Eilenberg-Moore spectral sequence

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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.

Let k be a field and

H_\ast(-)=H_\ast(-,k)

denote homology with coefficients in k.

The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's original paper [EM] addresses this for singular homology.

Assume that p is a fibration and Ef is the pullback.

\begin{array}{c c c} E_f &\rightarrow & E \\ \downarrow & & \downarrow{p}\\ X &\rightarrow_{ f} &B\\ \end{array}

Theorem[EM] If B is simply connected, there is a convergent spectral sequence with

E^2_{\ast,\ast}=\text{Cotor}^{H_\ast(B)}_{\ast,\ast}(H_\ast(X),H_\ast(E))\Rightarrow H_\ast(E_f).

This spectral sequence arises from the study of [[differential graded]] objects (chain complexes), not spaces. Eilenberg and Moore's original construction works as follows. Let

S_\ast(-)=S_\ast(-,k)

be the singular chain functor with coefficients in k. By the Eilenberg-Zilber theorem, S_\ast(B) has a differential graded coalgebra structure over k with structure maps

S_\ast(B)\rightarrow_{\triangle} S_\ast(B\times B)\rightarrow_{\simeq}S_\ast(B)\otimes S_\ast(B).

The maps f and p induce maps of differential graded coalgebras

f_\ast : S_\ast(X)\rightarrow S_\ast(B)
p_\ast : S_\ast(E)\rightarrow S_\ast(B)

When composed with the diagonal map, these maps endow S_\ast(B) with differential graded comodule structures over

S_\ast(E) and S_\ast(X), with structure maps
S_\ast(X)\rightarrow_{\triangle} S_\ast(X)\otimes S_\ast(X)\rightarrow_{f_\ast\otimes 1} S_\ast(B)\otimes S_\ast(X)
S_\ast(E)\rightarrow_{\triangle} S_\ast(E)\otimes S_\ast(E)\rightarrow_{1 \otimes p_\ast} S_\ast(E)\otimes S_\ast(B).

It is now possible to construct the cobar resolution for

S_\ast(X)

as a differential graded S_\ast(B) comodule.

\mathcal{C}(S_\ast(X),S_\ast(B))=
\cdots\leftarrow_{\delta_2} \mathcal{C}_{-2}(S_\ast(X),S_\ast(B))\leftarrow_{\delta_1} \mathcal{C}_{-1}(S_\ast(X),S \ast(B))\leftarrow_{\delta_0} S_\ast(X)\otimes S_\ast(B),

where, in general

\mathcal{C}_{-n}(S_\ast(X),S_\ast(B))=S_\ast(X)\otimes \underbrace{S_\ast(B)\otimes \cdots \otimes S_\ast(B)}_{n}\otimes S_\ast(B),

with maps

\delta_n = \lambda_f\otimes\cdots\otimes 1 + \sum_{i=2}^n 1\otimes\cdots \otimes\triangle_i\otimes\cdots\otimes 1.

In the expression for δn, λf is the structure map for S_\ast(X) as a left S_\ast(B) comodule. This cobar resolution is a bicomplex whose total complex is denoted

\mathbf{\mathcal{C}}_\bullet.

Eilenberg and Moore showed the following:

[EM] Suppose we have

\begin{array}{c c c} E_f &\rightarrow & E \\ \downarrow & & \downarrow{p}\\ X &\rightarrow_{ f } &B\\ \end{array}

with B simply connected. Then there is a map

\Theta: \mathbf{\mathcal{C}}_{\bullet {\text{ }\Box_{S_\ast(B)}}}S_\ast(E)\rightarrow S_\ast(E_f,k)

that induces an isomorphism on homology

\Theta_\ast : Cotor^{S_\ast(B)}(S_\ast(X)S_\ast(E))\rightarrow H_\ast(E_f),

where \Box_{S_\ast(B)} is the cotensor product and Cotor is the derived functor for the cotensor product. For detailed definitions see [EM].

To calculate

H_\ast(\mathbf{\mathcal{C}}_{\bullet {\text{ }\Box_{S_\ast(B)}}}S_\ast(E)),

view

\mathbf{\mathcal{C}}_{\bullet {\text{ }\Box_{S_\ast(B)}}}S_\ast(E)

as a double complex.

For any bicomplex there are two filtrations [JM]; 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

E^2=Cotor^{H_\ast(B)}(H_\ast(X),H_\ast(E)).

These results have been refined in various ways. For example [Dwyer] refined the convergence results to include spaces for which

π1(B)

acts nilpotently on

Hi(Ef)

for all i\geq 0 and [Ship] 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.

[edit] References

  1. [EM] Eilenberg, Samuel; Moore, John C. Limits and spectral sequences. Topology 1 1962 1--23.
  2. [JM] McCleary, John A user's guide to spectral sequences. Second edition. Cambridge Studies in Advanced Mathematics, 58. Cambridge University Press, Cambridge, 2001. xvi+561 pp. ISBN: 0-521-56759-9
  3. [Ship] Shipley, Brooke E. Convergence of the homology spectral sequence of a cosimplicial space. Amer. J. Math. 118 (1996), no. 1, 179--207
  4. [Dwyer] Dwyer, William G. Exotic convergence of the Eilenberg-Moore spectral sequence. Illinois J. Math. 19 (1975), no. 4, 607--617.
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