🪡K-Theory Unit 4 – The Atiyah–Hirzebruch Spectral Sequence

Key Concepts and Definitions

• The Atiyah–Hirzebruch spectral sequence (AHSS) computes the generalized cohomology theory of a space $X$ using its cellular filtration
• Relates the ordinary cohomology of $X$ with coefficients in the cohomology theory $h^*$ to the generalized cohomology $h^*(X)$
• Consists of a sequence of pages $E_r^{p,q}$ converging to the associated graded module of a filtration on $h^*(X)$
  • The pages are obtained by taking cohomology of the previous page with respect to a certain differential
• The $E_2$ page of the AHSS is given by $E_2^{p,q} = H^p(X; h^q(pt))$, where $h^q(pt)$ denotes the value of the cohomology theory $h^*$ on a point
• The differentials on the $E_r$ page have bidegree $(r, 1-r)$, i.e., they map $E_r^{p,q}$ to $E_r^{p+r, q-r+1}$
• Convergence of the spectral sequence means that for sufficiently large $r$, the $E_r$ page stabilizes and its terms are the associated graded module of $h^*(X)$ with respect to a certain filtration

Historical Context and Development

• The AHSS was introduced by Michael Atiyah and Friedrich Hirzebruch in their 1961 paper "Vector bundles and homogeneous spaces"
• Motivated by the problem of computing the K-theory of a space using its cellular structure
• The construction of the AHSS relies on the notion of a filtered space and the associated spectral sequence of a filtered complex
• The AHSS can be seen as a generalization of the Leray–Serre spectral sequence, which computes the cohomology of a fibration
• The development of the AHSS was influenced by earlier work on spectral sequences, such as the Serre spectral sequence and the Adams spectral sequence
• The AHSS has been widely applied in various areas of algebraic topology, including K-theory, cobordism theory, and stable homotopy theory
• Subsequent work by other mathematicians has led to generalizations and extensions of the AHSS, such as the Atiyah–Hirzebruch–Segal spectral sequence and the Adams–Novikov spectral sequence

Constructing the Spectral Sequence

• Start with a generalized cohomology theory $h^*$ and a CW complex $X$
• Consider the skeletal filtration of $X$, given by the subspaces $X_n$ consisting of cells of dimension at most $n$
• The filtration gives rise to a spectral sequence $E_r^{p,q}$ converging to the associated graded module of $h^*(X)$ with respect to the filtration
• The $E_1$ page is given by $E_1^{p,q} = h^{p+q}(X_p, X_{p-1})$, the relative cohomology of the pairs $(X_p, X_{p-1})$
  • These relative cohomology groups can be computed using the long exact sequence of the pair and the excision theorem
• The $E_2$ page is obtained by taking the cohomology of the $E_1$ page with respect to the differential $d_1$, which is induced by the boundary maps in the long exact sequences
• The higher differentials $d_r$ on the $E_r$ page are determined by the requirement that $d_r \circ d_r = 0$ and the convergence to the associated graded module of $h^*(X)$
• The spectral sequence is said to collapse at the $E_r$ page if all the differentials $d_s$ for $s \geq r$ are zero
  • In this case, the $E_r$ page gives the associated graded module of $h^*(X)$

Properties and Structure

• The AHSS is a first-quadrant spectral sequence, meaning that $E_r^{p,q} = 0$ for $p < 0$ or $q < 0$
• The differentials $d_r$ on the $E_r$ page have bidegree $(r, 1-r)$, i.e., they map $E_r^{p,q}$ to $E_r^{p+r, q-r+1}$
• The $E_2$ page of the AHSS has a multiplicative structure given by the cup product in ordinary cohomology and the multiplication in the cohomology theory $h^*$
  • This multiplicative structure is compatible with the differentials, making the AHSS a spectral sequence of algebras
• The edge homomorphisms of the AHSS relate the generalized cohomology $h^*(X)$ to the ordinary cohomology of $X$ with coefficients in $h^*$
  • The edge homomorphism $h^n(X) \to H^n(X; h^0(pt))$ is given by the map induced by the inclusion of the 0-skeleton $X_0 \to X$
• The AHSS is natural with respect to maps of spaces, i.e., a map $f: X \to Y$ induces a map of spectral sequences $f_r: E_r(X) \to E_r(Y)$ compatible with the differentials and convergence
• The AHSS satisfies a Künneth formula, relating the spectral sequence for a product of spaces to the tensor product of the spectral sequences for the individual factors

Applications in K-Theory

• The AHSS is particularly useful in computing the K-theory of a space $X$, denoted $K^*(X)$
• For K-theory, the $E_2$ page of the AHSS is given by $E_2^{p,q} = H^p(X; K^q(pt))$, where $K^q(pt) = \mathbb{Z}$ for $q$ even and $0$ for $q$ odd
• The differentials in the AHSS for K-theory have a geometric interpretation in terms of vector bundles and their Chern classes
  • The first non-trivial differential $d_3$ is related to the second Stiefel–Whitney class of vector bundles
• The AHSS has been used to compute the K-theory of various spaces, such as projective spaces, Grassmannians, and flag varieties
• The AHSS can also be used to study the relationship between K-theory and other generalized cohomology theories, such as complex cobordism and elliptic cohomology
• In some cases, the AHSS for K-theory collapses at the $E_2$ page, providing a complete description of $K^*(X)$ in terms of the ordinary cohomology of $X$
  • This happens, for example, when $X$ is a toric variety or a flag variety

Computational Techniques

• To compute the differentials in the AHSS, one often uses the geometric interpretation of the differentials in terms of characteristic classes of vector bundles
• The Chern character, a ring homomorphism from K-theory to ordinary cohomology, can be used to relate the differentials in the AHSS for K-theory to operations in ordinary cohomology
• Spectral sequences can be computed using algebraic software packages, such as Macaulay2 or Sage
  • These tools can be particularly helpful in dealing with the algebraic complexity of the higher pages of the spectral sequence
• The comparison theorem for spectral sequences can be used to relate the AHSS to other spectral sequences, such as the Serre spectral sequence or the Adams spectral sequence
  • This can provide additional information about the differentials and the convergence of the spectral sequence
• In some cases, the differentials in the AHSS can be determined using the structure of the cohomology theory $h^*$ and the space $X$
  • For example, if $X$ is a sphere or a complex projective space, the differentials can often be computed using the Atiyah–Hirzebruch spectral sequence for the cohomology of the classifying space $BU(n)$

Related Theories and Extensions

• The AHSS has been generalized to the equivariant setting, where one considers spaces with a group action and equivariant cohomology theories
  • The equivariant AHSS computes the equivariant generalized cohomology of a space using its equivariant cellular filtration
• The Atiyah–Hirzebruch–Segal spectral sequence is a generalization of the AHSS that applies to generalized cohomology theories represented by ring spectra
  • It takes into account the multiplicative structure of the cohomology theory and converges to the completed tensor product of the cohomology of the space and the coefficients
• The Adams–Novikov spectral sequence is another generalization of the AHSS that uses complex cobordism theory instead of ordinary cohomology
  • It is particularly useful in studying the stable homotopy groups of spheres and the structure of the stable homotopy category
• The AHSS has been applied to the study of other generalized cohomology theories, such as topological modular forms (tmf) and algebraic K-theory
  • In these cases, the spectral sequence often provides valuable information about the relationship between the generalized cohomology theory and ordinary cohomology
• The construction of the AHSS has been extended to the setting of triangulated categories, where it is known as the Postnikov system
  • This generalization has applications in derived algebraic geometry and the study of motives

Challenges and Open Problems

• Computing the differentials in the AHSS can be a challenging problem, particularly on the higher pages of the spectral sequence
  • In many cases, the differentials are not known explicitly and require a deep understanding of the geometry and topology of the space and the cohomology theory
• Determining the convergence of the AHSS can also be difficult, especially when the spectral sequence does not collapse at a finite page
  • In some cases, the convergence can be established using additional structure, such as the Adams filtration or the slice filtration
• The AHSS has been used to study the K-theory of various spaces, but there are still many open questions about the K-theory of certain classes of spaces, such as algebraic varieties or infinite-dimensional spaces
• Extending the AHSS to other generalized cohomology theories, such as elliptic cohomology or motivic cohomology, is an active area of research
  • These extensions often require new ideas and techniques to handle the additional structure and complexity of the cohomology theories
• Understanding the relationship between the AHSS and other spectral sequences, such as the Adams spectral sequence or the motivic Adams spectral sequence, is an ongoing challenge
  • These relationships can provide new insights into the structure of generalized cohomology theories and their applications in homotopy theory and algebraic geometry