8.5 Eilenberg-Moore spectral sequence

The Eilenberg-Moore spectral sequence is a powerful tool in algebraic topology that connects the cohomology of fiber bundles to their base spaces and fibers. It uses homological algebra techniques to break down complex spaces into simpler components, making it easier to analyze their cohomological properties.

This spectral sequence is particularly useful for studying loop spaces and classifying spaces, providing insights into their cohomology rings and characteristic classes. It complements other spectral sequences like Serre and Adams, offering a unique perspective on the structure of topological spaces and their invariants.

Definition of Eilenberg-Moore spectral sequence

• The Eilenberg-Moore spectral sequence is a powerful tool in algebraic topology that relates the cohomology of a fiber bundle to the cohomology of its base space and fiber
• It provides a systematic way to compute the cohomology of a space by breaking it down into simpler pieces and analyzing the relationships between them
• The spectral sequence is constructed using the machinery of homological algebra, specifically the derived functors of the cobar construction

Algebraic topology origins

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• The Eilenberg-Moore spectral sequence was introduced by Samuel Eilenberg and John C. Moore in the 1960s as a tool for studying the cohomology of fiber bundles and loop spaces
• It builds upon the ideas of the Serre spectral sequence, which relates the cohomology of a fibration to the cohomology of its base space and fiber
• The spectral sequence has its roots in the study of algebraic topology, where it is used to compute invariants such as cohomology rings and characteristic classes

Homological algebra applications

• The construction of the Eilenberg-Moore spectral sequence relies heavily on techniques from homological algebra, such as derived functors and spectral sequences
• It utilizes the cobar construction, which is a functor that takes a coalgebra and produces a differential graded algebra
• The spectral sequence is obtained by applying the derived functors of the cobar construction to the cohomology of the fiber and base space
• Homological algebra provides the necessary tools to analyze the convergence and collapsing of the spectral sequence

Construction of spectral sequence

• The Eilenberg-Moore spectral sequence is constructed by considering a filtration of the cohomology of the total space of a fiber bundle
• The filtration is obtained by looking at the preimages of the cohomology classes in the base space under the projection map
• The associated graded modules of the filtration give rise to the pages of the spectral sequence

Filtrations and associated graded modules

• A filtration is a sequence of submodules of a module that captures the "layers" of the module
• In the context of the Eilenberg-Moore spectral sequence, the filtration is obtained by considering the preimages of the cohomology classes in the base space
• The associated graded modules of the filtration are the successive quotients of the submodules in the filtration
• These associated graded modules form the pages of the spectral sequence, with differentials arising from the cohomology of the fiber and the cobar construction

Convergence properties

• The Eilenberg-Moore spectral sequence is said to converge if the successive pages of the spectral sequence stabilize after a finite number of steps
• Convergence implies that the spectral sequence provides a way to compute the cohomology of the total space in terms of the cohomology of the base space and fiber
• The convergence of the spectral sequence depends on certain conditions, such as the connectivity of the spaces involved and the structure of the cohomology rings

Collapsing of spectral sequence

• In some cases, the Eilenberg-Moore spectral sequence may collapse at a certain page, meaning that all the differentials on that page and beyond are zero
• Collapsing of the spectral sequence implies that the cohomology of the total space can be completely determined by the information on the collapsing page
• Collapsing can occur due to various reasons, such as the triviality of the cohomology of the fiber or the presence of certain cohomology operations

Cohomology of fiber bundles

• The Eilenberg-Moore spectral sequence is particularly useful for computing the cohomology of fiber bundles, which are topological spaces that locally look like a product of a base space and a fiber
• The spectral sequence relates the cohomology of the total space of the fiber bundle to the cohomology of the base space and the fiber
• By analyzing the differentials and the convergence of the spectral sequence, one can obtain information about the cohomology ring structure and characteristic classes of the fiber bundle

Serre spectral sequence comparison

• The Serre spectral sequence is another important tool for computing the cohomology of fiber bundles, and it is closely related to the Eilenberg-Moore spectral sequence
• While the Serre spectral sequence uses the cohomology of the base space with coefficients in the cohomology of the fiber, the Eilenberg-Moore spectral sequence uses the cobar construction and derived functors
• The two spectral sequences provide different perspectives on the cohomology of fiber bundles and can be used in complementary ways to obtain more detailed information

Computation of cohomology rings

• The Eilenberg-Moore spectral sequence can be used to compute the cohomology rings of fiber bundles, which are graded algebras that capture the multiplicative structure of the cohomology groups
• By analyzing the differentials and the convergence of the spectral sequence, one can determine the generators and relations of the cohomology ring
• The cohomology ring provides valuable information about the topological properties of the fiber bundle, such as its homotopy type and characteristic classes

Characteristic classes and transgression

• Characteristic classes are cohomology classes that are associated with fiber bundles and provide information about their topological structure
• The Eilenberg-Moore spectral sequence can be used to study the transgression of characteristic classes, which is a map between the cohomology of the fiber and the cohomology of the base space
• Transgression plays a crucial role in understanding the relationship between the characteristic classes of the total space, base space, and fiber
• The spectral sequence provides a systematic way to compute the transgression and analyze its properties

Eilenberg-Moore spectral sequence for loop spaces

• The Eilenberg-Moore spectral sequence has important applications in the study of loop spaces, which are topological spaces of loops (i.e., continuous maps from the circle to a given space)
• The spectral sequence relates the cohomology of a loop space to the cohomology of the original space and provides insights into the structure of the loop space

Based loop space cohomology

• For a topological space $X$ with a basepoint $x_0$, the based loop space $\Omega X$ is the space of loops in $X$ that start and end at $x_0$
• The Eilenberg-Moore spectral sequence can be used to compute the cohomology of $\Omega X$ in terms of the cohomology of $X$
• The spectral sequence takes into account the multiplicative structure of the cohomology of $X$ and the cobar construction on the cohomology of $\Omega X$

Cohomology of classifying spaces

• The classifying space $BG$ of a topological group $G$ is a space that classifies principal $G$-bundles
• The Eilenberg-Moore spectral sequence provides a way to compute the cohomology of the classifying space $BG$ in terms of the cohomology of the group $G$
• This is particularly useful in the study of characteristic classes and the classification of fiber bundles

Hopf algebras and homology

• The cohomology of a loop space has a natural Hopf algebra structure, which is a algebraic structure that combines the properties of an algebra and a coalgebra
• The Eilenberg-Moore spectral sequence can be used to study the relationship between the Hopf algebra structure on the cohomology of the loop space and the homology of the original space
• This connection provides insights into the algebraic and topological properties of loop spaces and their associated fiber bundles

Connections to other spectral sequences

• The Eilenberg-Moore spectral sequence is part of a larger family of spectral sequences in algebraic topology that are used to compute various invariants of topological spaces
• It has close connections to other spectral sequences, such as the Leray-Serre spectral sequence, the Adams spectral sequence, and the Grothendieck spectral sequence
• Understanding the relationships between these spectral sequences provides a more comprehensive view of the computational tools available in algebraic topology

Leray-Serre spectral sequence

• The Leray-Serre spectral sequence is a generalization of the Serre spectral sequence that applies to fibrations with non-trivial local coefficients
• It relates the cohomology of the total space of a fibration to the cohomology of the base space and the cohomology of the fibers
• The Eilenberg-Moore spectral sequence can be viewed as a special case of the Leray-Serre spectral sequence when the local coefficients are trivial

Adams spectral sequence

• The Adams spectral sequence is a powerful tool for computing stable homotopy groups of topological spaces
• It is constructed using the Ext functor in the category of modules over the Steenrod algebra
• The Eilenberg-Moore spectral sequence and the Adams spectral sequence can be related through the study of cohomology operations and their relationship to homotopy theory

Grothendieck spectral sequence

• The Grothendieck spectral sequence is a general framework for studying the composition of derived functors in homological algebra
• It provides a way to compute the derived functors of a composite functor in terms of the derived functors of its components
• The Eilenberg-Moore spectral sequence can be viewed as a special case of the Grothendieck spectral sequence in the context of algebraic topology

Applications and examples

• The Eilenberg-Moore spectral sequence has numerous applications in algebraic topology and related fields, where it is used to compute important invariants of topological spaces
• Some notable applications include the computation of homotopy groups, the study of cohomology operations, and the analysis of Massey products and higher-order operations
• Examples of spaces where the Eilenberg-Moore spectral sequence has been successfully applied include loop spaces, classifying spaces, and certain fiber bundles

Computation of homotopy groups

• Homotopy groups are important invariants of topological spaces that capture the different ways in which spheres can be mapped into the space
• The Eilenberg-Moore spectral sequence can be used to compute the homotopy groups of a space by relating them to the cohomology of the loop space
• This approach has been particularly successful in the study of homotopy groups of spheres and other classical spaces in algebraic topology

Cohomology operations

• Cohomology operations are maps between cohomology groups that are natural with respect to continuous maps between spaces
• The Eilenberg-Moore spectral sequence can be used to study the behavior of cohomology operations in the context of fiber bundles and loop spaces
• By analyzing the differentials in the spectral sequence, one can obtain information about the structure of cohomology operations and their relationships to other invariants

Massey products and higher order operations

• Massey products are a generalization of the cup product in cohomology that capture higher-order multiplicative structures
• The Eilenberg-Moore spectral sequence can be used to study Massey products and their relationships to the cohomology of fiber bundles and loop spaces
• Higher-order cohomology operations, such as Steenrod squares and Dyer-Lashof operations, can also be analyzed using the Eilenberg-Moore spectral sequence
• The spectral sequence provides a framework for understanding the role of these operations in the computation of homotopy and cohomology groups