Finite-CW AHSS convergence does not automatically extend to infinite CW complexes

The convergence theorems of Cohomological Atiyah–Hirzebruch spectral sequence and Homological Atiyah–Hirzebruch spectral sequence are stated for finite CW complexes. Their proofs use boundedness of the skeletal filtration: for each fixed total degree only finitely many filtration stages can differ, so both the increasing and decreasing families of stable cycles and boundaries stabilize after finitely many steps. For an infinite-dimensional CW complex the skeletal filtration is unbounded, and for a general infinite CW complex the finite-CW argument cannot simply be invoked. An infinite but finite-dimensional CW complex still has a bounded skeletal filtration, so infinitude alone is not the obstruction. In the genuinely unbounded case one needs separate hypotheses and arguments, such as conditional or strong convergence together with the derived-limit analysis of the filtration. No convergence and no failure of convergence is asserted here for infinite complexes; this is a limitation of the stated theorems, not a counterexample.

Compare Davis–Kirk, §9.1, printed pp. 237–246, for the finite skeletal-filtration convergence statements and the additional hypotheses required in the infinite case.