The Atiyah–Hirzebruch spectral sequence is an important tool in algebraic topology, specifically in the computation of homotopy groups and cohomology theories. It provides a way to calculate the homology or cohomology of a space using a spectral sequence that is associated with a specific filtration. The original context for the spectral sequence primarily relates to complex vector bundles and characteristic classes.
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The skeletal filtration of a CW complex gives this spectral sequence, with differential of bidegree in cohomological grading. For a finite complex the filtration is finite and convergence is bounded. In a multiplicative theory the associated graded product is the ordinary cohomology product with coefficients in the theory's coefficient ring.