Deleted products of spaces which are unions of two simplexes

Abstract:

If $X$ is a space, the deleted product space, ${X^ \ast }$, is $X \times X - D$, where $D$ is the diagonal. If $Y$ is a space and $f$ is a continuous map from $X$ to $Y$, then $X_f^ \ast$ is the inverse image of ${Y^ \ast }$ under the map $f \times f$ taking $X \times X$ into $Y \times Y$. In this paper, we investigate the following questions: “What maps $f$ are such that $X_f^ \ast$ is homotopically equivalent to ${X^ \ast }$", and “What maps $f$ are such that $X_f^ \ast$ is homotopically equivalent to $f{(X)^ \ast }$?” If $X$ is the union of two nondisjoint simplexes and $f$ is a simplicial map from $X \times X$ such that $f|f(X)$ is one-to-one, we obtain necessary and sufficient conditions for $X_f^ \ast$ and $f{(X)^ \ast }$ to be homotopically equivalent. If $X$ is the union of nondisjoint simplexes $A$ and $B$ with $\dim B = 1 + \dim (A \cap B)$, we obtain necessary and sufficient conditions for ${X^ \ast }$ and $X_f^ \ast$ to be homotopically equivalent if $f$ is in the class of maps mentioned.