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 $E f$ 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 $δ n$ , $λ f$ 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

π 1 (B)

acts nilpotently on

H i (E f)

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

[EM] Eilenberg, Samuel; Moore, John C. Limits and spectral sequences. Topology 1 1962 1--23.
[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
[Ship] Shipley, Brooke E. Convergence of the homology spectral sequence of a cosimplicial space. Amer. J. Math. 118 (1996), no. 1, 179--207
[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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Categories: Wikipedia articles needing context | Wikipedia introduction cleanup | Spectral sequences | Homology theory

Eilenberg-Moore spectral sequence

From Wikipedia, the free encyclopedia

[edit] References

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