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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.

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) |

### Keywords:

invariants for 3-manifolds; instanton solutions of the Yang-Mills equations; homology 3-sphere; anti-self-dual curvature tensor; Flat SU(2) connections; representations of \(\pi _ 1(M)\) in SU(2); moduli spaces; Chern-Simons function on the space of connections; Casson invariant; Euler characteristic of Floer homology
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\textit{A. Floer}, Commun. Math. Phys. 118, No. 2, 215--240 (1988; Zbl 0684.53027)

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### References:

[1] | Adams, R.A.: Sobolev spaces. New York: Academic Press 1975 · Zbl 0314.46030 |

[2] | Arondzajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of the second order. J. Math. Pures Appl.36 (9), 235-249 (1957) · Zbl 0084.30402 |

[3] | Atiyah, M.F.: Instantons in two and four dimensions. Commun. Math. Phys.93, 437-451 (1984) · Zbl 0564.58040 · doi:10.1007/BF01212288 |

[4] | Atiyah, M.F.: New invariants for 3- and 4-dimensional manifolds. Preprint 1987 |

[5] | Atiyah, M.F., Drinfield, V., Hitchin, N., Manin, Y.I.: Construction of instantons. Phys. Lett. A65, 185 (1978) · Zbl 0424.14004 |

[6] | Atiyah, M.F., Singer, I.M.: The index of elliptic operators I. Ann. Math.87, 484-530 (1968); III. Ibid.87, 546-604 (1968) · Zbl 0164.24001 · doi:10.2307/1970715 |

[7] | Atiyah, M.F.: Index theory of skew adjoint Fredholm operators. Publ. Math. IHES37, 305-325 (1969) · Zbl 0194.55503 |

[8] | Braam, P.: Monopoles on three-manifolds, preprint, Oxford, 1987 |

[9] | Donaldson, S.K.: An application of gauge theory to the topology of 4-manifolds. J. Differ. Geom.18, 269-316 (1983) |

[10] | Donaldson, S.K.: Instantons and geometric invariant theory · Zbl 0581.14008 |

[11] | Donaldson, S.K.: The orientation of Yang-Mills moduli-spaces and four manifold topology. J. Differ. Geom.26, 397-428 (1987) · Zbl 0683.57005 |

[12] | Donaldson, S.K.: Polynomial invariants for smooth 4-manifolds. Preprint, Oxford, 1987 · Zbl 0656.57005 |

[13] | Fintushel, R., Stern, R.J.: Pseudofree orbifolds. Ann. Math.122, 335-346 (1985) · Zbl 0602.57013 · doi:10.2307/1971306 |

[14] | Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. (to appear) · Zbl 0674.57027 |

[15] | Floer, A.: The unregularized gradient flow for the symplectic action. Commun. Pure Appl. Math. (to appear) · Zbl 0633.53058 |

[16] | Floer, A.: Symplectic fixed points and holomorphic spheres. Preprint, CIMS, 1988 · Zbl 0633.53058 |

[17] | Freed, D., Uhlenbeck, K.K.: Instantons and four-manifolds. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0559.57001 |

[18] | Goldman, W.: The symplectic nature of the fundamental groups of surfaces. Adv. Math.54, 200-225 (1984) · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9 |

[19] | Hempel, J.: Three-manifolds. Ann. of Math. Studies 86. Princeton, NJ: Princeton, University Press 1967 |

[20] | H?rmander, L.: The analysis of linear differential operators. III. Berlin, Heidelberg, New York: Springer 1985 |

[21] | Husemoller, D.: Fibre bundles. Springer Graduate Texts in Math., Vol. 20. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0144.44804 |

[22] | Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc.16 (1967) |

[23] | Lockhard, R.B., McOwen, R.C.: Elliptic operators on noncompact manifolds. Ann. Sci. Norm. Sup. PisaIV-12, 409-446 (1985) · Zbl 0615.58048 |

[24] | Matic, G.: SO(3) connections and rational homology cobordism. Preprint, MIT, 1987 · Zbl 0662.57018 |

[25] | Maz’ja, V.G., Plamenevski, B.A.: Estimates onL p and H?lder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary problems in domains with singular points on the boundary. Math. Nachr.81, 25-82 (1978) [English Transl. In: AMS Translations Ser. 2,123, 1-56 (1984) · Zbl 0371.35018 · doi:10.1002/mana.19780810103 |

[26] | Milnor, J.: Lectures on theh-cobordism theorem. Math. Notes. Princeton, NJ: Princeton University Press 1965 · Zbl 0161.20302 |

[27] | Novikov, S.P.: Multivalued functions and functionals, an analogue of Morse theory. Sov. Math. Dokl.24, 222-226 (1981) · Zbl 0505.58011 |

[28] | Palais, R.S.: Foundations of global analysis. New York: Benjamin 1968 · Zbl 0164.11102 |

[29] | Quinn, F.: Transversal approximation on Banach manifolds. In: Proc. Symp. Pure Math. 15. Providence, RI: AMS 1970 · Zbl 0206.25705 |

[30] | Ray, D.B., Singer, I.M.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145-201 (1971) · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4 |

[31] | Smale, S.: Morse inequalities for dynamical systems. Bull. AMS66, 43-49 (1960) · Zbl 0100.29701 · doi:10.1090/S0002-9904-1960-10386-2 |

[32] | Smale, S.: On gradient dynamical systems. Ann. Math.74, 199-206 (1961) · Zbl 0136.43702 · doi:10.2307/1970311 |

[33] | Smale, S.: An infinite dimensional version of Sard’s theorem. Am. J. Math.87, 213-221 (1973) |

[34] | Spanier, E.: Algebraic topology. New York: McGraw-Hill 1966 · Zbl 0145.43303 |

[35] | Taubes, C.H.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differ. Geom.17, 139-170 (1982) · Zbl 0484.53026 |

[36] | Taubes, C.H.: Gauge theory on asymptotically periodic 4-manifolds. J. Differ. Geom.25, 363-430 (1987) · Zbl 0615.57009 |

[37] | Taubes, C.H.: Private communication |

[38] | Taubes, C.H.: A framework for Morse theory for the Yang-Mills functional. Preprint, Harvard, 1986 · Zbl 0665.58006 |

[39] | Taubes, C.H.: Preprint, 1987 |

[40] | Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11-29 (1982) · Zbl 0491.58032 · doi:10.1007/BF01947068 |

[41] | Uhlenbeck, K.K.: Connections withL p -bounds on curvature. Commun. Math. Phys.83, 31-42 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069 |

[42] | Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom.17, 661-692 (1982) · Zbl 0499.53056 |

[43] | Witten, E.: Topological quantum field theory. Commun. Math. Phys.117, 353-386 (1988) · Zbl 0656.53078 · doi:10.1007/BF01223371 |

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