Finite group actions on the moduli space of self-dual connections. I

Cho, Yong Seung

doi:10.1090/S0002-9947-1991-1010409-2

Finite group actions on the moduli space of self-dual connections. I
HTML articles powered by AMS MathViewer

by Yong Seung Cho PDF

Trans. Amer. Math. Soc. 323 (1991), 233-261 Request permission

Abstract:

Let $M$ be a smooth simply connected closed $4$-manifold with positive definite intersection form. Suppose a finite group $G$ acts smoothly on $M$. Let $\pi :E \to M$ be the instanton number one quaternion line bundle over $M$ with a smooth $G$-action such that $\pi$ is an equivariant map. We first show that there exists a Baire set in the $G$-invariant metrics on $M$ such that the moduli space $\mathcal {M}_ * ^G$ of $G$-invariant irreducible self-dual connections is a manifold. By utilizing the $G$-transversality theory of $\text {T}$. Petrie, we then identify cohomology obstructions to globally perturbing the full space ${\mathcal {M}_ * }$ of irreducible self-dual connections to a $G$-manifold when $G = {{\mathbf {Z}}_2}$ and the fixed point set of the ${\mathbf {Z}}_2$ action on $M$ is a nonempty collection of isolated points and Riemann surfaces.

References

M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531–545. MR 236951, DOI 10.2307/1970716
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
M. F. Atiyah and I. M. Singer, The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119–138. MR 279833, DOI 10.2307/1970756

Stability and isolation phenomena for Yang-Mills theory

79

Jean-Pierre Bourguignon and H. Blaine Lawson Jr., Yang-Mills theory: its physical origins and differential geometric aspects, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 395–421. MR 645750
Yong Seung Cho, Finite group actions on the moduli space of self-dual connections. II, Michigan Math. J. 37 (1990), no. 1, 125–132. MR 1042518, DOI 10.1307/mmj/1029004070
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056
S. K. Donaldson, Connections, cohomology and the intersection forms of $4$-manifolds, J. Differential Geom. 24 (1986), no. 3, 275–341. MR 868974
Ronald Fintushel and Ronald J. Stern, $\textrm {SO}(3)$-connections and the topology of $4$-manifolds, J. Differential Geom. 20 (1984), no. 2, 523–539. MR 788294
Ronald Fintushel and Ronald J. Stern, Pseudofree orbifolds, Ann. of Math. (2) 122 (1985), no. 2, 335–364. MR 808222, DOI 10.2307/1971306

Definite

-manifold

28

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and four-manifolds, Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1984. MR 757358, DOI 10.1007/978-1-4684-0258-2
Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066

Finite group actions on

Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German and Appendix One by R. L. E. Schwarzenberger; With a preface to the third English edition by the author and Schwarzenberger; Appendix Two by A. Borel; Reprint of the 1978 edition. MR 1335917
Nicolaas H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30. MR 179792, DOI 10.1016/0040-9383(65)90067-4
H. Blaine Lawson Jr., The theory of gauge fields in four dimensions, CBMS Regional Conference Series in Mathematics, vol. 58, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 799712, DOI 10.1090/cbms/058
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Thomas H. Parker, Gauge theories on four-dimensional Riemannian manifolds, Comm. Math. Phys. 85 (1982), no. 4, 563–602. MR 677998
Ted Petrie, Pseudoequivalences of $G$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 169–210. MR 520505
Ted Petrie and John D. Randall, Transformation groups on manifolds, Monographs and Textbooks in Pure and Applied Mathematics, vol. 82, Marcel Dekker, Inc., New York, 1984. MR 748850

The Atiyah-Singer index theorem

I. M. Singer, Some remarks on the Gribov ambiguity, Comm. Math. Phys. 60 (1978), no. 1, 7–12. MR 500248
I. M. Singer and J. A. Thorpe, The curvature of $4$-dimensional Einstein spaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355–365. MR 0256303
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
Ronald J. Stern, Instantons and the topology of $4$-manifolds, Math. Intelligencer 5 (1983), no. 3, 39–44. MR 737689, DOI 10.1007/BF03026571
Clifford Henry Taubes, Self-dual Yang-Mills connections on non-self-dual $4$-manifolds, J. Differential Geometry 17 (1982), no. 1, 139–170. MR 658473