Adams resolution explained

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 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^{[1] 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 = w edge ^{inf ty} _{k =1} w edge _{I _{k}} \Sigma ^{kHZ/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 … \\ \downarrow & & \downarrow & & \downarrow \\ 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

l{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 …

giving a free resolution of H ^{*(X)} as an l{A} _{p} -module.

E_*-Adams resolution

Because there are technical difficulties with studying the cohomology ring

E ^{*(E)} in general^{[2] 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)} = l{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} & \xleftarrow{g _{0}} & X _{1} & \xleftarrow{g _{1}} & X _{2} & \leftarrow … \\ \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 ifX _{s+1} = 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} Ext ^{t,u} (E _{*(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 Z \subset A \subset Q being finitely generated for which the unique ring map

\theta: 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} = 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) ⊗} ^{\pi _{*(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))} .

See also

• Adams spectral sequence
• Adams–Novikov spectral sequence
• Eilenberg–Maclane spectrum
• Hopf algebroid

References

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