An Extension Theorem for Mappings Into Simply Connected Spaces

The modern work on the classification and extension of mappings had its starting point in a paper' [5] of H. Hopf, classifying the mappings of an ndimensional complex into an n-sphere S'. A simple formulation and proof of this theorem were given by the author [11] as one of the first applications of the new cohomology theory. Hurewicz [6] generalized the theorem, replacing S' by any space whose homotopy groups below the nth vanish. An early theorem of the same sort where this condition failed was that of Hopf [4], on the mappings of S' into S2. The author in 1936 (in an unpublished manuscript) gave algebraic conditions for the homotopy of mappings of an n-complex into projective n-space for n = 2, 3. This was generalized by Robbins [9], considering mappings of a 2-complex into any space. A great simplification and generalization of this work has recently been obtained by Paul Olum. Pontrjagin [7] classified the mappings of a 3-complex into S2, and gave a corresponding extension theorem for a 4-complex; this was one of the first applications of the new products. (The same theorem was obtained independently by the author; see an abstract in the Bull. Am. Math. Soc., vol. 42 (1936), p. 338. The method used is contained in the first half of CM.) In another paper [8], the 3-dimensional homotopy group of a simply connected space was determined, with the help of algebraic methods. (Proofs were not given.) A new type of cup product, X * Y, was made use of. This product of Pontrjagin furnished the clue to the general Ai products of Steenrod [10]; using them, Steenrod classified the mappings of an (n + 1)complex into an n-sphere, and gave the corresponding extension theorems. The object of the present paper is to give an extension theorem in the case n = 2. Given any simplicial complex K of dimension 4 (whose vertices we always take as given in a fixed order), we give an algebraic criterion that a mapping f of the 2-dimensional part K2 of K into a simply connected space R be extendable over K. We assume first that f is deformed into a standard mapping, which may always be done. In the case that 7r2(R) has no elements of finite order, the formula was obtained by the author in 1940 (see the abstract in the Bull. Am. Math. Soc., vol. 46 (1940), p. 897; the formula was mistakenly stated for the general case).