Local gluing-map properties for SO(3) monopoles: Hypothesis 2.5 of Feehan–Leness and the unconditional SO(3)-monopole cobordism formula

The 2019 Advances in Mathematics paper by Feehan and Leness proves that every standard four-manifold of Seiberg–Witten simple type has superconformal simple type — implying the Fintushel–Stern lower bound on the number of Seiberg–Witten basic classes and, with their companion paper, Witten's conjecture relating Donaldson and Seiberg–Witten invariants — but every one of these results is conditional on an unproved analytic hypothesis (Hypothesis 2.5): that the local gluing maps for SO(3) monopoles previously constructed by the authors give continuous parametrizations of neighborhoods of the lower-moduli-space strata in the compactified monopole moduli space. Hypothesis 2.5 remains unproved in the published literature. The problem posed here is to prove Hypothesis 2.5 in full: establish the expected gluing-map properties of continuity with respect to Uhlenbeck limits, injectivity, and local surjectivity, together with the further conditions the hypothesis imposes (detailed in the Problem Statement below), thereby making the SO(3)-monopole link-pairing formula and its corollaries unconditional. Not part of this task: the determination of the universal cobordism coefficients without the $c_1(\mathfrak t)=0$ restriction, and the extension of Mochizuki's formula to all standard four-manifolds.

Background

For a closed, oriented, smooth four-manifold $X$, the two gauge-theoretic invariants of central importance are the Donaldson invariants, defined by counting anti-self-dual $SO(3)$ connections modulo gauge transformations — instantons, connections whose curvature two-form is anti-self-dual, i.e. orthogonal to the self-dual two-forms singled out by the orientation — and the Seiberg–Witten invariants, defined by counting solutions of the monopole equations modulo gauge: for a $\mathrm{spin}^c$ structure (a lift of the structure group of the orthonormal frame bundle to $Spin^c(4)$, providing complex spinor bundles on which the Dirac equation is defined), the monopole equations couple a unitary connection on the determinant line bundle to a spinor section through the Dirac operator and a quadratic moment map. A class $K\in H^2(X;\mathbb{Z})$ arising as the first Chern class of a $\mathrm{spin}^c$ structure with nonzero Seiberg–Witten invariant is a basic class; $X$ has Seiberg–Witten simple type if the invariant vanishes whenever the expected (index-theoretic) dimension of the monopole moduli space is positive. Following Feehan and Leness, a standard four-manifold is a closed, connected, oriented, smooth four-manifold with $b_1(X)=0$ (vanishing first Betti number) and $b^+(X)\ge 3$ odd, where $b^+(X)$ is the maximal dimension of a positive-definite subspace for the intersection pairing on $H^2(X;\mathbb{R})$. Witten's conjecture (1994) asserts that for such manifolds of Seiberg–Witten simple type the Donaldson series (the generating function assembled from all Donaldson invariants) is determined by the Seiberg–Witten invariants through an explicit universal formula; it is proved unconditionally for complex projective surfaces via Mochizuki's formula expressing Donaldson invariants as residues of integrals over Hilbert schemes of points (the Hilbert scheme of $\ell$ points on a surface is its natural compactified configuration space of $\ell$ points; Mochizuki 2009; Göttsche–Nakajima–Yoshioka 2011). Mariño, Moore, and Peradze (1999), from physics of $N=2$ supersymmetric field theory at superconformal points, predicted that every standard four-manifold of simple type satisfies superconformal simple type: the vanishing of the Seiberg–Witten series in degrees below $c(X)-3$, where $c(X)$ is the complexity (roughly $\dim H_2$ plus half the Euler characteristic); this condition implies the Fintushel–Stern conjectured lower bound $|B(X)/\{\pm1\}|\ge\lfloor c(X)/2\rfloor$ on the number of basic classes. The Feehan–Leness route to these statements runs through the moduli space $M_{\mathfrak t}$ of SO(3) monopoles for a $\mathrm{spin}^u$ structure $\mathfrak t$ (a $\mathrm{spin}^c$ structure tensored with a complex rank-two vector bundle): points of $M_{\mathfrak t}$ are solutions $(A,\Phi)$, modulo gauge, of the SO(3)-monopole equations — a unitary connection $A$ coupled through the Dirac operator to a spinor $\Phi$ — with the anti-self-dual (instanton) moduli space appearing where the spinor vanishes identically and the Seiberg–Witten moduli spaces appearing as reducible loci. The Uhlenbeck compactification $\bar M_{\mathfrak t}$, taken in the Uhlenbeck topology (the weak topology in which curvature is allowed to concentrate at points, i.e. to bubble), has boundary strata of the form $\bar M_{\mathfrak s}\times\mathrm{Sym}^\ell(X)$, where $\bar M_{\mathfrak s}$ is a compactified Seiberg–Witten moduli space and $\mathrm{Sym}^\ell(X)$, the $\ell$-fold symmetric product, records the locations of $\ell$ curvature bubbles. Their 2018 AMS Memoir, building on the local gluing maps constructed in their PU(2)-monopoles series — maps that reconstruct points of $\bar M_{\mathfrak t}$ near a boundary stratum from lower-level solution data plus bubble data, the analogue of the gluing constructions of Taubes, Donaldson–Kronheimer, and Morgan–Mrowka for instantons — derives the SO(3)-monopole link-pairing formula, expressing the link pairings of $\bar M_{\mathfrak t}$ (intersection pairings between homology cycles and small spheres linking the boundary strata) through Seiberg–Witten invariants and universal coefficients $a_{i,j,k}$. However, the construction of the local gluing maps near each stratum (built in the authors' 1999 preprint PU(2) monopoles III) is only assumed to satisfy Hypothesis 2.5, whose expected properties include continuity with respect to Uhlenbeck limits, injectivity, and surjectivity in the sense that every point of $\bar M_{\mathfrak t}$ sufficiently Uhlenbeck-close to $\bar M_{\mathfrak s}\times\Sigma$, for each smooth stratum $\Sigma\subset\mathrm{Sym}^\ell(X)$, lies in the image of at least one local gluing map. Analogous properties are known for anti-self-dual connections in special cases, but the monopole setting couples the connection equation to the spinor, and the boundary strata involve symmetric products with singular strata, so no existing theorem covers it. All cobordism-based results — superconformal simple type (Theorem 1.1), the Fintushel–Stern bound (Corollary 1.2), and Witten's conjecture for standard four-manifolds (Corollary 1.4, with the companion paper arXiv:1408.5085) — therefore remain conditional, as Manschot's 2023 Encyclopedia of Mathematical Physics survey still records.

Problem Statement

Prove Hypothesis 2.5 of Feehan–Leness (Advances in Mathematics 356 (2019), article 106817), in full: show that the local gluing maps for SO(3) monopoles constructed by the authors in arXiv:math/9907107 satisfy, for every $\mathrm{spin}^u$ structure $\mathfrak t$ (a $\mathrm{spin}^c$ structure tensored with a complex rank-two vector bundle) arising in the compactified moduli space $\bar M_{\mathfrak t}$ and every smooth stratum $\Sigma\subset\mathrm{Sym}^\ell(X)$ of the relevant symmetric product, the properties of (a) continuity with respect to Uhlenbeck limits, (b) injectivity, and (c) local surjectivity — every point of $\bar M_{\mathfrak t}$ sufficiently close to $\bar M_{\mathfrak s}\times\Sigma$ in the Uhlenbeck topology lies in the image of at least one local gluing map — so that each local gluing map gives a continuous parametrization of a neighborhood of $\bar M_{\mathfrak s}\times\Sigma$ in $\bar M_{\mathfrak t}$. Properties (a)–(c) are necessary but not sufficient for Hypothesis 2.5, which additionally requires the $S^1$-equivariance, homotopy-to-the-splicing-map, stratum-smoothness, collar-identity, and zero-locus-homeomorphism conditions detailed below; establishing all of these is part of the task. Hypothesis 2.5 is the 2019 paper's restatement of the memoir's Hypothesis 7.8.1, whose setting is fixed there: $(X,g)$ is a closed, connected, oriented, smooth Riemannian four-manifold with $b_1(X)=0$ and odd $b^+(X)>1$ (the corollaries of the 2019 paper restrict to standard four-manifolds, $b^+(X)\ge 3$ odd), with the metric $g$ fixed once and for all and the perturbation parameters $(\tau,\vartheta)$ of the SO(3)-monopole equations (memoir Eq. (2.1.10)) generic in the sense of the memoir's Theorem 2.1.1 and fixed thereafter; the gluing map $\boldsymbol\gamma_{\mathscr M}$, the virtual moduli space $\bar{\mathscr M}^{\mathrm{vir}}_{\mathfrak t,\mathfrak s}$, the obstruction pseudo-bundles and their sections, and the Uhlenbeck topology (memoir Definition 2.1.2) are the explicit objects constructed in the memoir's Chapters 2 and 6–7, and Hypothesis 7.8.1 requires $\boldsymbol\gamma_{\mathscr M}$ to be a continuous $S^1$-equivariant embedding homotopic through such embeddings to the constructed global splicing map $\boldsymbol\gamma'_{\mathscr M}$, smooth on each stratum, equal to the identity on $N_{\mathfrak t(\ell),\mathfrak s}(\delta)\times\mathrm{Sym}^\ell(X)$, and, on the zero-locus $\bar{\boldsymbol\chi}^{-1}(0)$ of the obstruction section, a homeomorphism onto an open neighborhood of $M_{\mathfrak s}\times\mathrm{Sym}^\ell(X)$ in $\bar M_{\mathfrak t}$. A complete proof, in the precise setting of the constructions and topologies fixed by the memoir An SO(3)-monopole cobordism formula relating Donaldson and Seiberg–Witten invariants (Mem. Amer. Math. Soc. 256 (2018), no. 1226, Sections 7.8–7.9 and 10.1.1), would make the SO(3)-monopole link-pairing formula unconditional, and thereby convert the conditional consequences of the source paper into theorems: superconformal simple type for all standard four-manifolds of Seiberg–Witten simple type (Theorem 1.1), the Fintushel–Stern lower bound $|B(X)/\{\pm1\}|\ge\lfloor c(X)/2\rfloor$ when $B(X)$ is non-empty and $c(X)\ge 3$ (Corollary 1.2), and, together with arXiv:1408.5085, Witten's conjecture relating Donaldson and Seiberg–Witten invariants for these manifolds (Corollary 1.4).

References

1. Paul M. N. Feehan and Thomas G. Leness, "The SO(3) monopole cobordism and superconformal simple type", Advances in Mathematics 356 (2019), article 106817. DOI: 10.1016/j.aim.2019.106817; arXiv:1408.5307, https://arxiv.org/abs/1408.5307
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