Abstract
This chapter presents Pontrjagin's method of associating a 2n-dimensional co-homology class, px, mod.4r, with every n-dimensional co-homology class, x, mod.2r. If f is a co-cycle, mod.2r, in the cohomology class x, then px is represented by the co-chain. The cohomology rings of a polyhedron, P, with integers reduced mod. m (m = 0, 1, 2,…) as coefficients, may be combined into a single ring by a method given by M. Bockstein. This ring is given additional algebraic structure by introducing a certain operator Δ and also the Pontrjagin squares. The chapter presents a proof of the conditions when (1) any such ring, which satisfies the general algebraic conditions appropriate to a finite, simply connected polyhedron of at most four dimensions, can be realized geometrically; and (2) two such polyhedra are of the same homotopy type when, their cohomology rings are properly isomorphic.
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