I'm confused by the following question:
$f:X\to Y$ is a weak homotopy equivalence, that is $f\_\*:\pi\_\*(X)\to \pi\_\*(Y)$ is an isomorphism for any dimensional homotopy groups. However, for the stable homotopy groups, is the homomorphism $f\_\*:\pi\_\*^s(X)\to \pi\_\*^s(Y)$ still an isomorphism?


Any comments are welcome! Many Thanks! 



Must a weak homotopy equivalence induce an isomorphism between stable homotopy groups?
Let's be precise about the question! I claim it is
not meaningful until you choose basepoints in X and
Y and restrict to based maps, since otherwise the 
suspension used to define the stable homotopy groups
is ambiguous. And then you might well get different
answers for different choices of basepoint: some might
be degenerate, others nondegenerate, in the same space.
Algebraic topologists tend to define away point-set
horrors such as degenerate basepoints. Alternatively,
growing a whisker on a bad basepoint gives a good one, and 
that is actually a cofibrant approximation for the h-model
structure ("classical" or "Strom") on based spaces. The answer 
is yes for based maps between nondegenerately based spaces.
I'm not even sure you have a suspension isomorphism for
reduced homology if the basepoint is degenerate, but you
certainly do if it is nondegenerate (e.g page 107 of
"A concise course in algebraic topology"). Now suspend
twice to get an isomorphism on homology between simply 
connected spaces, etc.


