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The stable topology of self-dual moduli spaces. (English) Zbl 0669.58005
P(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 $${\mathcal M}(P)$$.
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
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