ON THE SIGNATURE AND EULER CHARACTERISTIC OF CERTAIN 4-MANIFOLDS

Let M be a smooth closed connected oriented 4-manifold; we shall say t hat M satisfies Winkelnkemper's inequality when its signature, sigma(M ), and Euler characteristic, chi(M), are related by \sigma(M)\ less-th an-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. Winkelnkempe r's theorem [10], re-proved below, is that (1) is satisfied when the f undamental group pi1(M) is Abelian. In this note we generalize Winkeln kemper'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.