Adams resolution

In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is to take a connective spectrum of finite type $X$ and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in $H^{*} (X; ℤ / p)$ using Eilenberg–MacLane spectra.

This construction can be generalized using a spectrum $E$ , such as the Brown–Peterson spectrum $B P$ , or the complex cobordism spectrum $M U$ , and is used in the construction of the Adams–Novikov spectral sequence^{pg 49}.

Construction

The mod $p$ Adams resolution $(X_{s}, g_{s})$ for a spectrum $X$ is a certain "chain-complex" of spectra induced from recursively looking at the fibers of maps into generalized Eilenberg–Maclane spectra giving generators for the cohomology of resolved spectra^{pg 43}. By this, we start by considering the map

$\begin{matrix} X \\ ↓ \\ K \end{matrix}$

where $K$ is an Eilenberg–Maclane spectrum representing the generators of $H^{*} (X)$ , so it is of the form

$K = ⋁_{k = 1}^{\infty} \underset{I_{k}}{⋁} Σ^{k} H ℤ / p$

where $I_{k}$ indexes a basis of $H^{k} (X)$ , and the map comes from the properties of Eilenberg–Maclane spectra. Then, we can take the homotopy fiber of this map (which acts as a homotopy kernel) to get a space $X_{1}$ . Note, we now set $X_{0} = X$ and $K_{0} = K$ . Then, we can form a commutative diagram

$\begin{matrix} X_{0} & \leftarrow & X_{1} \\ ↓ \\ K_{0} \end{matrix}$

where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram

$\begin{matrix} X_{0} & \leftarrow & X_{1} & \leftarrow & X_{2} & \leftarrow \dots \\ ↓ & ↓ & ↓ \\ K_{0} & K_{1} & K_{2} \end{matrix}$

giving the collection $(X_{s}, g_{s})$ . This means

$X_{s} = Hofiber (f_{s - 1} : X_{s - 1} \to K_{s - 1})$

is the homotopy fiber of $f_{s - 1}$ and $g_{s} : X_{s} \to X_{s - 1}$ comes from the universal properties of the homotopy fiber.

Resolution of cohomology of a spectrum

Now, we can use the Adams resolution to construct a free $𝒜_{p}$ -resolution of the cohomology $H^{*} (X)$ of a spectrum $X$ . From the Adams resolution, there are short exact sequences

$0 \leftarrow H^{*} (X_{s}) \leftarrow H^{*} (K_{s}) \leftarrow H^{*} (Σ X_{s + 1}) \leftarrow 0$

which can be strung together to form a long exact sequence

$0 \leftarrow H^{*} (X) \leftarrow H^{*} (K_{0}) \leftarrow H^{*} (Σ K_{1}) \leftarrow H^{*} (Σ^{2} K_{2}) \leftarrow \dots$

giving a free resolution of $H^{*} (X)$ as an $𝒜_{p}$ -module.

E_*-Adams resolution

Because there are technical difficulties with studying the cohomology ring $E^{*} (E)$ in general^{pg 280}, we restrict to the case of considering the homology coalgebra $E_{*} (E)$ (of co-operations). Note for the case $E = H 𝔽_{p}$ , $H 𝔽_{p *} (H 𝔽_{p}) = 𝒜_{*}$ is the dual Steenrod algebra. Since $E_{*} (X)$ is an $E_{*} (E)$ -comodule, we can form the bigraded group

${Ext}_{E_{*} (E)} (E_{*} (𝕊), E_{*} (X))$

which contains the $E_{2}$ -page of the Adams–Novikov spectral sequence for $X$ satisfying a list of technical conditions^{pg 50}. To get this page, we must construct the $E_{*}$ -Adams resolution^{pg 49}, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form

$\begin{matrix} X_{0} & \overset{g_{0}}{\leftarrow} & X_{1} & \overset{g_{1}}{\leftarrow} & X_{2} & \leftarrow \dots \\ ↓ & ↓ & ↓ \\ K_{0} & K_{1} & K_{2} \end{matrix}$

where the vertical arrows $f_{s} : X_{s} \to K_{s}$ is an $E_{*}$ -Adams resolution if

$X_{s + 1} = Hofiber (f_{s})$ is the homotopy fiber of $f_{s}$
$E \land X_{s}$ is a retract of $E \land K_{s}$ , hence $E_{*} (f_{s})$ is a monomorphism. By retract, we mean there is a map $h_{s} : E \land K_{s} \to E \land X_{s}$ such that $h_{s} (E \land f_{s}) = i d_{E \land X_{s}}$
$K_{s}$ is a retract of $E \land K_{s}$
${Ext}^{t, u} (E_{*} (𝕊), E_{*} (K_{s})) = π_{u} (K_{s})$ if $t = 0$ , otherwise it is $0$

Although this seems like a long laundry list of properties, they are very important in the construction of the spectral sequence. In addition, the retract properties affect the structure of construction of the $E_{*}$ -Adams resolution since we no longer need to take a wedge sum of spectra for every generator.

Construction for ring spectra

The construction of the $E_{*}$ -Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum $E$ satisfying some additional hypotheses. These include $E_{*} (E)$ being flat over $π_{*} (E)$ , $μ_{*}$ on $π_{0}$ being an isomorphism, and $H_{r} (E; A)$ with $ℤ \subset A \subset ℚ$ being finitely generated for which the unique ring map

$θ : ℤ \to π_{0} (E)$

extends maximally. If we set

$K_{s} = E \land F_{s}$

and let

$f_{s} : X_{s} \to K_{s}$

be the canonical map, we can set

$X_{s + 1} = Hofiber (f_{s})$

Note that $E$ is a retract of $E \land E$ from its ring spectrum structure, hence $E \land X_{s}$ is a retract of $E \land K_{s} = E \land E \land X_{s}$ , and similarly, $K_{s}$ is a retract of $E \land K_{s}$ . In addition

$E_{*} (K_{s}) = E_{*} (E) \otimes_{π_{*} (E)} E_{*} (X_{s})$

which gives the desired $Ext$ terms from the flatness.

Relation to cobar complex

It turns out the $E_{1}$ -term of the associated Adams–Novikov spectral sequence is then cobar complex $C^{*} (E_{*} (X))$ .

References

1 2 3 4 Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.
↑ Adams, J. Frank (John Frank) (1974). Stable homotopy and generalised homology. Chicago: University of Chicago Press. ISBN 0-226-00523-2. OCLC 1083550.

[:0-1] 1 2 3 4 Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.

[:02-2] Adams, J. Frank (John Frank) (1974). Stable homotopy and generalised homology. Chicago: University of Chicago Press. ISBN 0-226-00523-2. OCLC 1083550.