Eilenberg–Moore spectral sequence
In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's original paper addresses this for singular homology.
Let be a field and
denote singular homology and singular cohomology with coefficients in k, respectively.
Consider the following pullback Ef of a continuous map p:
A frequent question is how the homology of the fiber product Ef, relates to the ones of B, X and E. For example, if B is a point, then the pullback is just the usual product E × X. In this case the Künneth formula says
However this relation is not true in more general situations. The Eilenberg−Moore spectral sequence is a device which allows the computation of the (co)homology of the fiber product in certain situations.
The Eilenberg−Moore spectral sequences generalizes the above isomorphism to the situation where p is a fibration of topological spaces and the base B is simply connected. Then there is a convergent spectral sequence with
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