Cocycle formulas for homotopy classification; maps into projective and lens spaces

Abstract

of these problems for the case of the second obstruction. It is our purpose here to show how these invariants rnay be explicitly computed in terms of cocycle formulas, simplicial or singular, provided that a cocycle formula is known for the relevant cohomology operation which enters. We apply these formulas then to give complete homotopy classification theorems for mappings of complexes (or manifolds) into real and complex projective spaces and into lens spaces. By a homotopy classification theorem for the mappings of X into Y is meant a theorem which provides an effective procedure for determining whether two arbitrarily given maps f, g: X-> Y are homotopic or not. Although many results are known which yield an enumeration of homotopy classes, the literature of topology contains extremely few such classification theorems(2) and our results will add somewhat to this list. Since these results for mappings into projective and lens spaces are perhaps of most general interest here, we shall give them first in Part I below, deferring all proofs to Part II. Enumeration theorems for these mappings have been given before, but the hoinotopy classification has been known only in certain fortuitous special cases; references will be found at the beginning of each of the sections of Part I. In the statement of the theorems of Part I we use rather extensively the notion of the difference homomorphism (f- g) * of two mappings f and g. A detailed discussion of these difference homomorphisms will be found in [8,