in Feehan–Leness (arXiv:math/9907107) give, for each Spinc\mathrm{Spin}^c structure s\mathfrak{s} and each smooth stratum Σ⊂Symℓ(X)\Sigma\subset\mathrm{Sym}^\ell(X), a continuous parametrization of a neighborhood of Ms×ΣM_\mathfrak{s}\times\Sigma (with MsM_\mathfrak{s} the Seiberg–Witten monopole moduli space for s\mathfrak{s}) in Mˉt\bar{M}_t, the Uhlenbeck compactification of the SO(3)-monopole moduli space in which curvature may concentrate at up to ℓ\ell points, recorded by Symℓ(X)\mathrm{Sym}^\ell(X) — specifically, (i) continuity of the local gluing maps with respect to Uhlenbeck limits, (ii) injectivity, and (iii) surjectivity in the sense that points of Mˉt\bar{M}_t sufficiently close (in the Uhlenbeck topology) to Ms×ΣM_\mathfrak{s}\times\Sigma lie in the image of at least one local gluing map. An affirmative resolution makes the SO(3)-monopole cobordism formula unconditional, and hence — combined with the published conditional results of Feehan–Leness (Adv. Math. 356 (2019), articles 106817 and 106821) — establishes Witten's conjecture for every standard four-manifold of Seiberg–Witten simple type, the superconformal simple-type property for all such manifolds, and the Fintushel–Stern lower bound on Seiberg–Witten basic classes. A rigorous demonstration that the constructed local gluing maps fail any of the three properties would instead overturn this route to Witten's conjecture.

References

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