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On the symmetries of the fake $${\mathbb{C}}P^ 2$$. (English) Zbl 0602.57027
The author shows that for each odd prime p there is a locally smoothable $${\mathbb{Z}}_ p$$-action on the Chern manifold Ch (which is homotopy equivalent to $${\mathbb{C}}P^ 2$$ but admits no smooth structure). This is in so far interesting as he had shown in joint work with P. Vogel that no such action exists for $$p=2$$ [Asymmetric 4-dimensional manifolds, Duke Math. J. 53, 759-764 (1986)]. The proof uses standard techniques from topological surgery theory (which can be applied to dimension 4 by Freedman’s work).
Reviewer: M.Krech

##### MSC:
 57S25 Groups acting on specific manifolds 57S17 Finite transformation groups 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
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##### References:
 [1] Freedman, M.: The topology of four-dimensional manifolds. J. Diff. Geom.17, 357-453 (1982) · Zbl 0528.57011 [2] Freedman, M., Quinn, F.: Topology of 4-manifolds. Ann. Math. Studies (to appear) · Zbl 0705.57001 [3] Kirby, R., Siebenmann, L.: Foundationals essays on topological manifolds, smoothings and triangulations. Ann. Math. Studies, No. 88. Princeton: Princeton University Press 1977 · Zbl 0361.57004 [4] Kwasik, S., Vogel, P.: Asymmetric four-dimensional manifolds. Duke Math. J., to appear · Zbl 0669.57022 [5] Lashof, R., Taylor, L.: Smoothing theory and Freedman’s work on four manifolds. Proc. Conf. Algebraic Topology Aarhus 1982. Notes Math. Vol. 1051. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0562.57008 [6] Miller, H.R., Priddy, S.B., (Eds.): Proc. Northw. Homotopy Conf. Contemporary Mathematics, Vol. 19, American Mathematical Society, 1983 [7] Quinn, F.: Ends of maps III: dimensions 4 and 5. J. Differ. Geom.17, 503-521 (1982) · Zbl 0533.57009 [8] Schultz, R.: Proc. Conf. Group Actions on Manifolds. Contemporary Mathematics, Vol. 36, 1983 [9] Wall, C.T.C.: Surgery on compact manifolds. London, New York: Academic Press 1970 · Zbl 0219.57024
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