Some relations between semigroups of polyhedra

Abstract:

All spaces are CW-complexes of finite type. The notation “=” means homotopy equivalence. The following theorems are proved: (1) If $X \vee A = Y \vee A$ then there exists $T$ such that $\Omega X \times T = \Omega Y \times T$, (2) $X \times A = Y \times A$ implies that there is a $T$ such that $\Sigma X \vee T = \Sigma Y \vee T$. A partial converse is also proved. As a corollary we get that if $X \vee A = Y \vee A$ then ${\pi _ \ast }(X) = {\pi _ \ast }(Y)$ and if $X \times A = Y \times A$ then ${h_ \ast }(X) = {h_ \ast }(Y)$ for many homology theories.