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 $H^*(X;\mathbb{Z}/p)$ using Eilenberg–MacLane spectra.

This construction can be generalized using a spectrum $E$ , such as the Brown–Peterson spectrum $BP$ , or the complex cobordism spectrum $MU$ , 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{{Cite book|last=Ravenel, Douglas C.|url=https://www.worldcat.org/oclc/316566772|title=Complex cobordism and stable homotopy groups of spheres|date=1986|publisher=Academic Press|isbn=978-0-08-087440-1|location=Orlando|oclc=316566772}}^{pg 43}. By this, we start by considering the map

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

where

K

is an Eilenberg–Maclane spectrum representing the generators of

H^*(X)

, so it is of the form

$K = \bigvee_{k=1}^\infty \bigvee_{I_k} \Sigma^kH\mathbb{Z}/p$

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

$\begin{matrix}
X_0 & \leftarrow & X_1 \\
\downarrow & & \\
K_0
\end{matrix}$

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

$\begin{matrix}
X_0 & \leftarrow & X_1 & \leftarrow & X_2 & \leftarrow \cdots \\
\downarrow & & \downarrow & & \downarrow \\
K_0 & & K_1 & & K_2
\end{matrix}$

giving the collection

(X_s,g_s)

. This means

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

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 $\mathcal{A}_p$ -resolution of the cohomology $H^*(X)$ of a spectrum $X$ . From the Adams resolution, there are short exact sequences

$0 \leftarrow H^*(X_s) \leftarrow H^*(K_s) \leftarrow H^*(\Sigma X_{s+1}) \leftarrow 0$

which can be strung together to form a long exact sequence

$0 \leftarrow H^*(X) \leftarrow H^*(K_0) \leftarrow H^*(\Sigma K_1)
\leftarrow H^*(\Sigma^2 K_2) \leftarrow \cdots$

giving a free resolution of

H^*(X)

as an

\mathcal{A}_p

-module.

''E''<sub>*</sub>-Adams resolution

Because there are technical difficulties with studying the cohomology ring $E^*(E)$ in general{{Cite book|last=Adams, J. Frank (John Frank)|url=https://www.worldcat.org/oclc/1083550|title=Stable homotopy and generalised homology|date=1974|publisher=University of Chicago Press|isbn=0-226-00523-2|location=Chicago|oclc=1083550}}^{pg 280}, we restrict to the case of considering the homology coalgebra $E_*(E)$ (of co-operations). Note for the case $E = H\mathbb{F}_p$ , $H\mathbb{F}_{p*}(H\mathbb{F}_p) =\mathcal{A}_*$ is the dual Steenrod algebra. Since $E_*(X)$ is an $E_*(E)$ -comodule, we can form the bigraded group

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

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

$\begin{matrix}
X_0 & \xleftarrow{g_0} & X_1 & \xleftarrow{g_1} & X_2 & \leftarrow \cdots \\
\downarrow & & \downarrow & & \downarrow \\
K_0 & & K_1 & & K_2
\end{matrix}$

where the vertical arrows

f_s: X_s \to K_s

is an

E_*

-Adams resolution if

$X_{s+1} = \text{Hofiber}(f_s)$ is the homotopy fiber of $f_s$
$E \wedge X_s$ is a retract of $E\wedge K_s$ , hence $E_*(f_s)$ is a monomorphism. By retract, we mean there is a map $h_s:E \wedge K_s \to E \wedge X_s$ such that $h_s(E\wedge f_s) = id_{E \wedge X_s}$
$K_s$ is a retract of $E \wedge K_s$
$\text{Ext}^{t,u}(E_*(\mathbb{S}), E_*(K_s)) = \pi_u(K_s)$ 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 $\pi_*(E)$ , $\mu_*$ on $\pi_0$ being an isomorphism, and $H_r(E; A)$ with $\mathbb{Z} \subset A \subset \mathbb{Q}$ being finitely generated for which the unique ring map

$\theta:\mathbb{Z} \to \pi_0(E)$

extends maximally.

If we set

$K_s = E \wedge F_s$

and let

$f_s: X_s \to K_s$

be the canonical map, we can set

$X_{s+1} = \text{Hofiber}(f_s)$

Note that

E

is a retract of

E \wedge E

from its ring spectrum structure, hence

E \wedge X_s

is a retract of

E \wedge K_s = E \wedge E \wedge X_s

, and similarly,

K_s

is a retract of

E\wedge K_s

. In addition

$E_*(K_s) = E_*(E)\otimes_{\pi_*(E)}E_*(X_s)$

which gives the desired

\text{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

Category:Algebraic topology

Category:Homological algebra

Category:Topology

Adams resolution

Construction

= Resolution of cohomology of a spectrum =

''E''<sub>*</sub>-Adams resolution

= Construction for ring spectra =

== Relation to cobar complex ==

See also

References