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On the topological classification of pseudofree group actions on 4-manifolds: II. (English) Zbl 0864.57012

For Part I see [the second named author, Math. Z. 217, 335-366 (1994; Zbl 0813.57030)].
Let \(G\) be a finite cyclic group which acts (orientation preserving) locally linearly and pseudofreely on a closed oriented simply connected 4-manifold \(M\) with a nonempty fixed point set. Let \(O(\lambda)\) be the quotient of \(\operatorname{Aut} (H_2(M), \lambda)\) modulo those automorphisms inducing the identity on the Tate cohomology group \(\widehat H^0 (G;H_2(M))\). Similarly defined are the groups \(O(\lambda_r)\) for \(\lambda_r= \lambda\oplus rH\), where \(H\) is the hyperbolic form on \(\mathbb{Z}[G]^2\) and \(O_{st}(\lambda)= O(\lambda_r)\) with \(r\gg 0\). Then it follows that \[ O(\lambda) \subseteq O(\lambda_1) \subseteq O_{st} (\lambda) \subseteq O_*(\lambda), \] where the stable \(O\)-group \(O_*(\lambda)\) is the group of determinant \(\pm 1\) isometries of the associated quadratic form on \(\widehat H^0 (G;H_2 (M))\). The authors prove the following stability result of the equivariant intersection form \(\lambda\) of the \(G\)-action on \(M\): \[ O(\lambda_1)= O_{st} (\lambda)= O_*(\lambda). \] Then they obtain a topological rigidity theorem which states that two locally linear, pseudofree actions on \(M\), with the equivariant intersection forms indefinite and of rank at least 3 at each irreducible character, are topologically conjugate by an orientation preserving homeomorphism if and only if their oriented local representations at the corresponding fixed points are linearly equivalent.

MSC:

57M60 Group actions on manifolds and cell complexes in low dimensions
19G38 Hermitian \(K\)-theory, relations with \(K\)-theory of rings
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R91 Equivariant algebraic topology of manifolds

Citations:

Zbl 0813.57030
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References:

[1] Atiyah, M. F. and Bott, R.: A Lefschetz fixed point formula for elliptic complexes, II. Applications, Ann. of Math. 88 (1968), 451–491. · Zbl 0167.21703
[2] Bass, H.: Unitary algebraic K-theory, in: Algebraic K-theory III. Hermitian K-Theory and Geometric Applications, Lecture Notes in Math. 343, Springer-Verlag, Berlin, Heidelberg, New York, 1973, pp. 57–265. · Zbl 0299.18005
[3] Brown, K. S.: Cohomology of Groups, Springer-Verlag, New York, Heidelberg, Berlin, 1982. · Zbl 0584.20036
[4] Donaldson, S. K.: An application of guage theory in four dimensional topology, J. Differential Geom. 18 (1983), 279–315. · Zbl 0507.57010
[5] Edmonds, A. L. and Ewing, J. H.: Locally linear group actions on the complex projective plane, Topology 28 (1989), 211–223. · Zbl 0682.57021
[6] Edmonds, A. L. and Ewing, J. H.: Realizing forms and fixed point data in dimension four, Amer. J. Math. 114 (1992), 1103–1126. · Zbl 0766.57020
[7] Hahn, A. and O’Meara, O. T.: The Classical Groups and K-Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1989.
[8] Hambleton, I. and Riehm, C.: Splitting of hermitian forms over group rings, Invent. Math. 45 (1978), 19–33. · Zbl 0377.15010
[9] Higman, G.: The units of group rings, Proc. London Math. Soc. (2) 46 (1940), 231–248. · Zbl 0025.24302
[10] Hirzebruch, F. and Zagier, D.: The Atiyah–Singer Index Theorem and Elementary Number Theory, Publish or Perish, Boston, 1971.
[11] Jacobinski, H.: Genera and decompositions of lattices over orders, Acta. Math. 121 (1968), 1–29. · Zbl 0167.04503
[12] Lee, R. and Wilczyński, D. M.: Locally flat 2-spheres in simply connected 4-manifolds, Comment. Math. Helv. 65 (1990), 388–412; Correction, Ibid. 67 (1992), 334–335. · Zbl 0723.57015
[13] Lee, R. and Wilczyński, D. M.: Representing homology classes by locally flat 2-spheres, K-Theory 7 (1993), 333–367. · Zbl 0805.57014
[14] O’Meara, O.T.: Introduction to Quadratic Forms, 3rd edn, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
[15] Shimura, G.: Arithmetic of unitary groups, Ann. of Math. 79 (1964), 369–409. · Zbl 0144.29504
[16] Swan, R.: K-Theory of Finite Groups and Orders, Lecture Notes in Math. 149, Springer-Verlag, Berlin, Heidelberg, New York, 1970. · Zbl 0205.32105
[17] Wall, C. T. C.: On the classification of hermitian forms, I. Rings of algebraic integers, Compositio Math. 22, (1970) 425–451. · Zbl 0211.07602
[18] Wilczyński, D. M.: Group actions on the Chern manifold, Math. Ann. 281 (1988), 333–339. · Zbl 0627.57025
[19] Wilczyński, D. M.: Periodic maps on simply connected four-manifolds, Topology 30 (1991), 55–65. · Zbl 0715.57018
[20] Wilczyński, D. M.: On the topological classification of pseudofree group actions on 4-manifolds: I, Math. Z. 217 (1994), 335–366. · Zbl 0813.57030
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