Finite isometry groups of 4-manifolds with positive sectional curvature

Abstract

Let M be an oriented compact Riemannian 4-manifold with positive sectional curvature. Let G be a finite subgroup of the isometry group of M. We prove that, if G is a finite group of order |G| ≥ 3 10 500 , then (i) G is isomorphic to a subgroup of PU (3) if |G| is odd; (ii) G contains an index at most 2 normal subgroup which is isomorphic to a subgroup of SO(5) or PU (3) if |G| is even, and M is not homeomorphic to S 4 . Moreover, M is homeomorphic to CP 2 if G is non-abelian of odd order.

On nonnegatively curved 4-manifolds with discrete symmetry

“…It can be shown as in Proposition 2.1 (or Proposition 3.3 in [3]) that if m is greater than or equal to 61 then the Z m can have at most 5 isolated xed points. In particular, if the xed point set is isolated, then χ(M ) is just equal to the number of isolated xed points which is less than or equal to 5.…”

“…The proof of Proposition 2.1 easily follows from Corollary 2.6 in [3] and Proposition 2.7 in [3]. Actually, by Corollary 7 in [16] Proposition 2.1 holds even for any prime m 41.…”

“…Recently Fang proved the same result without any assumption that m is a prime (cf. Proposition 3.3 in [3]). So, we have: Proposition 2.1.…”

“…By Corollary 2.6 in [3], the 5-extent xt 5 (X) is less than π 3 , where X is the quotient space of the unit sphere in T Q 0 M by the Z m /(H 0 H 1 . .…”

Positive curvature and discrete abelian symmetry

“…In turn, each torus is a product of circles and therefore contains a product of cyclic groups. Previous work on positively and non-negatively curved 4-manifolds with discrete symmetries can be found in Yang [43], Hicks [23], Fang [12], and Kim and Lee [29], and in higher dimensions in Fang and Rong [14], Su and Wang [37], and Wang [40]. For all of these results, the focus has been on choosing a cyclic group or a product of cyclic groups, where each cyclic group is of large enough order to produce a fixed point.…”