The stable topology of self-dual moduli spaces.

*(English)*Zbl 0669.58005P(M,G) denotes a principal G-bundle (with M a compact, oriented Riemannian 4-manifold and G a compact, simple Lie group). \({\mathcal B}(P)\) is the associated manifold of (gauge-equivalence classes of) smooth connections and \({\mathcal M}(P)\) the submanifold of (classes of) self-dual connections. Atiyah and Jones and Donaldson have considered the topology of \({\mathcal M}(P)\) particularly in the special case of \((M=S^ 4,G=SU(2),SU(3))\) in the context of the topology of 4-manifolds. The author showed [J. Differ. Geom. 19, 337-392 (1984; Zbl 0551.53040)], using techniques inspired by Morse theory, that, in this special case, \({\mathcal M}(P)\) is path-connected.

In the present paper, those techniques are considerably extended to compute information about the topology of \({\mathcal M}(P)\) in the general case. Specifically, the author considers the maps \((i_ P)_*\) on homotopy and homology induced by the inclusion \(i_ P: {\mathcal M}(P)\hookrightarrow {\mathcal B}(P)\) (as suggested by Donaldson’s program). Theorem 1 states that \((i_ P)_*\) is epimorphic on all homotopy/homology groups up to a certain dimension determined only by the Pontrjagin number \(k(P)=p_ 1(Ad P)\) associated to P. Theorem 2 asserts the existence of a homotopy equivalence between \({\mathcal B}(P)\) and \({\mathcal B}(P')\) whenever \(k(P)-k(P')\) is an integer multiple of C(G) (an integer depending solely on the bundles structure group; \(C(SU(2)=4).\) One consequence of the latter result is that \({\mathcal M}(P)\hookrightarrow {\mathcal B}(P)\) becomes a weak homotopy equivalence on taking direct limits (with respect to k(P)\(\to \infty)\). This is described as “topological stability” for the moduli spaces.

A careful and highly technical Morse-theoretic analysis is required because the hypotheses of the Palais-Smale and Ljusternik-Shnirelman procedures just fail. A rough technical summary follows. Observing that \({\mathcal M}(P)=a^{-1}(0)\) where \[ a(A)=\int_{M}| (1-*)F_ A|^ 2 \] is the Yang-Mills functional (modulo some multiple of k(P)), the author considers a min-max procedure for the function \(a_{{\mathcal F}}\) \[ a_{{\mathcal F}}=\inf_{U\in {\mathcal F}}\{\sup_{b\in U}a(b)\} \] where \({\mathcal F}\) is a family of compact subsets of \({\mathcal B}(P)\) which are homotopy invariant with respect to the neighbourhood \({\mathcal B}_{\epsilon}(P)=\{b\in {\mathcal B}(P):\quad a(b)<\epsilon \}\) of \({\mathcal M}(P)\) in \({\mathcal B}(P).\)

In the special case, \((S^ 4,SU(2))\), one can construct a strong deformation retract of \({\mathcal B}_{\epsilon_ 0}(P)\) (some P- independent \(\epsilon_ 0>0)\) onto \({\mathcal M}(P)\). A careful analysis of the obstructions to the existence of such a retract enables the author to conclude that the topological properties mentioned above survive the generalisation to arbitrary pairs (M,G).

Contents: introduction and preliminaries, topology of \({\mathcal B}(P)\) when \(k(P)\gg 0\), topology of \({\mathcal B}_{\epsilon}(P)\) for \(\epsilon\) small, the obstruction bundle and, finally, homotopy equivalences relative to \({\mathcal M}(P)\).

In the present paper, those techniques are considerably extended to compute information about the topology of \({\mathcal M}(P)\) in the general case. Specifically, the author considers the maps \((i_ P)_*\) on homotopy and homology induced by the inclusion \(i_ P: {\mathcal M}(P)\hookrightarrow {\mathcal B}(P)\) (as suggested by Donaldson’s program). Theorem 1 states that \((i_ P)_*\) is epimorphic on all homotopy/homology groups up to a certain dimension determined only by the Pontrjagin number \(k(P)=p_ 1(Ad P)\) associated to P. Theorem 2 asserts the existence of a homotopy equivalence between \({\mathcal B}(P)\) and \({\mathcal B}(P')\) whenever \(k(P)-k(P')\) is an integer multiple of C(G) (an integer depending solely on the bundles structure group; \(C(SU(2)=4).\) One consequence of the latter result is that \({\mathcal M}(P)\hookrightarrow {\mathcal B}(P)\) becomes a weak homotopy equivalence on taking direct limits (with respect to k(P)\(\to \infty)\). This is described as “topological stability” for the moduli spaces.

A careful and highly technical Morse-theoretic analysis is required because the hypotheses of the Palais-Smale and Ljusternik-Shnirelman procedures just fail. A rough technical summary follows. Observing that \({\mathcal M}(P)=a^{-1}(0)\) where \[ a(A)=\int_{M}| (1-*)F_ A|^ 2 \] is the Yang-Mills functional (modulo some multiple of k(P)), the author considers a min-max procedure for the function \(a_{{\mathcal F}}\) \[ a_{{\mathcal F}}=\inf_{U\in {\mathcal F}}\{\sup_{b\in U}a(b)\} \] where \({\mathcal F}\) is a family of compact subsets of \({\mathcal B}(P)\) which are homotopy invariant with respect to the neighbourhood \({\mathcal B}_{\epsilon}(P)=\{b\in {\mathcal B}(P):\quad a(b)<\epsilon \}\) of \({\mathcal M}(P)\) in \({\mathcal B}(P).\)

In the special case, \((S^ 4,SU(2))\), one can construct a strong deformation retract of \({\mathcal B}_{\epsilon_ 0}(P)\) (some P- independent \(\epsilon_ 0>0)\) onto \({\mathcal M}(P)\). A careful analysis of the obstructions to the existence of such a retract enables the author to conclude that the topological properties mentioned above survive the generalisation to arbitrary pairs (M,G).

Contents: introduction and preliminaries, topology of \({\mathcal B}(P)\) when \(k(P)\gg 0\), topology of \({\mathcal B}_{\epsilon}(P)\) for \(\epsilon\) small, the obstruction bundle and, finally, homotopy equivalences relative to \({\mathcal M}(P)\).

Reviewer: P.Bryant

##### MSC:

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

53C05 | Connections, general theory |

57R22 | Topology of vector bundles and fiber bundles |