A connected CW-complex $X$ is contractible iff the homotopy groups $\pi_n(X)$ are zero for all $n\geq 1$.
What (if any?) is the analogous statement for the vanishing of all stable homotopy groups $\pi_n^S(X)$ for all $n\geq 1$ ?
More generally, what is (if any?) the analogue of Whitehead's theorem in the stable world?
Thanks.
Their is a version of Whitehead's theorem in the category of Spectra. On needs to define the notion of CW-spectra. Once this is done Whiteheads theorem may be proved. Hatcher does this in some notes on the Adams Spectral sequence http://www.math.cornell.edu/~hatcher/SSAT/SSch2.pdf. The discussion of CW spectra begins on page 7 and Whitehead's theorem appears on page 10.