The topology of instanton moduli spaces. I: The Atiyah-Jones conjecture.

*(English)*Zbl 0816.55002In a seminal paper [M. F. Atiyah and J. D. S. Jones, Commun. Math. Phys. 61, 97-118 (1978; Zbl 0387.55009)], Atiyah and Jones set out a number of questions about the topology of moduli spaces of Yang-Mills instantons over the 4-sphere, which have led to a great deal of research over the past decade. For each \(k\geq 0\) the moduli space \({\mathcal M}_ k\) of “framed” \(\text{SU}(2)\) instantons of Chern class \(k\) is a manifold of dimension \(8k\), which is naturally embedded in an infinite-dimensional space \({\mathcal C}_ k\) of all the framed connections. Each of the spaces \({\mathcal C}_ k\) has the weak homotopy type of the third loop space \(\Omega^ 3 \text{SU}(2)\) and the circle of questions revolves around the induced inclusion maps \(\theta_{p,k} : \pi_ q({\mathcal M}_ k) \to \pi_ p(\Omega^ 3(\text{SU}(2))\). Atiyah and Jones conjectured, partly in analogy with results of Segal on the topology of spaces of rational functions [G. Segal, Acta Math. 143, 39-72 (1979; Zbl 0427.55006)], that for fixed \(p\) the map \(\theta_{q,k}\) should be an isomorphism when \(k\) is large. The proof of this Atiyah-Jones conjecture is the central result of the paper under review. More precisely, the authors prove that \(\theta_{p,k}\) is an isomorphism when \(p \leq q(k) = [k/2] - 2\).

Attacks on the Atiyah-Jones conjecture have largely followed two lines: one using analytical methods inspired by the ideas of Morse theory and developed largely by Taubes, and the other based on more explicit geometric descriptions of the moduli spaces. Using the first approach, C. H. Taubes [J. Differ. Geom. 29, No. 1, 163-230 (1989; Zbl 0669.58005)] proved a “stable” version of the conjecture (in fact this is a more general result, for any 4-manifold and structure group). The stable result bears on the direct limit of the moduli spaces formed using a sequence of maps – whose definition involves some arbitrary choices, but unique up to homotopy – \(i_ k : {\mathcal M}_ k \to {\mathcal M}_{k + 1}\). Given this, the Atiyah-Jones conjecture reduces to showing that \(i_ k\) induces an isomorphism on homotopy groups in dimensions up to \(q(k)\). The paper under review follows the second route, relying on stratifications of the moduli spaces obtained from complex geometry. If a complex structure is fixed on \(\mathbb{R}^ 4\) the moduli spaces can be identified with moduli spaces of holomorphic bundles over \({\mathbb{C}\mathbb{P}}^ 2\), trivialised on the line at infinity. Following earlier work of Hurtubise, the authors consider the pencil of complex lines in \({\mathbb{C}\mathbb{P}}^ 2\) through the origin. Each point of the moduli space \({\mathcal M}_ k\) gives rise to \(k\) “jumping lines” in this pencil (counted with multiplicity) – the lines over which the corresponding bundle is holomorphically non-trivial. Hurtubise showed that the bundle could be recovered from its jumping lines together with data determined over infinitesimal neighbourhoods of the lines. This gives rise to a description of the moduli space in terms of a configuration of points on \({\mathbb{C}}{\mathbb{P}}^ 1\) “labelled” by points in manifolds \(Q_ m\), which encode this infinitesimal data. The manifolds \(Q_ m\) are stratified with strata indexed by increasing step functions, and this induces a stratification \(\{S_ \alpha\}\) of the moduli space \({\mathcal M}_ k\).

A stratification of a manifold gives rise to a spectral sequence, converging to the homology of the total space. In the case at hand the authors obtain a spectral sequence with \(E_ 1\) term \[ \bigoplus_{\alpha} H_ * (\Sigma^{2d(\alpha)} ({\mathcal S}_ \alpha)), \] converging to \(H_ *({\mathcal M}_ k)\), where \(2d(\alpha)\) is the codimension of the stratum \(S_ \alpha\). Moreover this is compatible with the inclusion map \(i_ k : {\mathcal M}_ k \to {\mathcal M}_{k + 1}\). The central topological argument in the proof is to show that the map induced by \(i_ k\) on the \(E_ 1\) terms in the spectral sequences is an isomorphism in a range of dimensions. This uses techniques developed to study the homotopy theory of second loop spaces and configurations spaces of labelled points in \(\mathbb{C}\). This proves the version of the Atiyah- Jones conjecture with homology in place of homotopy groups. Hurtubise has shown previously that the fundamental groups of the \({\mathcal M}_ k\) are \(\mathbb{Z}/2\), as for \(\Omega^ 3 \text{SU}(2)\). The final step is to deduce the homotopy results by examining the homology of the universal covers.

The results have been announced in Bull. Am. Math. Soc., New Ser. 26, No. 2, 317-321 (1992; Zbl 0748.32014).

Attacks on the Atiyah-Jones conjecture have largely followed two lines: one using analytical methods inspired by the ideas of Morse theory and developed largely by Taubes, and the other based on more explicit geometric descriptions of the moduli spaces. Using the first approach, C. H. Taubes [J. Differ. Geom. 29, No. 1, 163-230 (1989; Zbl 0669.58005)] proved a “stable” version of the conjecture (in fact this is a more general result, for any 4-manifold and structure group). The stable result bears on the direct limit of the moduli spaces formed using a sequence of maps – whose definition involves some arbitrary choices, but unique up to homotopy – \(i_ k : {\mathcal M}_ k \to {\mathcal M}_{k + 1}\). Given this, the Atiyah-Jones conjecture reduces to showing that \(i_ k\) induces an isomorphism on homotopy groups in dimensions up to \(q(k)\). The paper under review follows the second route, relying on stratifications of the moduli spaces obtained from complex geometry. If a complex structure is fixed on \(\mathbb{R}^ 4\) the moduli spaces can be identified with moduli spaces of holomorphic bundles over \({\mathbb{C}\mathbb{P}}^ 2\), trivialised on the line at infinity. Following earlier work of Hurtubise, the authors consider the pencil of complex lines in \({\mathbb{C}\mathbb{P}}^ 2\) through the origin. Each point of the moduli space \({\mathcal M}_ k\) gives rise to \(k\) “jumping lines” in this pencil (counted with multiplicity) – the lines over which the corresponding bundle is holomorphically non-trivial. Hurtubise showed that the bundle could be recovered from its jumping lines together with data determined over infinitesimal neighbourhoods of the lines. This gives rise to a description of the moduli space in terms of a configuration of points on \({\mathbb{C}}{\mathbb{P}}^ 1\) “labelled” by points in manifolds \(Q_ m\), which encode this infinitesimal data. The manifolds \(Q_ m\) are stratified with strata indexed by increasing step functions, and this induces a stratification \(\{S_ \alpha\}\) of the moduli space \({\mathcal M}_ k\).

A stratification of a manifold gives rise to a spectral sequence, converging to the homology of the total space. In the case at hand the authors obtain a spectral sequence with \(E_ 1\) term \[ \bigoplus_{\alpha} H_ * (\Sigma^{2d(\alpha)} ({\mathcal S}_ \alpha)), \] converging to \(H_ *({\mathcal M}_ k)\), where \(2d(\alpha)\) is the codimension of the stratum \(S_ \alpha\). Moreover this is compatible with the inclusion map \(i_ k : {\mathcal M}_ k \to {\mathcal M}_{k + 1}\). The central topological argument in the proof is to show that the map induced by \(i_ k\) on the \(E_ 1\) terms in the spectral sequences is an isomorphism in a range of dimensions. This uses techniques developed to study the homotopy theory of second loop spaces and configurations spaces of labelled points in \(\mathbb{C}\). This proves the version of the Atiyah- Jones conjecture with homology in place of homotopy groups. Hurtubise has shown previously that the fundamental groups of the \({\mathcal M}_ k\) are \(\mathbb{Z}/2\), as for \(\Omega^ 3 \text{SU}(2)\). The final step is to deduce the homotopy results by examining the homology of the universal covers.

The results have been announced in Bull. Am. Math. Soc., New Ser. 26, No. 2, 317-321 (1992; Zbl 0748.32014).

Reviewer: S.K.Donaldson (Oxford)

##### MSC:

55P35 | Loop spaces |

58D27 | Moduli problems for differential geometric structures |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

14D20 | Algebraic moduli problems, moduli of vector bundles |

32G13 | Complex-analytic moduli problems |