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An instanton-invariant for 3-manifolds. (English) Zbl 0684.53027
In this seminal paper the author introduces new invariants for 3- manifolds defined by the instanton solutions of the Yang-Mills equations. Let M be an oriented homology 3-sphere and choose a Riemannian metric on M. Let X be the Riemannian product $$M\times {\mathbb{R}}$$. Any finite energy solution of the SU(2) instanton equations over X (i.e. connection on an SU(2) bundle over X with anti-self-dual curvature tensor, and whose curvature lies in $$L^ 2)$$ is asymptotically flat at the ends of X. Flat SU(2) connections over M correspond to representations of $$\pi_ 1(M)$$ in SU(2). Suppose for the moment that there is only a finite set $$S\cup \{\theta \}$$ of conjugacy classes of such representations, and also that apart from the trivial representation $$\theta$$ these are “acyclic”, in the sense that the cohomology of M in the coefficient system defined by the associated flat $${\mathfrak su}(2)$$ bundle is trivial. Then the equivalence classes of instantons, modulo bundle automorphism and the action of translations on X, are parametrised by moduli spaces $${\mathcal M}_{\alpha \beta}$$, for $$\alpha$$,$$\beta\in S\cup \{\theta \}$$. Generically these will be countable unions of smooth manifolds. There is a function d: $$S\to {\mathbb{Z}}/8$$ such that the components of $${\mathcal M}_{\alpha,\beta}$$ have dimension d($$\alpha)$$-d($$\beta)$$-1, mod 8. In this situation the author’s invariants are groups formed from the cohomology of a complex whose chains are freely generated by the set S: $$C_*=Maps(S,{\mathbb{Z}})$$, which is $${\mathbb{Z}}/8$$-graded by d. The differential $$\partial$$ in the complex is defined so that the matrix element $$\partial_{\alpha,\beta}$$ of a pair $$\alpha$$ $$\beta$$ with $$d(\alpha)=d(\beta)+1$$ is given by the number of elements in the 0- dimensional piece of $${\mathcal M}_{a\beta}$$, counted with appropriate signs. (To formulate this definition the author shows first that, in this situation, the 0-dimensional piece must be compact.) The key result is that the homology groups $$I_*(M)$$ are independent of the metric on M which is used, and this $${\mathbb{Z}}/8$$-graded group is the author’s new invariant. More generally he defines the groups $$I_*(M)$$ even in a situation where there are, for example, infinite sets of representations by suitably perturbing the condition for the flatness of a vector bundle and the instanton equations. Then he shows that the groups so defined are independent of the perturbation. The key to understanding these ideas is the author’s observation that the instanton equations over X can be regarded as the gradient flow equations for the Chern-Simons function on the space of connections over M. From this point of view his complex is analogous to the Morse description of the homology of a finite dimensional manifold.
The author’s groups are closely related to the Casson invariant of a 3- manifold, especially through the differential geometric definition worked out by Taubes of Casson’s invariant [C. H. Taubes, “Casson’s invariant and gauge theory”, J. Differ. Geom. 31, No.2, 547-599 (1990)]. The Euler characteristic of Floer homology groups is twice the Casson invariant of M.
Reviewer: S.K.Donaldson

##### MSC:
 53C05 Connections, general theory 57N10 Topology of general $$3$$-manifolds (MSC2010) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 81T08 Constructive quantum field theory 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
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