Smooth group actions on 4-manifolds and Seiberg–Witten theory

Abstract

Let X be a smooth closed spin 4-manifold with the first Betti number b 1( X)=0 and signature σ( X)≤0. In this paper we use Seiberg-Witten theory theory to prove that (i) If X admits an odd type Z 2 p action preserving the spin structure, then b 2(X)≥ 5 4 |σ(X)|+2p+2, provided b 2 +(X; Z 2)≠b 2 +− 1 8 |σ(X)| , b + 2(X; Z 2 p )>0 and in addition, b + 2(X; Z 2 p )<b 2 + if p≥2, where Z 2⊂ Z 2 p . (ii) If H 1(X, Z 2)=0 and X admits an even type involution τ preserving the spin structure, then the number of isolated fixed points of τ, say k, is divisible by 8 and satisfies that k≤8(b 2 +(X;τ)−1)+ 1 2 σ(X)