8.4 Adams spectral sequence

The Adams spectral sequence is a powerful tool in algebraic topology for computing stable homotopy groups. It's constructed by applying the Ext functor to cohomology with coefficients in the Steenrod algebra, bridging algebra and topology.

This sequence uses Adams resolutions to compute topological Ext groups, which form its E2 term. Convergence depends on spectrum connectivity and filtration completeness. The vanishing line allows finite computations in specific degree ranges.

Construction of Adams spectral sequence

• The Adams spectral sequence a powerful tool in algebraic topology used to compute stable homotopy groups of topological spaces and spectra
• Constructed by applying the algebraic Ext functor to the cohomology of a space or spectrum with coefficients in a suitable ring (usually the Steenrod algebra)

Algebraic Ext functor

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• Derived functor of the Hom functor in the category of modules over a ring
• Measures the failure of the Hom functor to be exact
• Can be computed using projective resolutions or injective resolutions
• Yields a sequence of abelian groups $Ext^n_R(M,N)$ for $R$-modules $M$ and $N$

Topological Ext functor

• Analogous to the algebraic Ext functor but defined in the category of spectra
• Takes as input a ring spectrum $E$ and $E$-module spectra $M$ and $N$
• Yields a sequence of abelian groups $Ext^n_E(M,N)$ which are the $E_2$ term of the Adams spectral sequence
• Can be computed using Adams resolutions (projective resolutions in the category of spectra)

Comparison of algebraic and topological Ext

• For a space or spectrum $X$, the algebraic $Ext$ over the Steenrod algebra $Ext^*_{A_p}(H^*(X;\mathbb{F}_p),\mathbb{F}_p)$ is isomorphic to the topological $Ext$ over the Eilenberg-MacLane spectrum $Ext^*_{H\mathbb{F}_p}(H\mathbb{F}_p \wedge X, H\mathbb{F}_p)$
• This isomorphism is a key ingredient in setting up the Adams spectral sequence

Adams resolution

• A projective resolution in the category of spectra used to compute the topological Ext functor
• Constructed inductively by taking the fiber of a map from the previous stage to a product of Eilenberg-MacLane spectra
• The $E_1$ term of the Adams spectral sequence the cohomology of the Adams resolution

Filtration of Adams resolution

• The Adams resolution naturally equipped with a filtration by the degree of the resolution
• This filtration induces a filtration on the homotopy groups of the original spectrum
• The associated graded of this filtration the $E_\infty$ term of the Adams spectral sequence

Convergence of Adams spectral sequence

• The Adams spectral sequence not always convergent, meaning that the $E_\infty$ term may not be isomorphic to the associated graded of the filtration on the homotopy groups
• Convergence depends on the connectivity of the spectrum and the completeness of the filtration

E2 term

• The $E_2$ term of the Adams spectral sequence the cohomology of the $E_1$ term with respect to the differential $d_1$
• Isomorphic to the topological $Ext$ functor $Ext^{s,t}_{H\mathbb{F}_p}(H\mathbb{F}_p \wedge X, H\mathbb{F}_p)$
• The $s$ grading the homological degree and the $t$ grading the internal degree

Convergence theorem

• If $X$ a connective spectrum (i.e., its homotopy groups vanish in negative degrees), then the Adams spectral sequence converges to the $p$-completion of the homotopy groups of $X$: $\{E_r^{s,t}\} \Rightarrow \pi_{t-s}(X)_p^\wedge$

Completeness of filtration

• The filtration on the homotopy groups of $X$ induced by the Adams resolution complete if $X$ is connective
• Completeness means that the filtration is Hausdorff (the intersection of all filtration stages is zero) and exhaustive (the union of all filtration stages is the entire group)

Connectivity of resolution

• The connectivity of the Adams resolution increases with the degree of the resolution
• This ensures that the $E_\infty$ term has a vanishing line of slope 1, above which all groups are zero
• The vanishing line implies that for a fixed degree $t-s$, there are only finitely many non-zero terms contributing to $\pi_{t-s}(X)$

Vanishing line

• The vanishing line a line of slope 1 in the $(s,t)$-plane above which the $E_\infty$ term of the Adams spectral sequence is zero
• Existence of the vanishing line a consequence of the increasing connectivity of the Adams resolution
• Allows for the computation of the homotopy groups of $X$ in a range of degrees using only a finite portion of the Adams spectral sequence

Computation with Adams spectral sequence

• The Adams spectral sequence a powerful tool for computing stable homotopy groups, but its use requires a good understanding of its structure and convergence properties
• Computation typically involves a combination of algebraic and geometric techniques

Massey products

• Massey products higher order cohomology operations that can detect non-trivial differentials in the Adams spectral sequence
• Defined using the cochain-level structure of the Adams resolution
• Can provide information about the differentials and extensions in the spectral sequence
• Example: the Toda bracket, a three-fold Massey product, detects the first non-trivial differential in the Adams spectral sequence for the sphere spectrum

Differential analysis

• Differentials in the Adams spectral sequence often determined by a combination of algebraic and geometric arguments
• Algebraic methods include the use of Massey products, power operations, and the module structure over the Steenrod algebra
• Geometric methods include the use of cohomology operations, the comparison to other spectral sequences (e.g., the Atiyah-Hirzebruch spectral sequence), and the analysis of the mapping cone and fiber of maps between spectra

Hidden extensions

• The Adams spectral sequence only computes the associated graded of the filtration on the homotopy groups, not the actual homotopy groups
• Extensions in the filtration can lead to "hidden" extensions in the homotopy groups that are not visible in the $E_\infty$ term
• These hidden extensions can be detected using algebraic methods (e.g., the Leibniz rule for Massey products) or geometric methods (e.g., the analysis of the mapping cone)

Comparison to other spectral sequences

• The Adams spectral sequence can be compared to other spectral sequences in stable homotopy theory, such as the Atiyah-Hirzebruch spectral sequence and the Serre spectral sequence
• These comparisons can provide additional information about the differentials and extensions in the Adams spectral sequence
• Example: the comparison to the Atiyah-Hirzebruch spectral sequence can detect differentials in the Adams spectral sequence that are not visible using algebraic methods alone

Examples of computations

• The Adams spectral sequence has been used to compute the stable homotopy groups of spheres in a large range of dimensions
• Other examples include the computation of the stable homotopy groups of Moore spaces, Eilenberg-MacLane spaces, and other spectra of interest in algebraic topology
• The computation of the Adams spectral sequence for the sphere spectrum a central problem in stable homotopy theory and has led to the development of many new techniques and ideas

Applications of Adams spectral sequence

• The Adams spectral sequence has numerous applications in algebraic and differential topology, as it provides a powerful tool for computing stable homotopy groups and studying the structure of the stable homotopy category

Stable homotopy groups of spheres

• One of the main applications of the Adams spectral sequence the computation of the stable homotopy groups of spheres $\pi_*^S(S^0)$
• The first few stable homotopy groups of spheres are well-known: $\pi_0^S(S^0) = \mathbb{Z}$, $\pi_1^S(S^0) = \mathbb{Z}/2$, $\pi_2^S(S^0) = \mathbb{Z}/2$, $\pi_3^S(S^0) = \mathbb{Z}/24$
• The Adams spectral sequence has been used to compute many more stable homotopy groups of spheres, although the computations become increasingly difficult in higher dimensions

Hopf invariant one problem

• The Hopf invariant one problem asks whether there exist maps $f: S^{2n-1} \to S^n$ with Hopf invariant one for $n > 2$
• The Adams spectral sequence can be used to show that such maps exist only for $n = 2, 4, 8$, corresponding to the existence of the complex, quaternionic, and octonionic Hopf fibrations
• The proof uses the structure of the Adams spectral sequence for the sphere spectrum and the analysis of differentials and extensions

Vector fields on spheres

• The Adams spectral sequence can be used to study the problem of determining the maximum number of linearly independent vector fields on a sphere
• The existence of a non-vanishing vector field on $S^n$ equivalent to the existence of a map $S^n \to S^{n-1}$ with Hopf invariant one
• The Adams spectral sequence can be used to show that such maps exist only in dimensions $n = 1, 3, 7$, corresponding to the existence of the complex, quaternionic, and octonionic Hopf fibrations

Immersions vs embeddings

• The Adams spectral sequence can be used to study the difference between immersions and embeddings of manifolds
• An immersion of a manifold $M^n$ into $\mathbb{R}^{n+k}$ a map $f: M^n \to \mathbb{R}^{n+k}$ that is locally an embedding, while an embedding a map that is globally injective
• The Adams spectral sequence can be used to show that every immersion of $S^n$ into $\mathbb{R}^{n+k}$ is regularly homotopic to an embedding if and only if $n < 2k+1$

Exponents in stable homotopy

• The exponent of a stable homotopy group $\pi_n^S(X)$ the smallest positive integer $e$ such that $e \cdot x = 0$ for all $x \in \pi_n^S(X)$
• The Adams spectral sequence can be used to study the exponents of stable homotopy groups, particularly for the sphere spectrum
• For example, the Adams spectral sequence can be used to show that the exponent of $\pi_{2^k-2}^S(S^0)$ is divisible by $2^k$ for all $k \geq 3$, which related to the existence of the Kervaire invariant one elements in the stable homotopy groups of spheres