3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.