- Eilenberg-Moore spectral sequence
In
mathematics , in the field ofalgebraic topology , the Eilenberg-Moore spectral sequence addresses the calculation of thehomology group s of apullback over afibration . Thespectral sequence formulates the calculation from knowledge of the homology of the remaining spaces.Samuel Eilenberg andJohn C. Moore 's original paper addresses this forsingular homology .Motivation
Let be a field and
:denote
singular homology andsingular 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:"H"∗("Ef") = "H"∗("X"×"E") ≅ "H"∗("X") ⊗k "H"∗("E").
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.
tatement
The Eilenberg-Moore spectral sequences generalizes the above isomorphism to the situation where "p" is a
fibration of topological spaces and the base "B" issimply connected . Then there is a convergent spectral sequence with:This is a generalization insofar as the zeroethTor 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::
Indications on the proof
The spectral sequence arises from the study of differential graded objects (
chain complex es), 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 theEilenberg-Zilber theorem , has a differential gradedcoalgebra structure over withstructure 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
comodule s, they endow with differential graded comodule structures over and , 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". Thetotal 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 thederived functor for the cotensor product.To calculate
:,
view
:
as a
double complex .For any bicomplex there are two
filtration s (see Harv|McCleary|2001 or thespectral 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 Harv|Dwyer|1975 refined the convergence results to include spaces for which
:
acts
nilpotent ly on:
for all and Harv|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 Smith's original work Harv|Smith|1970 or the introduction in Harv|Hatcher|2002).
References
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* | year=1970 | volume=134
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