On Homotopy K 3 Surfaces

Abstract

By a surface we shall mean a compact complex manifold of complex dimension 2. A surface is said to be regular if its first Betti number vanishes. A K 3 surface is defined to be a regular surface of which the first Chern class vanishes. Every K 3 surface is diffeomorphic to a non-singular quartic surface in a complex projective 3-space (see [1], Theorem 13). Thus there is a unique diffeomorphic type of K 3 surface. By a homotopy K 3 surface we mean a surface of the oriented homotopy type of K 3 surface. The purpose of this paper is to study the structure of homotopy K 3 surfaces and prove the following theorem: Any homotopy K 3 surface is either a K 3 surface or a regular elliptic surface of geometric genus 1. A regular elliptic surface S of geometric genus 1 is a homotopy K 3 surface if and only if S satisfies the following three conditions: (i) The number of multiple fibres of S does not exceed two. (ii) The multiplicities of the multiple fibres of S are odd. (iii) In the case in which S has two multiple fibres, their multiplicities are relatively prime.