Duality and S-theory

1. Given topological spaces X and Fand two continuous mappings ƒ o and jfi from X to F (denoted by fo, fi : X—> F) we say that ƒ o is homotopic to jfi (denoted by fo—fi) if there exists a continuous family of continuous mappings ht: X—+Y for O^t^l such that ho=fo and h =/ i . Intuitively fo<^fi if the map ƒ o can be continuously deformed into the map jfi. I t is easily seen [4; 7] that the relation of homotopy thus defined is reflexive, symmetric, and transitive and, therefore, partitions the set of continuous maps from X to Y into disjoint equivalence classes called homotopy classes. We denote the set of homotopy classes of mappings from X to F by [X, F ] , and if ƒ: X—>Y, then [ƒ] will denote the homotopy class of/. I t is of fundamental importance in present day topology to determine the structure of [X, F ] . Specifically, we would like to have a method of determining whether two given mappings are homotopic, and also we would like to determine how many elements there are in [X, F ] . In many cases our information is so limited that we do not even know whether [X, Y] contains more than one element. (In the above when X is contractible we assume F is arcwise connected.) I t is clear that if either X or F is a contractible space then there is exactly one homotopy class of maps from X to F. In particular if either l o r F is a cell of any dimension, the structure of [X, Y] is completely known as this set consists of a single element. Perhaps the most natural spaces to consider after the cells are the spheres. We denote by S the unit sphere in a euclidean space of (w + 1) dimensions. We want to discuss the homotopy classes 5—»F and Z » S n .