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 diffeomorphism or homeomorphism types of manifolds of dimension >= 5. In this paper, a modification of this theory is presented, where instead of fixing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are fixed, 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 CP^n and one can use the modified theory to obtain information about their diffeomorphism type although the homotopy classification is not known. The theory reduces this classification result to the determination of complete intersections in a certain bordism group. The restrictions are: If d = d_1 ... d_r is the total degree of a complete intersection X^n_{d_1,..., d_r} of complex dimension n, then the assumption is, that for all primes p with p(p-1) <= n+1, the total degree d is divisible by p^{[(2n+1)/(2p-1)]+1}. Theorem A. Two complete intersections X^n_{d_1,.,d_r} and X^n_{d'_1,\ldots , d'_s} of complex dimension n>2 fulfilling the assumption above for the total degree are diffeomorphic if and only if the total degrees, the Pontrjagin classes and the Euler characteristics agree.