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
k
H\ast(-)=H\ast(-,k)
H\ast(-)=H\ast(-,k)
Consider the following pullback
Ef
\begin{array}{ccc}Ef& → &E\ \downarrow&&\downarrow{p}\ X& → &B\ \end{array}
A frequent question is how the homology of the fiber product,
Ef
E x X
*(E | |
H | |
f) |
=H*(X x E)\congH*(X) ⊗ kH*(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.
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
\ast,\ast | |
E | |
2 |
\ast,\ast | |
=Tor | |
H\ast(B) |
(H\ast(X),H\ast(E)) ⇒
\ast(E | |
H | |
f). |
Dually, we have the following homology spectral sequence:
2 | |
E | |
\ast,\ast |
H\ast(B) | |
=Cotor | |
\ast,\ast |
(H\ast(X),H\ast(E)) ⇒ H\ast(Ef).
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
S\ast(-)=S\ast(-,k)
k
S\ast(B)
k
S\ast(B)\xrightarrow{\triangle}S\ast(B x B)\xrightarrow{\simeq}S\ast(B) ⊗ S\ast(B).
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
f
p
f\ast\colonS\ast(X) → S\ast(B)
p\ast\colonS\ast(E) → S\ast(B)
In the language of comodules, they endow
S\ast(E)
S\ast(X)
S\ast(B)
S\ast(X)\xrightarrow{\triangle}S\ast(X) ⊗ S\ast(X)\xrightarrow{f\ast ⊗ 1}S\ast(B) ⊗ S\ast(X)
S\ast(X)
S\ast(B)
l{C}(S\ast(X),S\ast(B))= … \xleftarrow{\delta2}l{C}-2(S\ast(X),S\ast(B))\xleftarrow{\delta1}l{C}-1(S\ast(X),S\ast(B))\xleftarrow{\delta0}S\ast(X) ⊗ S\ast(B),
where the n-th term
l{C}-n
l{C}-n(S\ast(X),S\ast(B))=S\ast(X) ⊗ \underbrace{S\ast(B) ⊗ … ⊗ S\ast(B)}n ⊗ S\ast(B).
The maps
\deltan
λf ⊗ … ⊗ 1+
n | |
\sum | |
i=2 |
1 ⊗ … ⊗ \trianglei ⊗ … ⊗ 1,
λf
S\ast(X)
S\ast(B)
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
l{C
The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map
\Theta\colonl{C
that induces a quasi-isomorphism (i.e. inducing an isomorphism on homology groups)
\Theta\ast\colon
S\ast(B) | |
\operatorname{Cotor} |
(S\ast(X)S\ast(E)) → H\ast(Ef),
where
\Box | |
S\ast(B) |
To calculate
H\ast(l{C
view
l{C
as a double complex.
For any bicomplex there are two filtrations (see 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
E2=\operatorname{Cotor}
H\ast(B) | |
(H\ast(X),H\ast(E)).
These results have been refined in various ways. For example, refined the convergence results to include spaces for which
\pi1(B)
acts nilpotently on
Hi(Ef)
for all
i\geq0
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 or the introduction in .