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

Abstract

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 Z p k -action with prime p ⩾ p ( n ) (a constant depending only on n) and k satisfies any one of the following conditions: (i) k ⩾ n − 4 8 and n ≠ 12 , 18 or 20; (ii) k ⩾ n − 2 10 , and n ≡ 0 mod 4 with n ≠ 12 or 20; (iii) k ⩾ n + 4 12 , and n ≡ 0 , 4 or 12 mod 20 with n ≠ 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].