On the Theory of Obstructions

Abstract

N. E. Steenrod has solved1 the homotopy classification problem for maps K -+ S', where n > 2 and K is an (n + 1)-dimensional polyhedron. His solution is stated in terms of separation cochains and depends on a certain theorem concerning obstructions (cf. (6.1) below). The latter, likewise the homotopy classification theorem, has been extended by M. M\. Postnikov and also, for n = 2, by Hassler Whitney to the case of maps in an (n 1)-connected space X, which is arbitrary except that 7r1(X) is finitely generated.2 The main purpose of this note is to show how the theory of composite chain systems ([11], ?4) and the secondary boundary operator can be used in proving the theorems of Postnikov and Whitney. Our conditions are more restrictive than theirs, since we eventually confine ourselves to finite complexes. This is because our definitions of the squaring operations (?5 below) do not apply to infinite complexes. However the same theorems can be proved, by an elaboration of our methods, for maps of a finite complex in an arbitrary (n 1)-connected space. This was done in an earlier draft. But the simplification due to the relation (5.9) below, for which the image space is also required to be a finite complex, seems to justify the loss of generality. In revising the first draft I have been greatly helped by a series of discussions with Steenrod, who suggested the use of the difference homomorphism and the relation (5.9).