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 let and denote singular homology and singular cohomology with coefficients in k, respectively.
Consider the following pullback of a continuous map p:
A frequent question is how the homology of the fiber product, , relates to the homology of B, X and E. For example, if B is a point, then the pullback is just the usual product . 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
This is a generalization insofar as the zeroeth Tor functor is just the tensor product and in the above special case the cohomology of the point B is just the coefficient field k (in degree 0).
Dually, we have the following homology spectral sequence:
The spectral sequence arises from the study of differential graded objects ( chain complexes), not spaces. The following discusses the original homological construction of Eilenberg and Moore. The cohomology case is obtained in a similar manner.
Let
be the singular chain functor with coefficients in . By the Eilenberg–Zilber theorem, has a differential graded coalgebra structure over with structure maps
In down-to-earth terms, the map assigns to a singular chain s: Δn → B the composition of s and the diagonal inclusion B ⊂ B × B. Similarly, the maps and induce maps of differential graded coalgebras
, .
In the language of comodules, they endow and with differential graded comodule structures over , with structure maps
and similarly for E instead of X. It is now possible to construct the so-called cobar resolution for
as a differential graded comodule. The cobar resolution is a standard technique in differential homological algebra:
where the n-th term is given by
The maps are given by
where is the structure map for as a left comodule.
The cobar resolution is a bicomplex, one degree coming from the grading of the chain complexes S∗(−), the other one is the simplicial degree n. The total complex of the bicomplex is denoted .
The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map
that induces a quasi-isomorphism (i.e. inducing an isomorphism on homology groups)
where is the cotensor product and Cotor (cotorsion) is the derived functor for the cotensor product.
To calculate
view
as a double complex.
For any bicomplex there are two filtrations (see John McCleary ( 2001) or the spectral sequence of a filtered complex); in this case the Eilenberg−Moore spectral sequence results from filtering by increasing homological degree (by columns in the standard picture of a spectral sequence). This filtration yields
These results have been refined in various ways. For example, William G. Dwyer ( 1975) refined the convergence results to include spaces for which
acts nilpotently on
for all and Brooke Shipley ( 1996) further generalized this to include arbitrary pullbacks.
The original construction does not lend itself to computations with other homology theories since there is no reason to expect that such a process would work for a homology theory not derived from chain complexes. However, it is possible to axiomatize the above procedure and give conditions under which the above spectral sequence holds for a general (co)homology theory, see Larry Smith's original work ( Smith 1970) or the introduction in ( Hatcher 2002).
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 let and denote singular homology and singular cohomology with coefficients in k, respectively.
Consider the following pullback of a continuous map p:
A frequent question is how the homology of the fiber product, , relates to the homology of B, X and E. For example, if B is a point, then the pullback is just the usual product . 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
This is a generalization insofar as the zeroeth Tor functor is just the tensor product and in the above special case the cohomology of the point B is just the coefficient field k (in degree 0).
Dually, we have the following homology spectral sequence:
The spectral sequence arises from the study of differential graded objects ( chain complexes), not spaces. The following discusses the original homological construction of Eilenberg and Moore. The cohomology case is obtained in a similar manner.
Let
be the singular chain functor with coefficients in . By the Eilenberg–Zilber theorem, has a differential graded coalgebra structure over with structure maps
In down-to-earth terms, the map assigns to a singular chain s: Δn → B the composition of s and the diagonal inclusion B ⊂ B × B. Similarly, the maps and induce maps of differential graded coalgebras
, .
In the language of comodules, they endow and with differential graded comodule structures over , with structure maps
and similarly for E instead of X. It is now possible to construct the so-called cobar resolution for
as a differential graded comodule. The cobar resolution is a standard technique in differential homological algebra:
where the n-th term is given by
The maps are given by
where is the structure map for as a left comodule.
The cobar resolution is a bicomplex, one degree coming from the grading of the chain complexes S∗(−), the other one is the simplicial degree n. The total complex of the bicomplex is denoted .
The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map
that induces a quasi-isomorphism (i.e. inducing an isomorphism on homology groups)
where is the cotensor product and Cotor (cotorsion) is the derived functor for the cotensor product.
To calculate
view
as a double complex.
For any bicomplex there are two filtrations (see John McCleary ( 2001) or the spectral sequence of a filtered complex); in this case the Eilenberg−Moore spectral sequence results from filtering by increasing homological degree (by columns in the standard picture of a spectral sequence). This filtration yields
These results have been refined in various ways. For example, William G. Dwyer ( 1975) refined the convergence results to include spaces for which
acts nilpotently on
for all and Brooke Shipley ( 1996) further generalized this to include arbitrary pullbacks.
The original construction does not lend itself to computations with other homology theories since there is no reason to expect that such a process would work for a homology theory not derived from chain complexes. However, it is possible to axiomatize the above procedure and give conditions under which the above spectral sequence holds for a general (co)homology theory, see Larry Smith's original work ( Smith 1970) or the introduction in ( Hatcher 2002).