The Homotopy Theory of Coincidences
F. B. Fuller · Annals of Mathematics · 1954
Let f and g be two maps from a complex K to a manifold M. A coincidence of f and g is a point of K where f and g assume the same value. The question to be considered is this: When can f and g be deformed into maps f' and g' which have no coincidence? Let L be a subcomplex of K containing no coincidence of g and f, and let us allow only those deformations of f and g which keep L free of coincidences. f and g determine a map (f, g) mapping K into the product manifold M X M: (f, g)x = (f(x), g(x)). Because of our requirements on L(f, g) maps the pair (K, L) into the pair (M X M, M X M D), where D is the diagonal of M X M. Then it is readily seen that the pair of maps f and g can be freed of coincidences by a pair of homotopies keeping L free of coincidences if, and only if, the single map (f, g) can be deformed, that is to say if (f, g) is homotopic to a map (f, g)' such that (f, g)'K c M X M D. If (K, L) and (M X M, M X M D) are replaced by pairs of the same homotopy types and the map (f, g) is replaced by a corresponding map, then it follows quite generally that the new map can be deformed only if (f, g) can be. Hence we shall be interested in (K, L) only as a homotopy type. However, as the following example shows, M will not stand such rough handling. Any two maps of a complex into a point have a coincidence, but if the point is replaced by a 1-cell, which has the same homotopy type, the two maps can be freed of coincidences. Thus the homotopy type of (M X M, M X M D) depends on more than the homotopy type of M. The example shows that the question of coincidences of maps into a general complex would probably be unmanageable. However the theory of fixed points, where one map is an identity map and only the other map can be deformed, is successful in general spaces. The theory of fixed points should not, therefore, be regarded as a special case of the coincidence problem formulated here. For example, the Brouwer theorem asserts that any map of a closed cell into itself has a fixed point, but it is not true that two maps of a cell into itself must have a coincidence. The coincidence problem and its partial solution will be expressed in terms of the theory of obstructions to deformations. The p-dimensional obstruction to deforming (f, g) is a subset D'(f, g) of the cohomology group H'(K, L; irp(M X M, M X M D)) (if M is simply connected and of dimension > 3). The pskeleton of K can be freed of coincidences by homotopies of f and g keeping L free of coincidences if, and only if, 0 is among the set of cocycles DY(f, g). In the