Deleted Products of Spaces Which Are Unions of Two Simplexesc)

If A1 is a space, the deleted product space, X*, is Xx X— D, where D is the diagonal. If y is a space and /isa continuous map from X to Y, then X* is the inverse image of Y* under the map fxf taking Xx X into Kx y. In this paper, we investigate the following questions: "What maps/are such that X* is homotopically equivalent to X*", and "What maps/are such that X* is homotopically equivalent to f(X)* ?" If A' is the union of two nondisjoint simplexes and / is a simplicial map from Xx X such that f\f(X) is one-to-one, we obtain necessary and sufficient conditions for X* and f(X)* to be homotopically equivalent. If X is the union of nondisjoint simplexes A and B with dim S=l + dim (A r> B), we obtain necessary and sufficient conditions for X* and Xr* to be homotopically equivalent if/is in the class of maps mentioned.