The is a powerful tool in algebraic topology. It connects generalized cohomology theories of a space to its ordinary cohomology, providing a systematic way to compute complex cohomology groups.
This method uses a of the space by its skeleta in a CW complex structure. The resulting spectral sequence starts with ordinary cohomology and converges to the desired generalized cohomology, revealing relationships between different cohomology theories.
Definition of Atiyah-Hirzebruch spectral sequence
The Atiyah-Hirzebruch spectral sequence (AHSS) is a powerful tool in algebraic topology that relates the generalized cohomology theories of a space to its ordinary cohomology
It provides a systematic way to compute the generalized cohomology groups of a space, such as K-theory or cobordism, in terms of its ordinary cohomology groups and certain additional data
The AHSS is named after and Friedrich Hirzebruch, who introduced it in the 1960s as a generalization of the Serre spectral sequence
Construction of AHSS
Filtration of CW complex
Top images from around the web for Filtration of CW complex
Frontiers | Mechanism and Control of Meiotic DNA Double-Strand Break Formation in S. cerevisiae View original
Is this image relevant?
Narrow equilibrium window for complex coacervation of tau and RNA under cellular conditions | eLife View original
Is this image relevant?
Cortical Activation Patterns Evoked by Temporally Asymmetric Sounds and Their Modulation by ... View original
Is this image relevant?
Frontiers | Mechanism and Control of Meiotic DNA Double-Strand Break Formation in S. cerevisiae View original
Is this image relevant?
Narrow equilibrium window for complex coacervation of tau and RNA under cellular conditions | eLife View original
Is this image relevant?
1 of 3
Top images from around the web for Filtration of CW complex
Frontiers | Mechanism and Control of Meiotic DNA Double-Strand Break Formation in S. cerevisiae View original
Is this image relevant?
Narrow equilibrium window for complex coacervation of tau and RNA under cellular conditions | eLife View original
Is this image relevant?
Cortical Activation Patterns Evoked by Temporally Asymmetric Sounds and Their Modulation by ... View original
Is this image relevant?
Frontiers | Mechanism and Control of Meiotic DNA Double-Strand Break Formation in S. cerevisiae View original
Is this image relevant?
Narrow equilibrium window for complex coacervation of tau and RNA under cellular conditions | eLife View original
Is this image relevant?
1 of 3
The construction of the AHSS begins with a filtration of the space X by its skeleta in a CW complex structure
Each skeleton Xn is obtained by attaching n-dimensional cells to the previous skeleton Xn−1
This filtration gives rise to a sequence of inclusions X0⊂X1⊂⋯⊂Xn⊂⋯⊂X, where X0 is a discrete set of points and X=⋃n=0∞Xn
Spectral sequence of filtered complex
The filtration of X induces a filtration on the generalized cohomology theory h∗(X), where h∗ can be any cohomology theory (ordinary cohomology, K-theory, cobordism, etc.)
This filtered complex gives rise to a spectral sequence {[Erp,q](https://www.fiveableKeyTerm:erp,q)}, where each term Erp,q is a cohomology group that approximates the generalized cohomology group hp+q(X)
The spectral sequence starts with the E2 page, where E2p,q≅Hp(X;hq(pt)), the ordinary cohomology of X with coefficients in the generalized cohomology of a point
Convergence of AHSS
Conditions for convergence
Under suitable conditions, the AHSS converges to the generalized cohomology of the space X
The is guaranteed if X is a finite-dimensional CW complex and the cohomology theory h∗ satisfies certain axioms (e.g., the dimension axiom and the wedge axiom)
In this case, there exists an r0 such that for all r≥r0, the differentials [dr](https://www.fiveableKeyTerm:dr) are zero, and the spectral sequence collapses at the Er0 page
Limit of spectral sequence
The limit of the spectral sequence, denoted by E∞p,q, is isomorphic to the associated graded group of the filtered generalized cohomology hp+q(X)
There is a filtration 0⊂Fp+1hp+q(X)⊂Fphp+q(X)⊂⋯⊂F0hp+q(X)=hp+q(X), where Fphp+q(X)/Fp+1hp+q(X)≅E∞p,q
To obtain the actual generalized cohomology groups hp+q(X), one needs to solve extension problems that relate the successive quotients in the filtration
Applications of AHSS
Computation of generalized cohomology theories
The AHSS provides a powerful tool for computing generalized cohomology theories, such as K-theory, cobordism, and stable homotopy groups
By using the ordinary cohomology of the space X and the known generalized cohomology of a point, one can systematically compute the generalized cohomology groups of X
This is particularly useful when the generalized cohomology theory is difficult to compute directly, but its values on a point are known
Relationship between cohomology theories
The AHSS also reveals intricate relationships between different cohomology theories
For example, the AHSS for complex K-theory relates the complex K-theory of a space to its ordinary cohomology with coefficients in the ring of Laurent polynomials Z[u,u−1], where u has degree 2
Similarly, the AHSS for real K-theory involves the ordinary cohomology with coefficients in Z2[η,α,λ]/(2η,η3,ηα,α2−4λ), where η, α, and λ have degrees 1, 4, and 8, respectively
Differentials in AHSS
Structure of differentials
The differentials in the AHSS, denoted by dr:Erp,q→Erp+r,q−r+1, are homomorphisms that relate the terms in the spectral sequence
The differentials satisfy the condition dr∘dr=0, which means that the composition of any two consecutive differentials is zero
The structure of the differentials is determined by the generalized cohomology theory h∗ and the specific space X under consideration
Computation of differentials
Computing the differentials in the AHSS is a crucial step in determining the limit of the spectral sequence and the generalized cohomology groups
The differentials can be computed using various techniques, such as the geometric boundary theorem, the Atiyah-Hirzebruch differentials, and the comparison with other spectral sequences
In some cases, the differentials can be determined by the naturality of the AHSS with respect to certain maps between spaces
Extensions in AHSS
Extension problems
Once the spectral sequence collapses at the E∞ page, the next step is to solve extension problems to determine the actual generalized cohomology groups
The extension problems arise because the filtration of hp+q(X) by the subgroups Fphp+q(X) may not split, meaning that hp+q(X) may not be a direct sum of the quotients E∞p,q
Solving extension problems involves determining the possible ways to "glue together" the quotients E∞p,q to obtain the generalized cohomology groups hp+q(X)
Solving extension problems
Extension problems can be solved using various algebraic techniques, such as the use of the universal coefficient theorem, the Bockstein spectral sequence, and the Adams operations
In some cases, the extensions may be uniquely determined by the structure of the E∞ page and the properties of the generalized cohomology theory
However, in other cases, there may be multiple possible extensions, and additional information (e.g., the ring structure of the generalized cohomology) may be needed to determine the correct extension
Naturality of AHSS
Functoriality of AHSS
The AHSS is functorial with respect to maps between spaces
Given a map f:X→Y between two CW complexes, there is an induced map of spectral sequences f∗:Er∗,∗(X)→Er∗,∗(Y) that is compatible with the differentials and the limit
The functoriality of the AHSS allows for the comparison of generalized cohomology theories across different spaces and the study of the behavior of generalized cohomology under maps
Naturality with respect to maps
The naturality of the AHSS can be used to compute differentials and solve extension problems
For example, if f:X→Y is a map that induces an isomorphism on ordinary cohomology, then the induced map f∗ on the AHSS is an isomorphism from the E2 page onward
This can be used to transfer differentials and extensions from one space to another, simplifying computations
Variants of AHSS
Homological version
In addition to the cohomological AHSS, there is also a homological version that relates the generalized homology theories of a space to its ordinary homology
The homological AHSS is constructed using a filtration of the space by its coskeleta, and it converges to the generalized homology groups under suitable conditions
The homological AHSS is particularly useful for studying theories like bordism and stable homotopy groups
Multiplicative version
For generalized cohomology theories with a multiplicative structure (e.g., K-theory and cobordism), there is a multiplicative version of the AHSS that takes into account the ring structure
The multiplicative AHSS is constructed using a filtered ring spectrum, and the differentials and extensions are required to be compatible with the multiplicative structure
The multiplicative AHSS provides additional information about the ring structure of the generalized cohomology groups and can simplify computations in some cases
Comparison with other spectral sequences
Serre spectral sequence
The Serre spectral sequence is another powerful tool in algebraic topology that relates the cohomology of a fiber bundle to the cohomology of its base space and fiber
The AHSS can be viewed as a generalization of the Serre spectral sequence, where the role of the fiber bundle is played by the skeletal filtration of the space
In some cases, the AHSS can be used in conjunction with the Serre spectral sequence to compute generalized cohomology groups of certain fiber bundles
Adams spectral sequence
The Adams spectral sequence is a spectral sequence that computes the stable homotopy groups of a space using its mod p cohomology
The AHSS and the Adams spectral sequence are related by the Atiyah-Hirzebruch-Whitehead spectral sequence, which is a spectral sequence that relates the generalized homology of a space to its stable homotopy groups
The comparison between the AHSS and the Adams spectral sequence can provide insights into the relationship between generalized cohomology theories and stable homotopy theory
Examples and calculations
Classical cohomology theories
The AHSS has been successfully applied to compute the generalized cohomology groups of various spaces using classical cohomology theories like ordinary cohomology, K-theory, and cobordism
For example, the AHSS can be used to compute the complex K-theory of projective spaces, Grassmannians, and flag varieties
The AHSS has also been used to study the relationship between different classical cohomology theories and to prove important results like the Atiyah-Hirzebruch formula for the Todd genus
Extraordinary cohomology theories
The AHSS has also been applied to compute the generalized cohomology groups of spaces using extraordinary cohomology theories, such as elliptic cohomology, tmf (topological modular forms), and Morava K-theory
These extraordinary cohomology theories are related to important problems in algebraic topology, number theory, and mathematical physics
The use of the AHSS in these contexts has led to significant advances in our understanding of these theories and their applications to other areas of mathematics
Key Terms to Review (17)
Atiyah-Hirzebruch spectral sequence: The Atiyah-Hirzebruch spectral sequence is a mathematical tool used to compute the homology and cohomology of topological spaces by utilizing the properties of K-theory. This sequence connects the geometry of the space with algebraic invariants, allowing for computations that might be difficult using traditional methods. It forms an essential bridge between algebraic topology and K-theory, enabling deeper insights into the structure of vector bundles over a space.
Cohomological Dimension: Cohomological dimension is a concept in algebraic topology that measures the complexity of a space or a module by determining the highest degree of cohomology groups that can be non-zero. This dimension provides insight into how 'complicated' a space is in terms of its topological features and relationships, linking directly to various important theories and results in cohomology.
Convergence: Convergence refers to the process by which a sequence of objects, often in a topological or algebraic context, approaches a limit or stable configuration. In various mathematical frameworks, such as spectral sequences, convergence is crucial for ensuring that the derived objects ultimately yield consistent and meaningful results, allowing mathematicians to make conclusions about the structure and properties of spaces under study.
D_r: In cohomology theory, $d_r$ represents the differential in the $r$-th page of a spectral sequence. It is a key operation that connects elements of one filtration or grading to another, allowing for the computation of cohomology groups and other algebraic invariants. The role of $d_r$ is crucial in understanding how information propagates through the spectral sequence as it converges towards a final result.
Differential: In mathematics, a differential is an operator that describes the rate at which a function changes as its input changes. It captures how functions evolve and is crucial in various contexts, including the study of homology and cohomology. Differentials can be used to compute various properties of spaces and mappings, serving as a foundation for deeper tools like spectral sequences, which analyze complex structures by breaking them down into simpler components.
E_r^{p,q}: The notation e_r^{p,q} represents the entry at the (p,q) position in the 'r'-th page of the Atiyah-Hirzebruch spectral sequence, which is a powerful tool in algebraic topology used to compute homology or cohomology groups. Each entry in this table captures significant information about the structure of the underlying space and its associated topological properties, and it evolves through different pages as one applies the spectral sequence's differential maps.
E-page: An e-page is a term used in the context of spectral sequences, particularly the Atiyah-Hirzebruch spectral sequence, to refer to the page at which one computes the $E$-terms, which give crucial information about cohomology. The e-page specifically allows mathematicians to analyze the convergence of the spectral sequence and extract significant topological data from a given space.
Fibration: A fibration is a specific type of mapping between topological spaces that satisfies a certain lifting property, allowing for the lifting of homotopies and paths. In essence, fibrations help organize spaces into a structured framework where fibers can be studied individually while still understanding their relation to the base space. This concept is critical in the context of spectral sequences, as it allows for the analysis of how cohomological properties are preserved across different levels.
Filtration: Filtration is a mathematical concept that refers to a way of organizing or breaking down a structure into simpler parts, often used in the context of algebraic or topological objects. It allows for the systematic study of properties by examining substructures, making it a vital tool in various cohomological contexts, such as spectral sequences and their applications.
Fritz Hirzebruch: Fritz Hirzebruch was a prominent German mathematician known for his significant contributions to topology, particularly in the development of characteristic classes and the Hirzebruch-Riemann-Roch theorem. His work has been influential in bridging algebraic geometry and topology, and he played a key role in the creation of the Atiyah-Hirzebruch spectral sequence, which is a powerful tool for computing homology and cohomology groups.
Grothendieck's Theorem: Grothendieck's Theorem is a fundamental result in algebraic geometry and topology that establishes a deep connection between cohomology theories and K-theory. It is particularly known for providing a framework for understanding how various types of cohomological information can be expressed in terms of vector bundles and their classifications. This theorem plays a crucial role in many advanced concepts, including spectral sequences and the interplay between algebraic and topological structures.
Leray Spectral Sequence: The Leray Spectral Sequence is a powerful tool in algebraic topology that relates the cohomology of a space to the cohomology of its fibers and base spaces, particularly in the context of fibrations. It provides a systematic method to compute cohomology groups when dealing with maps between topological spaces, bridging the concepts of singular homology and sheaf cohomology. This sequence also extends to various types of spectral sequences, making it a versatile tool across different mathematical frameworks.
Michael Atiyah: Michael Atiyah was a renowned British mathematician known for his significant contributions to geometry and topology, particularly in the development of K-theory and the Atiyah-Hirzebruch spectral sequence. His work has had a lasting impact on various areas of mathematics, influencing both theoretical frameworks and practical applications.
Projective bundles: Projective bundles are geometric constructions that generalize the notion of projective space to vector bundles. They can be understood as the space of lines in a vector bundle, and they play a crucial role in algebraic geometry and topology, particularly in relation to the study of vector bundles over a base space and their associated cohomological properties.
Sheaf Cohomology: Sheaf cohomology is a mathematical tool used to study the global properties of sheaves on topological spaces through the use of cohomological techniques. It allows for the calculation of the cohomology groups of a sheaf, providing insights into how local data can give rise to global information, which connects with several important concepts in algebraic topology and algebraic geometry.
Singular Cohomology: Singular cohomology is a mathematical tool used in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, allowing us to study its global properties through the use of singular simplices. This concept connects the geometric aspects of spaces with algebraic structures, providing insights into various topological features such as holes and connectivity.
Topological k-theory: Topological k-theory is a branch of algebraic topology that studies vector bundles over topological spaces using cohomological methods. It provides a way to classify vector bundles and relates them to stable homotopy theory, making it a powerful tool in understanding the geometric and topological properties of spaces. This theory is deeply connected to the Atiyah-Hirzebruch spectral sequence, which provides a computational framework for deriving invariants of topological spaces, and it plays a crucial role in K-theory, which extends the concepts of vector bundles to more abstract settings.