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RESEARCH PRODUCT

On the signature of four-manifolds with universal covering spin

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In this note we study closed oriented 4-manifolds whose universal covering is spin and ask whether there are restrictions on the divisibility of the signature. Since any natural number appears as the signature of a connected sum of r 2,s, without the assumption on the universal covering there cannot exist any restrictions. Certainly, the most famous such restriction was proved by Rohlin in [10], where he showed that the signature a of a smooth 4-dimensional spin manifold is divisible by 16 (compare part (2) of our Main Theorem for a new proof). The Kummer surface K shows that this is the best possible general result. Dividing by a certain free holomorphic involution on K, one obtains the Enriques surface (compare [1]) which by construction has signature 8 and fundamental group 7//2. Furthermore, Hitchin showed in [5] that there exists an antiholomorphic free involution on the Enriques surface. We will refer to the quotient as the Hitchin manifold which then has signature 4 and fundamental group 7//2 x 7//2. Rohlin's theorem admits a nice generalization to nonspin 4-manifolds, compare [4, Theorem 6.3]: