TITLE: Are there non-contractible spaces $A$ and $B$ such that $A \wedge B$ is contractible?
QUESTION [5 upvotes]: The question is in the title : Can we find spaces $A$ and $B$, each non contractible, such that their smash product $A \wedge B$, i.e. the homotopy cofibre of $A \vee B \to A \times B$, is a contractible space ?

REPLY [6 votes]: I just saw this in Ravenel's "orange book"! Let $X$ be any simply connected CW complex whose reduced homology is all torsion. Let $X_{(p)}$ be the p-localization for a prime $p$. Then for primes $p\neq q$, $X_{(p)}\wedge X_{(q)}$ is contractible. But neither $X_{(p)}$ nor $X_{(q)}$ is contractible if $H_*(X)$ has $p$ and $q$ torsion.