Suppose that $f:A\rightarrow B$ is a homotopy equivalence (both $A$ and $B$ are CW complexes), and $Y$ is a CW complex. Then is it true that the induced map
$f\wedge Id:A\wedge Y\rightarrow B\wedge Y$
is also a homotopy equivalence?
More generally, I want to answer the question: if $f$ is an $n$ connected map, that is, it induces isomorphisms $\pi_i(A)\cong\pi_i(B)$ for all $i<n$, then is the map $f\wedge Id$ also $n$ connected? i.e. is $\pi_i(A\wedge Y)\cong\pi_i(B\wedge Y)$ for all $i<n$?