Topological classification of 4-dimensional complete intersections

Abstract

Let X,,(d) C C P "+r denote a complete intersection, the transversal intersection of r hypersurfaces in C P ~+r defined by r homogeneous polynomials of degrees (d l , . . . ,dr) =: d, with dld2...d,. =: d the total degree. It is well-known that the diffeomorphism type of X,,(d) is determined by n and d. In [7] and [8], Libgober and Wood showed that in dimension n -~ 2, there exist k distinct multidegrees ibr any integer k 6 N such that the corresponding complete intersections are all diffeomorphic. For n = 1,3, the diffeomorphism classification of Xn(d) is well-known by surface theory and the classification of 1-connected six-manifolds [12] respectively. For 7z = 2, at least the topological classification is known by M. Freedman's results. For n >_ 4, it is still an outstanding open problem to classifl' the X~(d) by some topological invariants computed from d = ( d l , . . . , dr) and n. In this paper, we will classify X4(d) up to homeomorphism. Our main result is: