Homotopy types of the deleted product of unions of two simplexes

Abstract:

If X is a space, let ${X^\ast } = X \times X - D$, where D is the diagonal. If f is a map on X to a space Y, let $X_f^\ast = \{ (x,y) \in {X^\ast }|f(x) \ne f(y)\}$. In this paper we continue our investigation, begun in [6], of the homotopy types of ${X^\ast }$ and $X_f^\ast$, and of a question due to Brahana [1, p. 236], as to when the homotopy types of ${X^\ast }$ and $X_f^\ast$ are the same. If X is the union of two nondisjoint simplexes, and if f is a simplicial map on X, we are able, using results and techniques developed in [6], to express the homotopy types of ${X^\ast }$ and $X_f^\ast$ in terms of spheres, and then to determine when the homotopy types of these spaces are the same.