The Hopf conjecture for positively curved manifolds with discrete abelian group actions

Let M be a closed even n-manifold of positive sectional curvature. The main result asserts that the Euler characteristic of M is positive, if M   admits an isometric Zpk-action with prime p⩾p(n)p⩾p(n) (a constant depending only on n) and k   satisfies any one of the following conditions: (i) k⩾n−48 and n≠12n≠12, 18 or 20; (ii) k⩾n−210, and n≡0 mod 4n≡0 mod 4 with n≠12n≠12 or 20; (iii) k⩾n+412, and n≡0,4n≡0,4 or 12 mod 2012 mod 20 with n≠20n≠20. This generalizes some results in [T. Püttmann, C. Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc. Amer. Math. Soc. 130 (2002) 163–166; X. Rong, Positively curved manifolds with almost maximal symmetry rank, Geom. Dedicata 59 (2002) 157–182; X. Rong, X. Su, The Hopf conjecture for positively curved manifolds with abelian group actions, Comm. Cont. Math. 7 (2005) 121–136].