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 [math]\displaystyle{ X }[/math] and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in [math]\displaystyle{ H^*(X;\mathbb{Z}/p) }[/math] using Eilenberg–MacLane spectra. This construction can be generalized using a spectrum [math]\displaystyle{ E }[/math], such as the Brown–Peterson spectrum [math]\displaystyle{ BP }[/math], or the complex cobordism spectrum [math]\displaystyle{ MU }[/math], and is used in the construction of the Adams–Novikov spectral sequence[1]pg 49.
Construction
The mod [math]\displaystyle{ p }[/math] Adams resolution [math]\displaystyle{ (X_s,g_s) }[/math] for a spectrum [math]\displaystyle{ X }[/math] 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[1]pg 43. By this, we start by considering the map
[math]\displaystyle{ \begin{matrix} X \\ \downarrow \\ K \end{matrix} }[/math]
where [math]\displaystyle{ K }[/math] is an Eilenberg–Maclane spectrum representing the generators of [math]\displaystyle{ H^*(X) }[/math], so it is of the form
[math]\displaystyle{ K = \bigwedge_{k=1}^\infty \bigwedge_{I_k} \Sigma^kH\mathbb{Z}/p }[/math]
where [math]\displaystyle{ I_k }[/math] indexes a basis of [math]\displaystyle{ H^k(X) }[/math], 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 [math]\displaystyle{ X_1 }[/math]. Note, we now set [math]\displaystyle{ X_0 = X }[/math] and [math]\displaystyle{ K_0 = K }[/math]. Then, we can form a commutative diagram
[math]\displaystyle{ \begin{matrix} X_0 & \leftarrow & X_1 \\ \downarrow & & \\ K_0 \end{matrix} }[/math]
where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram
[math]\displaystyle{ \begin{matrix} X_0 & \leftarrow & X_1 & \leftarrow & X_2 & \leftarrow \cdots \\ \downarrow & & \downarrow & & \downarrow \\ K_0 & & K_1 & & K_2 \end{matrix} }[/math]
giving the collection [math]\displaystyle{ (X_s,g_s) }[/math]. This means
[math]\displaystyle{ X_s = \text{Hofiber}(f_{s-1}:X_{s-1} \to K_{s-1}) }[/math]
is the homotopy fiber of [math]\displaystyle{ f_{s-1} }[/math] and [math]\displaystyle{ g_s:X_s \to X_{s-1} }[/math] 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 [math]\displaystyle{ \mathcal{A}_p }[/math]-resolution of the cohomology [math]\displaystyle{ H^*(X) }[/math] of a spectrum [math]\displaystyle{ X }[/math]. From the Adams resolution, there are short exact sequences
[math]\displaystyle{ 0 \leftarrow H^*(X_s) \leftarrow H^*(K_s) \leftarrow H^*(\Sigma X_{s+1}) \leftarrow 0 }[/math]
which can be strung together to form a long exact sequence
[math]\displaystyle{ 0 \leftarrow H^*(X) \leftarrow H^*(K_0) \leftarrow H^*(\Sigma K_1) \leftarrow H^*(\Sigma^2 K_2) \leftarrow \cdots }[/math]
giving a free resolution of [math]\displaystyle{ H^*(X) }[/math] as an [math]\displaystyle{ \mathcal{A}_p }[/math]-module.
E*-Adams resolution
Because there are technical difficulties with studying the cohomology ring [math]\displaystyle{ E^*(E) }[/math] in general[2]pg 280, we restrict to the case of considering the homology coalgebra [math]\displaystyle{ E_*(E) }[/math] (of co-operations). Note for the case [math]\displaystyle{ E = H\mathbb{F}_p }[/math], [math]\displaystyle{ H\mathbb{F}_{p*}(H\mathbb{F}_p) =\mathcal{A}_* }[/math] is the dual Steenrod algebra. Since [math]\displaystyle{ E_*(X) }[/math] is an [math]\displaystyle{ E_*(E) }[/math]-comodule, we can form the bigraded group
[math]\displaystyle{ \text{Ext}_{E_*(E)}(E_*(\mathbb{S}), E_*(X)) }[/math]
which contains the [math]\displaystyle{ E_2 }[/math]-page of the Adams–Novikov spectral sequence for [math]\displaystyle{ X }[/math] satisfying a list of technical conditions[1]pg 50. To get this page, we must construct the [math]\displaystyle{ E_* }[/math]-Adams resolution[1]pg 49, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form
[math]\displaystyle{ \begin{matrix} X_0 & \xleftarrow{g_0} & X_1 & \xleftarrow{g_1} & X_2 & \leftarrow \cdots \\ \downarrow & & \downarrow & & \downarrow \\ K_0 & & K_1 & & K_2 \end{matrix} }[/math]
where the vertical arrows [math]\displaystyle{ f_s: X_s \to K_s }[/math] is an [math]\displaystyle{ E_* }[/math]-Adams resolution if
- [math]\displaystyle{ X_{s+1} = \text{Hofiber}(f_s) }[/math] is the homotopy fiber of [math]\displaystyle{ f_s }[/math]
- [math]\displaystyle{ E \wedge X_s }[/math] is a retract of [math]\displaystyle{ E\wedge K_s }[/math], hence [math]\displaystyle{ E_*(f_s) }[/math] is a monomorphism. By retract, we mean there is a map [math]\displaystyle{ h_s:E \wedge K_s \to E \wedge X_s }[/math] such that [math]\displaystyle{ h_s(E\wedge f_s) = id_{E \wedge X_s} }[/math]
- [math]\displaystyle{ K_s }[/math] is a retract of [math]\displaystyle{ E \wedge K_s }[/math]
- [math]\displaystyle{ \text{Ext}^{t,u}(E_*(\mathbb{S}), E_*(K_s)) = \pi_u(K_s) }[/math] if [math]\displaystyle{ t = 0 }[/math], otherwise it is [math]\displaystyle{ 0 }[/math]
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 [math]\displaystyle{ E_* }[/math]-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 [math]\displaystyle{ E_* }[/math]-Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum [math]\displaystyle{ E }[/math] satisfying some additional hypotheses. These include [math]\displaystyle{ E_*(E) }[/math] being flat over [math]\displaystyle{ \pi_*(E) }[/math], [math]\displaystyle{ \mu_* }[/math] on [math]\displaystyle{ \pi_0 }[/math] being an isomorphism, and [math]\displaystyle{ H_r(E; A) }[/math] with [math]\displaystyle{ \mathbb{Z} \subset A \subset \mathbb{Q} }[/math] being finitely generated for which the unique ring map
[math]\displaystyle{ \theta:\mathbb{Z} \to \pi_0(E) }[/math]
extends maximally. If we set
[math]\displaystyle{ K_s = E \wedge F_s }[/math]
and let
[math]\displaystyle{ f_s: X_s \to K_s }[/math]
be the canonical map, we can set
[math]\displaystyle{ X_{s+1} = \text{Hofiber}(f_s) }[/math]
Note that [math]\displaystyle{ E }[/math] is a retract of [math]\displaystyle{ E \wedge E }[/math] from its ring spectrum structure, hence [math]\displaystyle{ E \wedge X_s }[/math] is a retract of [math]\displaystyle{ E \wedge K_s = E \wedge E \wedge X_s }[/math], and similarly, [math]\displaystyle{ K_s }[/math] is a retract of [math]\displaystyle{ E\wedge K_s }[/math]. In addition
[math]\displaystyle{ E_*(K_s) = E_*(E)\otimes_{\pi_*(E)}E_*(X_s) }[/math]
which gives the desired [math]\displaystyle{ \text{Ext} }[/math] terms from the flatness.
Relation to cobar complex
It turns out the [math]\displaystyle{ E_1 }[/math]-term of the associated Adams–Novikov spectral sequence is then cobar complex [math]\displaystyle{ C^*(E_*(X)) }[/math].
See also
- Adams spectral sequence
- Adams–Novikov spectral sequence
- Eilenberg–Maclane spectrum
- Hopf algebroid
References
- ↑ 1.0 1.1 1.2 1.3 Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. https://www.worldcat.org/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. https://www.worldcat.org/oclc/1083550.
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