Surgery and Duality

Abstract

Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension‚ 5. In this paper, a modiflcation of this theory is presented, where instead of flxing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are flxed, where k is at least [n=2]. A particularly attractive example which illustrates the concept is given by complete intersections. By the Lefschetz hyperplane theorem, a complete intersection of complex dimension n has the same n-skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known. The theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group. This was under certain restrictions carried out in [Tr]. The restrictions are: If d = d1¢:::¢dr is the total degree of a complete intersection X n1;:::;dr of complex dimension n, then the assumption is, that for all primes p with p(pi1)• n+1, the total degree d is divisible by p [(2n+1)=(2pi1)]+1 . Theorem A. Two complete intersections X n