ON THE SIGNATURE AND EULER CHARACTERISTIC OF CERTAIN 4-MANIFOLDS

Let M be a smooth closed connected oriented 4-manifold; we shall say that M satisfies Winkelnkemper's inequality when its signature, sigma(M), and Euler characteristic, chi(M), are related by \sigma(M)\ less-than-or-equal-to chi(M). This inequality is trivially true for manifolds M with first Betti number b1(M) less-than-or-equal-to 1. Winkelnkemper's theorem [10], re-proved below, is that (1) is satisfied when the fundamental group pi1(M) is Abelian. In this note we generalize Winkelnkemper's result to more general fundamental groups. We shall also see that most manifolds with a geometric structure satisfy Winkelnkemper's inequality. Except where geometric structures enter in Section 1, we could consider topological manifolds instead of smooth ones.