The order of the differentials in the Atiyah-Hirzebruch spectral sequence

LetF _* be the homology theory corresponding to a spectrumF and consider the Atiyah-Hirzebruch spectral sequenceE _s,t ² H _s (X; _t F) F _s+t (X) for a bounded below spectrum (or CW-complex)X. This paper shows that the images of the differentials d _s,t ^r :E _s,t ^r E _s ^{–r,t+r–1r} in this spectral sequence are always torsion groupsof finite exponent and that this exponent isbounded in a very universal way: we prove the existence of integersR _r forr2 such thatR _r d _s,t ^r =0 for any spectrumF, for any bounded below spectrumX and for all integersr2,s andt. The interesting point is that these upper boundsR _r for the additive order of the differentials d _s,t ^r dependonly onr, and that the result holds without any hypothesis on the spectrumF. In certain special cases, this implies that the spectral sequence collapses and even that the extension problems given by itsE -term are trivial.