Spaces of multiplicative maps between highly structured ring spectra

A.Lazarev

Department of Mathematics, Univ. of Bristol, Bristol, BS8 1TW, UK. email A.Lazarev@bristol.ac.uk

Abstract

We uncover a somewhat unexpected connection between spaces of multiplicative maps between A_{∞} ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping only once. We also prove that any multiplicative cohomology operation in complex cobordisms theory MU canonically lifts to an A_{∞} map MU → MU. This implies, in particular, that the Brown-Peterson spectrum BP splits off MU as an A_{∞}-ring spectrum.

Key words: Topological Hochschild cohomology,derivations, S-algebras, complex cobordisms.

1 Introduction

The main purpose of the present work is to provide a workable method for computing the homotopy type of spaces of A_{∞}-maps between A_{∞} ring spectra (or S-algebras in the terminology of [8]). We make substantial use of the previous results by the author in [10] and for reader’s convenience a brief summary of these is given in Section 2. In Section 3 we collect miscellaneous technical results concerning function spectra and topological Hochschild cohomology. Some of these results are surely known to experts but never have been written down. The formula (3) (base change) deserves special mention. While easy to prove it is extremely convenient when computing with various spectral sequences.

Our first main result (Theorem 4.3) essentially states that the mapping space between two S-algebras A and B is determined after taking based loops by the spectrum of topological derivations Der(A, B). Therefore the problem of computing higher homotopy groups of this mapping space is a problem of stable homotopy which turns out to be quite amenable, particularly because in many cases Der(A, B) can be reduced to THH(A, B), the topological Hochschild cohomology of A with values in B.

The computation of the zeroth homotopy group of the mapping space is, of course, a completely different story. We give a simple answer in the special case when A is a connective S-algebra while B is coconnective (that is, with vanishing homotopy in positive dimensions). This is Theorem 4.8.

Even though the problem of computing homotopy classes of S-algebra maps A → B is essentially unstable it does lend itself to analysis by methods of obstruction theory developed in [10]. Our second main result (Theorem 5.4) demonstrates that any multiplicative cohomology operation in complex cobordism theory MU canonically (even uniquely in an appropriate sense) lifts to an S-algebra self-map of MU. This is used to show that for an S-algebra E belonging to a fairly large class of complex-oriented theories any multiplicative operation MU → E lifts to an S-algebra map. Another corollary is that the Brown-Peterson spectrum BP splits off MU localized at p as an S-algebra.

The paper is written in the language of S-modules of [8] and we routinely use the results and terminology of the cited reference.

Notations. In Sections 2 and 3 we work in the category of modules or algebras over a fixed q-cofibrant commutative S-algebra R, the smash product ∧ is always understood as ∧_{R}. In Section 4 we specialize R = S. The free R-algebra on an R-module M is denoted by T (M). The space of maps between two R-algebras A and B is denoted by F_{R−alg}(A, B). We use the symbol [A, B]_{a} ([A, B]_{c}) for the set of homotopy classes of associative Ralgebra maps between A and B (the set of homotopy classes of commutative S-algebra maps respectively). Finally we denote by Mult(E, F) the set of multiplicative (up to homotopy) maps between ring spectra E and F. To distinguish between strict isomorphisms and weak equivalences we will use, as a rule, the symbol ′′ ≅′′ for the former and ′′ ≃′′ for the latter.

2 Topological derivations and topological singular extension of S-algebras

In this short section we give an overview of some of the author’s results from [10] which will be needed later on.

Let A be a q-cofibrant R-algebra and M an A-bimodule. Then the R-module A ∨ M has the obvious structure of an R-algebra (“square-zero extension” of A). Consider the set [A, A ∨ M]_{a/A} of homotopy classes of R-algebra maps from A to A ∨ M which commute with the projection onto A.

Theorem 2.1 There exists an A-bimodule Ω_{A} and a natural in M isomorphism

[A, A ∨ M]_{a/A} ≅ [Ω_{A}, M]_{A−bimod}

where the right hand side denotes the homotopy classes of maps in the category of A-bimodules.

Definition 2.2 The topological derivations R-module of A with values in M is the function R-module F_{A∧A^{op}}(Ω_{A}, M). We denote it by Der(A, M) and its ith homotopy group by Der^{−i}(A, M).

The A-bimodule Ω_{A} is constructed as the homotopy fibre of the multiplication map A ∧ A → A. There exists the following homotopy fibre sequence of R-modules:

THH(A, M) → M → Der(A, M)  (1)

Here THH(A, M) is the topological Hochschild cohomology spectrum of A with values in M, THH(A, M) := F_{A∧A^{op}}(A, A).

If A is a commutative R-algebra and the left and right A-module structures on M agree then the sequence (1) splits giving a canonical weak equivalence THH(A, M) ≃ Σ^{−1}Der(A, M) ∨ M.

Suppose we are given a topological derivation d : A → A ∨ M. Consider the following homotopy pullback diagram

X →  A
↓  ↓
A →^{d} A ∨ M

Here the rightmost downward arrow is the canonical inclusion of a retract. Then we have the following homotopy fibre sequence of R-modules:

Σ^{−1}M → X → A  (2)

Definition 2.3 The homotopy fibre sequence (2) is called the topological singular extension associated with the derivation d : A → A ∨ M.

Theorem 2.4 Let Σ^{−1}M → X → A be a singular extension of R-algebras associated with a derivation d : A → A ∨ M and f : B → A a map of R-algebras. Then f lifts to an R-algebra map B → X iff a certain element in Der^{0}(B, M) is zero. Assuming that a lifting exists the map

F_{R−alg}(B, X) → F_{R−alg}(B, A)

has homotopy fibre over the point f ∈ F_{R−alg}(B, A) weakly equivalent to Ω^{∞}Der(B, Σ^{−1}M) (the 0th space of the spectrum Der(X, Σ^{−1}M)).

Theorem 2.5 Assume that R is connective and A is a connective R-algebra. Then the Postnikov tower of A

A_{0} = Hπ_{0}A ← A_{1} ← … ← A_{n} ← A_{n+1} ← …

is one of R-algebras. Moreover the homotopy fibre sequences

A_{n} ← A_{n+1} ← Σ^{n+1}Hπ_{n+1}A

are topological singular extension.

3 Base change and topological derivations of supplemented R-algebras

In this section we discuss topological Hochschild cohomology and topological derivatons of supplemented R-algebras and the behaviour of the forgetful map l in the hypercohomology spectral sequence. This material is largely parallel to [10], section 9 and so most of the proofs will be omitted.

We’ll start with some general lemmas.

Lemma 3.1 Let A, B, C be R-algebras and M, N, L be an A ∧ B-module, an C ∧A^{op}-module and a B ∧C-module respectively. Then there is a natural isomorphism of R-modules:

F_{A∧B}(M, F_{C}(N, L)) ≅ F_{C∧B}(N ∧_{A} M, L).

Proof It suffices to check the above equivalence for M = A ∧ B ∧ M̃ and N = C ∧ A^{op} ∧ Ñ. We have:

F_{A∧B}(M, F_{C}(N, L)) ≅ F_{A∧B}(A ∧ B ∧ M̃, F_{C}(C ∧ A^{op} ∧ Ñ, L)

≅ F (M̃, F (A^{op} ∧ Ñ, L) ≅ F (M̃ ∧ A^{op} ∧ Ñ, L).

Likewise,

F_{C∧B}(N ∧_{A}M, L) ≅ F_{C∧B}(C∧A^{op}∧Ñ ∧_{A}A∧B∧M̃, L) ≅ F (A^{op}∧Ñ ∧M̃, L)

and we are done.

Corollary 3.2 Setting in the above lemma N = C, we get

F_{C∧B}(C ∧_{A} M, L) ≅ F_{A∧B}(M, F_{C}(C, L)) ≅ F_{A∧B}(M, L)  (3)

We will refer to the isomorphism (3) as the base change. Related formulae are found in [8], 3.6.

Corollary 3.3 If an A-bimodule B is also an R-algebra, then T HH(A, B) ≅ F_{B∧A^{op}}(B, B). In particular, T HH(A, B) is an R-algebra under the composition product. Indeed, we have the following isomorphisms of R-modules:

T HH(A, B) ≅ F_{A∧A^{op}}(A, B) ≅ F_{A∧A^{op}}(A, F_{B}(B, B))
≅ F_{B∧A^{op}}(B ∧_{A} A, B) ≅ F_{B∧A^{op}}(B, B)

We now discuss topological Hochschild cohomology of supplemented Ralgebras. Let A be a q-cofibrant R-algebra. We say that A is supplemented if it is supplied with an R-algebra morphism ϵ : A → B. Denote by Ω_{A}^{B} the homotopy fibre of the map B ∧ A → B that determines the structure of a right A-module structure on B. We will assume without loss of generality that Ω_{A}^{B} is a q-cofibrant right A-module. Recall that the module of differentials Ω_{A} for A is defined from the homotopy fibre sequence

Ω_{A} → A ∧ A → A

where the second arrow is the multiplication map. Smashing this fibre sequence on the left with B over A we get the fibre sequence

B ∧_{A} Ω_{A} → A → B

That shows that Ω_{A}^{B} is weakly equivalent to B ∧_{A} Ω_{A} as an R-module. Further, base change gives a weak equivalence

F_{B∧A^{op}}(Ω_{A}^{B}, B) ≃ F_{B∧A^{op}}(B ∧_{A} Ω_{A}, B) ≃ F_{A∧A^{op}}(Ω_{A}, B) ≅ Der(A, B).

Recall from [10] that there is a “universal derivation” d : A → Ω_{A} which is defined as the composite map

A → A ∨ Ω_{A} → Ω_{A}

where the first map is the map of algebras over A adjoint to the identity map Ω_{A} → Ω_{A} and the second map is the projection. The universal derivation allows one to define the forgetful map l : Der(A, B) → F (A, B) as the composite map

Der(A, B) ≃ F_{A∧A^{op}}(Ω_{A}, B) → F (Ω_{A}, B) → F (A, B)

where the last map is induced by d. In terms of Ω_{A}^{B} the forgetful map l admits the following description. The cofibre sequence Ω_{A}^{B} → B ∧ A → B splits so there is a weak equivalence of R-modules A ≃ B ∨ Ω_{A}^{B}. Denote by \overline{d} : A → Ω_{A}^{B} the projection on the wedge summand. Then d coincides with the composite map:

Der(A, B) = F_{B∧A^{op}}(Ω_{A}^{B}, B) → F (Ω_{A}^{B}, B) → F (A, A)

the first arrow being the forgetful map and the second one is induced by \overline{d}.

Next we discuss the behaviour of the forgetful map l in the hypercohomology spectral sequence. To do this we need to review algebraic Hochschild cohomology for supplemented algebras. The exposition will be somewhat sketchy since it is parallel to [10], section 9.

Definition 3.4 Let A_{∗} be a graded algebra over a graded commutative algebra supplied with an R_{∗}-algebra map ϵ : A_{∗} → B_{∗} (supplementation). Then algebraic Hochschild cohomology of A_{∗} with coefficients in B_{∗} is defined as

HH_{R_{∗}}^{∗}(A_{∗}, B_{∗}) = Ext_{B_{∗}⊗_{R∗}^{L}A_{∗}^{op}}(B_{∗}, B_{∗})

where ⊗_{R_{∗}}^{L} denotes the derived tensor product

Remark 3.5 If A_{∗} is flat as an R_{∗}-module, then this definition is equivalent to the standard one found in, e.g. [6]

We also have a generalization of the standard complex which computes Hochschild cohomology. Let Ã_{∗} be a differential graded augmented R_{∗}- algebra which is quasiisomorphic to A_{∗} and R_{∗}-projective. Denote ϵ̃: A_{∗} → B_{∗} its supplementation. Consider the bar-resolution of the right Ã_{∗}-module B_{∗} ; here and later on ⊗ stands for ⊗_{R_{∗}}

B_{∗}←B_{∗} ⊗ Ã_{∗} ← B_{∗} ⊗ Ã_{∗} ⊗ Ã_{∗} ← …  (4)

with the usual bar differential

d(b⊗ã_{1}⊗…⊗ã_{n}) = bϵ̃(ã_{1})⊗ã_{2} …⊗ã_{n}+Σ(−1)^{i}b⊗ã_{1}⊗…⊗ã_{i}ã_{i+1}⊗…⊗ã_{n}.

This is actually a bicomplex since Ã is a differential graded algebra. Applying the functor HomÃ_{∗}^{op}(?, B_{∗}) to (4) we get the standard Hochschild cohomology (bi)complex

C^{ij}(A_{∗}, B_{∗}) = Hom^{i}(Ã_{∗}^{⊗j}, B_{∗}).

Now define the module of differentials Ω_{A_{∗}}^{B_{∗}} from the following short exact sequence

0 → Ω_{A_{∗}}^{B_{∗}} → B_{∗} ⊗ Ã_{∗}→B_{∗} → 0

Clearly Ω_{A_{∗}}^{B_{∗}} is quasiisomorphic as a right Ã_{∗}-module to the truncated barresolution:

B_{∗} ⊗ Ã_{∗}^{⊗2} ← B_{∗} ⊗ Ã_{∗}^{⊗3} ← …  (5)

The universal derivation d : Ã_{∗} → Ω_{A_{∗}}^{B∗} is induced by the map Ã_{∗} → B_{∗}⊗Ã_{∗}, ã → 1 ⊗ ϵ̃(ã) − ã ⊗ 1. If we take the complex (5) as a model for Ω_{A_{∗}}^{B_{∗}} then the universal derivation d is a map of complexes

d : Ã → {B_{∗} ⊗ Ã_{∗}^{⊗2} → B_{∗} ⊗ Ã_{∗}^{⊗3} → …}

where Ã is considered to be a complex concentrated in degree 0 and d(ã) = −1 ⊗ ã ⊗ 1.

Further define algebraic derivations of A_{∗} with coefficients in B_{∗} as

Der_{R_{∗}}^{∗}(A_{∗}, B_{∗}) = Ext_{B_{∗}⊗A_{∗}^{op}}^{∗}(Ω_{A_{∗}}^{B_{∗}}, B_{∗}).

Then the (truncated) standard resolution (5) provides a (bi)complex for computing Der^{∗}(A_{∗}, R_{∗}):

\overline{C}^{∗}(A_{∗}, B_{∗}) : Hom(Ã_{∗}, B_{∗}) → Hom(Ã_{∗}^{⊗2}, B_{∗}) → ….

(This is indeed a bicomplex, the additional differential being induced from the internal differential in Ã_{∗}). As in the topological case the universal derivation determines the forgetful map

l_{alg} : Der_{R_{∗}}^{∗}(A_{∗}, B_{∗}) = Ext_{B_{∗}⊗Ã_{∗}^{op}}^{∗}(Ω_{A_{∗}/R_{∗}}, B_{∗}) → Hom^{∗}(Ã_{∗}, B_{∗}).

Then we have the obvious Proposition 3.6 The forgetful map

Der_{R_{∗}}^{∗}(A_{∗}, B_{∗}) → Hom^{∗}(Ã_{∗}, B_{∗}) = Ext_{R_{∗}}^{∗}(A_{∗}, B_{∗})

is induced by the projection \overline{C}^{∗}(A_{∗}, B_{∗}) → Hom^{∗}(Ã_{∗}, B_{∗}) times (−1)

Returning to our topological situation we have the following result which is analogous to Proposition 9.3 in [10]:

Proposition 3.7 Let A, B be R-algebras, A → B is an R-algebra map. Suppose that the Kunneth spectral sequence for π_{∗}(B ∧ A^{op}) collapses multiplicatively so we have a ring isomorphism

π_{∗}(B ∧ A^{op}) = T or_{∗}^{R_{∗}}(B_{∗}, A_{∗}^{op})

Then there are the following spectral sequences

1E_{2}^{∗∗} = Der_{R_{∗}}^{∗}(A_{∗}, B_{∗}) = Ext_{T or_{∗}^{R∗}(B_{∗},A_{∗}^{op})}^{∗}(Ω_{A_{∗}}^{B_{∗}}, R_{∗}) ⇒ Der^{∗}(A, B);
2E_{2}^{∗∗} = Ext_{R_{∗}}^{∗}(A_{∗}, B_{∗}) ⇒ [A, B]^{∗}.

Furhermore, the forgetful map l : Der^{∗}(A, B) → [A, B]^{∗} induces a map of spectral sequences ^{1}E_{∗}^{∗∗} → ^{2}E_{∗}^{∗∗} which on the level of E_{2}-terms gives the forgetful map l_{alg} := Der_{R_{∗}}^{∗}(A_{∗}, B_{∗}) → Ext_{R_{∗}}^{∗}(A_{∗}, B_{∗}).

4 Mappings spaces via derivations

In this section we show that for two R-algebras A and B the higher homotopy groups of the space R − alg(A, B) can be reduced to the computation of certain topological derivations. This is important because in many cases topological derivations can be further reduced to topological Hochschild cohomology which is an essentially stable object, so that one could apply standard methods of homological algebra for computation.

Now consider the R-module A ∨ Σ^{−d}A, d > 0. It can be supplied canonically with the structure of an R-algebra over A so that Σ^{−d}A is a “squarezero ideal”. Denote this R-algebra by A_{d}.

Let us also introduce the algebra A(d) := A^{S^{d}}, the cotensor of A and the d-sphere S^{d}. Then A(d) ≅ F (RΣ^{∞}S_{+}^{d}, A) as an R-module (here R stands for the free R-module functor. The structure of an R-algebra on A(d) is induced by the R-algebra structure on A and the topological diagonal S^{d} → S^{d} ×S^{d}. The coefficient rings of A(d) and A_{d} are both isomorphic to the exterior algebra Λ_{A_{∗}}(y) where y has degree −d. There is also a weak equivalence of R-modules:

A(d) ≃ A ∨ Σ^{−d}A ≅ A_{d}.

Notice that both A(d) and A_{d} are R-algebras over A, that is there exist maps of R-algebras A(d) → A and A_{d} → A. (The first map is induced by choosing a base point in S^{d}, the second map is the canonical projection).

Theorem 4.1 The R-algebras A(d) and A_{d} are weakly equivalent in the category of R-algebras.

Proof. First consider the case A = R. Since R is an R(d)-module it makes sense to consider self-maps of R in the category of R(d)-modules. Notice that R(d) is actually a commutative R-algebra so we need not distinguish between left and right R(d)-modules..

Lemma 4.2

π_{∗}F_{R(d)}(R, R) = R_{∗}[[x]]

where the element x has degree d − 1.

Proof. Assume first that R = S, the sphere spectrum. We need this special case because the connectiveness of S will be used. If R is connective this step could be skipped. Consider the spectral sequence

E_{2}^{∗∗} = Ext_{S(d)_{∗}}^{∗∗}(S_{∗}, S_{∗}) = Ext_{Λ_{S∗}(y)}^{∗∗}(S_{∗}, S_{∗}) = S_{∗}[[x]] ⇒ π_{∗}F_{S(d)}(S, S).

Here the element x has degree d − 1 This spectral sequence collapses for dimensional reasons. By Boardman’s criterion [3] it converges strongly to its target which is complete with respect to the (cobar) filtration. Since this filtration coincides with the x-adic filtration on the associated graded S_{∗}-module we conclude that

π_{∗}F_{S(d)}(S, S) = S_{∗}[[x]].

Notice that the fact that S is connective was used to show the collapse of our spectral sequence. For instance if d = 1 the elements x^{k} are located along the diagonal and the whole spectral sequence E_{2}^{∗∗} lies above it.

Now let R be an arbitrary commutative S-algebra. Take the spectral sequence

′E_{2}^{∗∗} = Ext_{R(d)_{∗}}^{∗∗}(R_{∗}, R_{∗}) = Ext_{Λ_{R∗}(y)}^{∗∗}(R_{∗}, R_{∗}) = R_{∗}[[x]] ⇒ π_{∗}F_{R(d)}(R, R) and notice that the unit map S → R determines the map of spectral sequences E_{2}^{∗∗} → ′E_{2}^{∗∗} taking x to x. It follows that ′E_{2}^{∗∗} collapses proving our claim.

Let us now return to the proof of the theorem; recall that we are still handling the special case A = R. Consider the set of maps R(d) → R_{d} in the homotopy category of R-algebras over R. By Theorem 2.1 this set is an abelian group of topological derivations of R(d) with values in Σ^{−d}R. Since R(d) is a commutative R-algebra there is a canonical splitting T HH^{∗}(R(d), R) ≃ R ∨ Der^{∗−1}(R(d), R). So the computation of Der^{∗}(R(d), R) reduces to the computation of T HH^{∗}(R(d), R). Further Corollary 3.3 provides an isomorphism

T HH^{∗}(R(d), R) ≅ F_{R(d)}(R, R).

It follows from Lemma 4.2 that the spectral sequence

E_{2}^{ij} = Der_{R_{∗}}(R(d)_{∗}, R_{∗}) = Ext_{R_{∗}}^{∗}(Ω_{R(d)_{∗}}^{R_{∗}}, R_{∗}) ⇒ Der^{∗}(R(d), R)

collapses and

Der^{∗−1}(R(d), R) = R_{∗}[[x]]/R_{∗}

Further,

[R(d), R]^{∗} = R_{∗} ⊕ Σ^{d}R_{∗} = Λ_{R_{∗}}(z)

where the symbol z has degree d (of course, we do not claim the existence of any multiplicative structure). The element z maps the wedge summand Σ^{−d}R of the R-module R(d) isomorphically to Σ^{−d}R, and the other wedge summand maps to zero. It follows from Proposition 3.7 that the image of x ∈ Der^{∗}(R(d), R) in [R(d), R]^{∗} under the forgetful map is z (up to an invertible factor).

In other words we proved that there exists a topological derivation of R(d) with values in R, that is a map in the homotopy category of R-algebras over R

R(d) → R_{d}  (6)

such that the wedge summand Σ^{−d}R of R(d) maps isomorphically onto the corresponding wedge summand of R_{d}. Therefore the map (6) is a weak equivalence of R-algebras and our theorem is proved (in the special case A = R). To get the general case consider the canonical map

R(d) ∧ A = F (RΣ^{∞}S_{+}^{d}, R) ∧ A → F (RΣ^{∞}S_{+}^{d}, A) = A^{d}  (7)

Since A is a q-cofibrant R-algebra the point-set level smash product F (RΣ^{∞}S_{+}^{d}, R) ∧ A represents the derived smash product. Further (7) is a weak equivalence since RΣ^{∞}S_{+}^{d} is a finite cell R-module and diagram chase shows that this is an R-algebra map. So we have the following equivalences of R-algebras:

A_{d} ≅ R_{d} ∧ A ≃ R(d) ∧ A ≃ A(d)

With this Theorem 4.1 is proved.

Now suppose that we have another R-algebra B and a map f : A → B of R-algebras. Then the pair (F_{R−alg}(A, B), f) is a pointed topological space. This space turns out to be closely related to Der(A, B). We have the following theorem:

Theorem 4.3 For a q-cofibrant algebra A and a map of R-algebras f : A → B the space of d-loops Ω^{d}(F_{R−alg}(A, B), f) is weakly equivalent to the space Ω^{∞}Der(A, Σ^{−d}B).

Proof. For two topological spaces X and Y we will denote the space of maps between them by T (X, Y) (T_{∗}(X, Y) in the pointed case). Then we have the following commutative diagram of spaces where both rows are homotopy fibre sequences:

T_{∗}(S^{d}, F_{R−alg}(A, B)) → T (S^{d}, F_{R−alg}(A, B)) → F_{R−alg}(A, B)
↓  ↓  ||
?  →  F_{R−alg}(A, B^{S^{d}})  → F_{R−alg}(A, B)

Here the horizontal rightmost arrows are both induced by the inclusion of the base point into S^{d}. Since the right and the middle vertical arrows are weak equivalences (even isomorphisms) it follows that the map T_{∗}(S^{d}, Map(A, B)) →? is a weak equivalence. But Theorem 4.1 tells us that the R-algebra B^{S^{d}} is weakly equivalent as an R-algebra to B ∨ Σ^{−d}B. In other words the term ? is weakly equivalent to the topological space of maps A → B ∨ Σ^{−d}B which commute with the projection onto A. Therefore ? is weakly equivalent to Ω^{∞}Der(A, Σ^{−d}B) and our theorem is proved.

Corollary 4.4 For a q-cofibrant algebra A and a map of R-algebras f : A → B the group π_{d}(F_{R−alg}A, B), f) for d > 0 is isomorphic to Der^{−d}(A, B)

Remark 4.5 One might wonder whether Theorem 4.3 remains true in the context of commutative S-algebras. The answer is no. The crucial point is the weak equivalence of S-algebras S ∨ S^{−1} and S^{S^{1}}. It is clear that π_{0}S ∧_{S∨S^{−1}} S is the divided power ring. However N.Kuhn and M.Mandell proved that π_{0}S∧_{S^{S^{1}}} S is the ring of numeric polynomials. Therefore S∨S^{−1} and S^{S^{1}} cannot be weakly equivalent as commutative S-algebras.

We see, that the space F_{R−alg}(A, B) when looped only once becomes an infinite loop space. This is somewhat surprising since F_{R−alg}(A, B) is hardly ever an infinite loop space itself. In particular the set of connected components of F_{R−alg}(A, B) does not have to be a group, let alone an abelian group. Therefore the connection between π_{0}F_{R−alg}(A, B) and Der^{0}(A, B) may be rather weak. For instance the set of homotopy classes of A_{∞} self-maps of the p-completed K-theory spectrum is the multiplicative group of p-adic integers whereas the corresponding topological derivations spectrum can be proved to be contractible. A generalization of this example is discussed in author’s work [11]. However there is some evidence for the following

Conjecture 4.6 For an R-algebra map f : A → B the connected component of f in F_{R−alg}(A, B) is weakly equivalent to the connected component of Ω^{∞}Der(A, B). In particular it is an infinite loop space.

To see why this conjecture has a chance of being true notice that the Whitehead products in π_{i}(F_{R−alg}(A, B)) determine via Theorem 4.3 various brackets in Der^{i}(A, B) and, for commutative A and B - also in T HH(A, B). No such brackets have been recorded so far and it seems likely that they should all vanish. This suggests that the connected component of f in F_{R−alg}(A, B) is an H-space.

There is another interesting question raised by Theorem 4.3. In recent work [12] J.McClure and J.Smith introduced the Gerstenhaber bracket on THH(A, A). Their work probably implies the existence of the bracket on Der(A, A). This is surely the case if A is commutative since then Der(A, A) splits off THH(A, A) as a wedge summand. Then via Theorem 4.3 a bracket is defined on π_{i}F_{R−alg}(A, A) for i > 0.

Conjecture 4.7 The bracket described above agrees with the Whitehead product on BF_{R−alg}(A, A), the classifying space of the monoid F_{R−alg}(A, A).

We see that the problem of computing π_{0}F_{R−alg}(A, B) differs sharply from computing higher homotopy groups. This problem is usually much harder, being essentially nonabelian. However there is one case when it is possible to give a complete general answer.

Theorem 4.8 Assuming that R is connective let A be a connective Ralgebra, and B a coconnective S-algebra (i.e. π_{i}B = 0 for i > 0). Then any π_{0}R-algebra map π_{0}A → π_{0}B lifts to a unique R-algebra map A → B so that the forgetful map [A, B]_{a} → Hom_{π_{0}R−alg}(π_{0}A, π_{0}B) is bijective.

Similarly if A is a commutative R-algebra and B is a commutative coconnective S-algebra then the forgetful map [A, B]_{c} → Hom_{π_{0}R−alg}(π_{0}A, π_{0}B) is bijective.

Proof. We will deal only with the associative case, the commutative one being completely analogous. Picking a system of generators and relations for the π_{0}R-algebra π_{0}A we construct the following pushout diagram in the category of R-algebras:

\begin{aligned}T (∐_{J} R) & → & R \\ ↓ & & ↓ \\ T (∐_{I} R) & → & A^{0} \end{aligned}  (8)

Here the sets I and J run respectively through the systems of generators and relations in π_{0}A. There is a canonical R-algebra map from A^{0} to A that induces an isomorphism on zeroth homotopy group. The R-algebra A^{0} is the zeroth skeleton of A in the category of R-algebras and (the CW - approximation of) A is obtained from A^{0} by attaching R-algebra cells in higher dimensions. Then induction up the CW -filtration of A shows that the the map A^{0} → A induces the isomorphism [A, B]_{a} ≅ [A^{0}, B].

Further applying the functor F_{R−alg}(?, B) to the diagram (8) we get the following homotopy pullback of topological spaces:

∏_{J} π_{0}B ←  pt
↑  ↑
∏_{I} π_{0}B ← F_{π_{0}R−alg}(A^{0}, B)

It follows that the space F_{π_{0}R−alg}(A^{0}, B) is homotopically discrete with the set of connected components Hom_{π_{0}R−alg}(π_{0}A, π_{0}B). With this Theorem 4.8 is proved.

The combination of Theorems 4.3 and 4.8 allows one to calculate homotopy groups of mapping spaces for a large number of S-algebras. For example by Bokstedt’s theorem (cf. [4]) π_{2i+1}T HH(HZ, HZ) = 0 whilst π_{2i}T HH(HZ, HZ) = Z/iZ. This gives

Corollary 4.9 The space F_{S−alg}(HZ, HZ) is connected. Furthermore π_{2i}F_{S−alg}(HZ, HZ) = 0 and π_{2i−1}F_{S−alg}(HZ, HZ) = Z/(i + 1).

5 Spaces of multiplicative self-maps of MU

In this section we study the homotopy groups of A_{∞}-maps from the complex cobordism spectum MU into itself. Our main result here is that any homotopy multiplicative operation MU → MU lifts canonically to an S-algebra map. We also calculate completely higher homotopy groups of S-algebra maps out of MU into an arbitrary MU-algebra E.

Before we state our main theorem we need to introduce the notion of Q-commutative S-algebras and Q-preferred S-algebra maps.

Definition 5.1 Let A be an S-algebra. We say that A is Q-commutative if A_{Q}, the rationalization of A is weakly equivalent to a commutative S-algebra.

Remark 5.2 Later on all Q-commutative S-algebras which we encounter will in fact be commutative. Notice, however, that it is not always the case. Denote by S[x_{i}] the free S-algebra on the S-module S_{S}^{2i}, the cell approximation of the 2i-dimensional sphere. Then clearly S_{Q}[x_{i}] is weakly equivalent to the free commutative S-algebra on S_{Q}^{2i}. Therefore S[x_{i}] is a Q-commutative S-algebra which is not commutative unless i = 0.

Consider two Q-commutative S-algebras A_{Q} and B_{Q}. We have the following maps:

k : [A_{Q}, B_{Q}]_{c} → [A_{Q}, B_{Q}]_{a} ← [A, B]_{a} : q.

Here k is the forgetful map and q is induced by rationalization.

Definition 5.3 A map f ∈ [A, B]_{a} is called Q-preferred if q(f) ∈ [A_{Q}, B_{Q}]_{a} is in the image of k. For an A-bimodule M an S-algebra derivation d : A → A ∨ M is called Q-preferred if d is Q-preferred as an S-algebra map.

In other words a map of S-algebras (or a topological derivation) is Qpreferred if it lifts to a map (to a derivation) of commutative S-algebras after rationalization.

Theorem 5.4 The forgetful map of monoids

[MU, MU]_{a} → Mult(MU, MU)

admits a unique section whose image consists of Q-preferred S-algebra maps.

The proof will be given below after a succession of lemmas.

Remark 5.5 The set Mult(MU, MU) is relatively well understood. One can describe it for example as the set of all MU_{∗}-algebra maps

MU_{∗}MU = MU_{∗}[t_{1}, t_{2}, …] → MU_{∗}.

Our next result is the computation of topological Hochschild cohomology of MU with coefficients in an MU-algebra E. Since there is a canonical splitting of spectra

THH^{∗}(MU, E) ≃ E ∨ Σ^{−1}Der(MU, E)

the combination of this result with Corollary 4.4 gives a complete calculation of higher homotopy groups of the based space F_{S−alg}(MU, E).

Proposition 5.6 For an MU-algebra E the following isomorphism holds

T HH^{∗}(MU, E) = Λ̂_{E_{∗}}(y_{1}, y_{2}, …)

where the hat denotes the completed exterior algebra and the exterior generator y_{i} has cohomological degree 2i − 1.

Proof. Consider the topological Hochschild homology S-module of MU with coefficients in E,

THH_{∗}(MU, E) = MU ∧_{MU∧MU} E.

We have the spectral sequence of E_{∗}-algebras

E_{∗∗}^{2} = T or_{∗∗}^{MU_{∗}MU}(MU_{∗}, E_{∗}) = E_{∗} ⊗ Λ(ỹ_{1}, ỹ_{2}, …) ⇒ T HH_{∗}(MU, E) (9)

This spectral sequence is multiplicative and since the differentials applied to the exterior generators ỹ_{i} are trivial for dimensional reasons we conclude that it collapses. Now the result for topological Hochschild cohomology follows by virtue of the universal coefficients formula and the isomorphism

THH^{∗}(MU, E) ≅ F_{E}(THH_{∗}, E).

Proposition 5.6 is proved.

Recall that we are using the notation S_{Q}[x_{i}] for the free commutative S-algebra on the S-module S_{Q}^{2i}, the rationalized 2i-sphere S-module. The coefficient ring of S_{Q}[x_{i}] is isomorphic to Q[x_{i}] where the polynomial generator x_{i} has degree 2i. Further denote the infinite smash power S_{Q}[x_{1}]^{∧∞} by S_{Q}[x_{1}, x_{2}, …].

Lemma 5.7 The commutative S-algebras MU_{Q} and by S_{Q}[x_{1}, x_{2}, …] are weakly equivalent.

Proof. The polynomial generators x_{i} ∈ MU_{Q∗} = Q[x_{1}, x_{2}, …] determine a collection of maps S_{Q}^{2i} → MU_{Q} and therefore a map of commutative algebras S_{Q}[x_{1}]^{∧∞} → MU_{Q} which is clearly a weak equivalence.

Definition 5.8 Let E be a ring spectrum (in the traditional up to homotopy sense) with multiplication m : E ∧ E → E and M an E-bimodule spectrum with the left action m_{l} : E ∧M → M and the right action m_{r} : M ∧E → M. We say that a map f : E → M is a primitive operation if f ∘ m and f ∧ m_{r} + m_{l} ∧ f are homotopic as maps from E ∧ E to M. The set of all primitive operation from E to M is denoted by P rim(E, M)

Remark 5.9 Perhaps it is more natural to use the term “derivation” instead of “primitive operation” but this term is already overworked in this paper.

The next lemma provides a description of topological derivations of MU with coefficients in HZ.

Lemma 5.10 There is the following isomorphism of graded abelian groups:

Der^{∗}(MU, HZ) ≅ Λ^{∗−1}(y_{1}, y_{2} …)/Z.

Under the forgetful map

l : Der^{∗}(MU, HZ) → [MU, HZ]^{∗} = Hom(Z[t_{1}, t_{2}, …], Z)

the elements y_{i} ∈ Der^{2i−2}(MU, HZ) correspond to the standard derivations ∂_{t_{i}} Moreover the elements y_{i} are Q-preferred topological derivations.

Proof. We have the spectral sequence

Der^{∗}(HZ_{∗}MU, Z) = Λ^{∗−1}(y_{1}, y_{2} …)/Z ⇒
Der_{HZ}^{∗}(HZ ∧ MU, HZ) = Der^{∗}(MU, HZ).

This spectral sequence clearly collapses. Next using Proposition 3.7 we see that the image of the element y_{i} under the forgetful map l in the group

[HZ ∧ MU, HZ]_{HZ−mod} = [MU, HZ]^{∗} = Hom(Z[t_{1}, t_{2}, …], Z)

is precisely the algebraic derivation ∂_{t_{i}} (up to elements of higher filtration). Since this image contains in the subgroup of primitive operations MU → HZ none of these elements of higher filtration are present.

To see that y_{i} are Q-preferred derivations let us introduce the notation CDer^{∗}(MU_{Q}, HQ) to denote topological derivations of MU_{Q} with values in HQ ≃ S_{Q} in the category of commutative S-algebras. (These commutative derivations are also known as topological Andre-Quillen cohomology, cf.[2]). Then since MU_{Q} is a commutative free S-algebra we see immediately that

CDer^{∗}(MU_{Q}, HQ) = Der^{∗}(MU_{Q∗}, Q) = Q < ∂_{x_{1}}, ∂_{x_{2}}, … >

the right hand side being the set of derivations (in the usual algebraic sense) of the algebra MU_{Q∗} with values in the rational numbers. It is spanned over the rationals by the derivations ∂_{x_{1}}, ∂_{x_{2}}, . . ..

On the other hand Der^{∗}(MU_{Q}, HQ) ≅ Λ_{Q}^{∗−1}(y_{1}, y_{2} …)/Q. We need to prove therefore that the forgetful map

CDer^{∗}(MU_{Q}, HQ) → Der^{∗}(MU_{Q}, HQ)

sends the elements ∂_{x_{i}} to y_{i}.

Since the commutative S-algebra HQ[x_{i}] ≃ S_{Q}[x_{i}] is free as a commutative S-algebra as well as an (associative) S-algebra it follows that

CDer^{∗}(H_{Q}[x_{i}], HQ) = Der^{∗}(Q[x_{i}], Q) = Q < ∂_{x_{i}} > .

There is a unique map of commutative S-algebras MU_{Q} → HQ[x_{i}] which corresponds to quotienting out the ideal (x_{1}, x_{2}, …, x_{i−1}x̂_{i}, x_{i+1}, …) in the coefficient ring of MU_{Q}. We have the following commutative diagram:

CDer^{∗}(HQ[x_{i}], HQ) ≅ Der^{∗}(Q[x_{i}], Q))
↓  ↓
CDer^{∗}(MU_{Q}, HQ) → Der^{∗}(MU_{Q}, HQ)

from which it is clear that the image of ∂_{x_{i}} in Der^{∗}(MU_{Q}, HQ) is y_{i} and the lemma is proved.

Corollary 5.11 The set of Q-preferred derivations of MU with values in HZ maps bijectively onto the set of primitive cohomology operations MU → HZ under the forgetful map Der^{∗}(MU, HZ) → [MU, HZ]^{∗}.

Lemma 5.12 Let A be an S-algebra and B a commutative S-algebra. Suppose that B has a structure of an A-bimodule via a map of S-algebras A → B. Then THH(A, B) has a structure of a B-bimodule and there is a canonical splitting of B-bimodules THH(A, B) ≃ B ∨ Σ^{−1}Der(A, B).

Proof. Consider the following sequence of S-algebra maps:

B → THH(B, B) ≅ F_{B∧B^{op}}(B, B) → F_{B∧A^{op}}(B, B) ≅ THH(A, B).

The first map exists because B is commutative, the middle map is induced by the S-algebra map A → B and the last equivalence is Corollary 3.3. The composite map B → THH(A, B) supplies THH(A, B) with a structure of a B-bimodule and splits the canonical map THH(A, B) → B. This proves our lemma.

Let us introduce the notation MU_{n} for the nth Postnikov stage of MU, Then MU_{n} is an S-algebra (even a commutative S-algebra).

Lemma 5.13 There is the following isomorphism of MU_{n∗}-modules:

T HH^{∗}(MU, MU_{n}) ≅ Λ_{MU_{n∗}}(y_{1}, y_{2}, …)

where MU_{n} is considered as an MU-bimodule via any (not necessarily central) map of S-algebras f : MU → MU_{n}.

Proof. Since MU_{n} is a commutative S-algebra the multiplication map MU_{n} ∧ MU_{n} →^{m} MU_{n} is an S-algebra map. Therefore the composition

MU ∧ MU_{n} ⟶^{f∧id} MU_{n} ∧ MU_{n} →^{m} MU_{n}

is also an S-algebra map. This gives the following isomorphisms of Smodules:

THH(MU, MU_{n}) ≅ F_{MU∧MU_{n}}(MU_{n}, MU_{n}) ≅
F_{MU_{n}}(MU_{n} ∧_{MU∧MU_{n}} MU_{n}, MU_{n}).

Therefore it is enough to show that

π_{∗}MU_{n} ∧_{MU∧MU_{n}} MU_{n} = Λ_{MU_{n∗}}(ỹ_{1}, ỹ_{2} …).

(The exterior generators y_{i} ∈ T HH^{∗}(MU, MU_{n}) will be dual to ỹ_{i}). Consider the spectral sequence

T or_{∗∗}^{MU_{∗}MU_{n}}(MU_{n∗}, MU_{n∗}) = Λ_{MU_{n∗}}(ỹ_{1}, ỹ_{2}, …) ⇒ π_{∗}MU_{n} ∧_{MU∧MU_{n}} MU_{n}

(10)

This spectral sequence is not multiplicative since the map f : MU → MU_{n} may not be central. However it is a spectral sequence of MU_{n∗}-modules.

Let us introduce another spectral sequence

T or_{∗∗}^{MU_{∗}MU}(MU_{∗}, MU_{∗}) = Λ_{MU_{∗}}(ỹ_{1}, ỹ_{2}, …) ⇒ π_{∗}MU ∧_{MU∧MU} MU (11)

Then the map f : MU → MU_{n} induces a map of spectral sequences (11)→(10). Further the spectral sequence (11) is multiplicative and collapses for that reason. Therefore in (10) all elements of the form ỹ_{i_{1}} ∧ỹ_{i_{2}} ∧… ỹ_{i_{k}} are permanent cycles and it follows that (10) collapses. Lemma 5.13 is proved.

Suppose as before that we have an MU-bimodule structure on MU_{n+1} via some S-algebra map f : MU → MU_{n}. Composing f with the canonical map in the Postnikov tower p_{n} : MU_{n+1} → MU_{n} we get an MU-bimodule structure on MU_{n} also. Then we have the following

Corollary 5.14 The induced map Der^{∗}(MU, MU_{n+1}) → Der^{∗}(MU, MU_{n}) is onto.

Proof. Indeed, the map

T HH^{∗}(MU, MU_{n+1}) = Λ_{MU_{n+1∗}}(y_{1}, y_{2}, …) →
Λ_{MU_{n∗}}(y_{1}, y_{2}, …) = T HH^{∗}(MU, MU_{n})

is clearly onto and our claim follows from Lemma 5.12.

Proof of Theorem 5.4. We start by outlining the strategy of the proof. Take a multiplicative operation f ∈ Mult(MU, MU). Define f_{n} : MU → MU_{n} as the the composition

MU →^{f} MU →^{p_{n}} MU_{n}.

(Recall that we denoted by p_{n} the canonical projection onto the nth Postnikov stage.) Then by Theorem 4.8 the map f_{0} : MU → MU_{0} = HZ is homotopic to a unique S-algebra map which we will denote by f̃_{0}. Proceeding by induction assume that there exists a unique Q-preferred S-algebra map f̃_{n} : MU → MU_{n} which is homotopic to f_{n} when considered as a map of S-modules. We will see that 1) f̃_{n} admits a Q-preferred lifting to an S-algebra map MU → MU_{n+1} and 2) there is a one-to-one correspondence between such liftings and the set of liftings of f̃_{n} to a homotopy multiplicative map MU → MU_{n+1}. In particular f_{n+1} being one of such liftings can be realized as a Q-preferred S-algebra lifting in a unique fashion.

We now proceed to realize the above program in detail. The first thing is to show that there exists a lifting of f̃_{n} in the category of S-algebras. The homotopy fibre sequence

Σ^{n+1}Hπ_{n+1}MU → MU_{n+1} → MU_{n}

is a topological singular extension by Theorem 2.5. Then Theorem 2.4 tells us that the obstruction to an S-algebra lifting f̃_{n} to MU_{n+1} is a certain element σ ∈ Der(MU, ΣHπ_{n+1}MU). Clearly a lift does exists after rationalization. Therefore the image of σ in Der^{0}(MU, Σ^{n+2}Hπ_{n+1}MU_{Q}) is zero. But Hπ_{n+1}MU is a wedge of suspensions of HZ and by Lemma 5.10 Der(MU, Hπ_{n+1}MU) has no torsion. It follows that σ = 0 in Der(MU, ΣHπ_{n+1}MU) and a lift of f̃_{n} exists (not necessarily Q-preferred). By Theorem 2.4 the homotopy fibre of the map

F_{S−alg}(MU, MU_{n+1}) → F_{S−alg}(MU, MU_{n})

over the point f̃_{n} ∈ F_{S−alg}(MU, MU_{n}) is weakly equivalent to the space Ω^{∞}Der(MU, Σ^{n+1}Hπ_{n+1}MU).

Therefore denoting by [MU, MU_{n+1}]_{a}^{lift} ⊂ [MU, MU_{n+1}]_{a} the set of homotopy classes of S-algebra maps MU → MU_{n+1} lifting f̃_{n} we have the following long exact sequence:

… → Der^{−1}(MU, Σ^{n+1}Hπ_{n+1}MU) → π_{1}F_{S−alg}(MU, MU_{n+1}) →

π_{1}F_{S−alg}(MU, MU_{n}) → Der^{0}(MU, Σ^{n+1}Hπ_{n+1}MU) → [MU, MU_{n+1}]_{a}^{lift} → pt

which is the same (by Theorem 4.3) as the long exact sequence

… → Der^{−1}(MU, Σ^{n+1}Hπ_{n+1}MU) → Der^{−1}(MU, MU_{n+1}) →

Der^{−1}(MU, MU_{n}) → Der^{0}(MU, Σ^{n+1}Hπ_{n+1}MU) → [MU, MU_{n+1}]_{a}^{lift} → pt.

By Corollary 5.14 the map

Der^{−1}(MU, MU_{n+1}) → Der^{−1}(MU, MU_{n})

is onto and we conclude that the map

Der^{0}(MU, Σ^{n+1}Hπ_{n+1}MU) → [MU, MU_{n+1}]_{a}^{lift}

is bijective. So the set of all possible lifts of f̃_{k} is in bijective correspondence with Der^{0}(MU, Σ^{n+1}Hπ_{n+1}MU). Clearly the set of all Q-preferred lifts corresponds under this isomorphism to the set of Q-preferred topological derivations of MU with values in Σ^{n+1}Hπ_{n+1}MU. By Corollary 5.11 these Qpreferred derivations are identified with the set of primitive operations from MU to Σ^{n+1}Hπ_{n+1}MU. So we established a one-to-one correspondence between the set of Q-preferred lifts of f̃_{n} and P rim(MU, Σ^{n+1}Hπ_{n+1}MU).

Now we examine the question of lifting the map f̃_{n} up to homotopy to a homotopy multiplicative map MU → MU_{n+1}. Clearly the homotopy class of any map of S-modules MU → MU_{k} is determined by its rationalization, i.e, the rationalization map [MU, MU_{k}] → [MU_{Q}, MU_{Qk}] is injective. It follows that the map

Mult(MU, MU_{k}) → Mult(MU_{Q}, MU_{kQ})

is also injective.

Further we have the following bijection (for any k)

Mult(MU_{Q}, MU_{kQ}) ≅ Hom_{rings}(MU_{Q∗}MU_{Qk∗})

Therefore there is a short exact sequence

0 → P rim(MU_{Q}, Σ^{n+1}Hπ_{n+1}MU_{Q}) = Der(MU_{Q∗}, π_{n+1}MU_{Q}) →

Mult(MU_{Q}, MU_{Qn+1}) → Mult(MU_{Q}, MU_{Qn}) → 0 (12)

Of course the last two terms are only sets. The exactness here means that Mult(MU_{Q}, MU_{Qn+1}) has a faithful action of P rim(MU_{Q}, Σ^{n+1}Hπ_{n+1}MU_{Q}) so that the quotient is isomorphic to Mult(MU_{Q}, MU_{Qn}).

Consider the diagram of cofibre sequences

Σ^{n+1}Hπ_{n+1}MU  → MU_{n+1} → MU_{n}
↓ ↓ ↓
Σ^{n+1}Hπ_{n+1}MU_{Qn+1} → MU_{Q} → MU_{Qn}

Taking into account the fact that MU_{k+1}^{∗}MU surjects onto MU_{k}^{∗}MU for any k we obtain a map of short exact sequences

0 → [MU, Σ^{n+1}Hπ_{n+1}MU] → [MU, MU_{n+1}] → [MU, MU_{n}] → 0
↓ ↓ ↓
0 → [MU_{Q}, Σ^{n+1}Hπ_{n+1}MU_{Q}] → [MU, MU_{Qn+1}] → [MU, MU_{Qn}] → 0

Notice that all downward maps are injections. Combining this with (12) we find that there is a short exact sequence

0 → P rim(MU, Σ^{n+1}Hπ_{n+1}MU) →
Mult(MU, MU_{n+1}) → Mult(MU, MU_{n}) → 0  (13)

That is the indeterminacy in lifting the map f̃_{n} : MU → MU_{n} is precisely the set of primitive cohomology operations P rim(MU, Σ^{n+1}Hπ_{n+1}MU). We see that the set of lifts of the map f̃_{n} to a a homotopy multiplicative map is in one-to-one correspondence with Q-preferred S-algebra lifts. This completes the inductive step and shows that the original homotopy multiplicative map f : MU → MU can be improved in a unique way to a Q-preferred S-algebra map.

So we succeded in finding a section i : Mult(MU, MU) → [MU, MU]_{a} of the forgetful map j : [MU, MU]_{a} → Mult(MU, MU) so that the image of i consists of Q-preferred S-algebra self-maps of MU. To see that i respects composition notice that for f, g ∈ Mult(MU, MU) the S-algebra map i(f)∘ i(g) is Q-preferred and j(i(f) ∘ i(g)) = f ∘ g. Since there is a unique Qpreferred S-algebra self-map whose image under j is f ∘ g we conclude that i(f) ∘ i(g) = i(f ∘ g). With this the proof of Theorem 5.4 is completed.

Remark 5.15 Using the Bousfield-Kan mapping space spectral sequence (cf. [5]) it is possible to calculate the set of all S-algebra self-maps of MU. However this approach leads to the identification of [MU, MU]_{a} only as a set, not as a monoid. It seems that the monoid structure on [MU, MU]_{a} should be related to the Gerstenhaber bracket on T HH^{∗}(MU, MU)

Now consider an S-algebra E with a fixed map of S-algebras f : MU → E Suppose that E satisfies the following condition:

(S) The unit map f : MU_{∗} → E_{∗} is surjective.

Remark 5.16 In [7] it was proved that a rather broad class of C-oriented spectra (namely those which are obtained by killing any regular ideal in the ring MU) can be supplied with MU-algebra structures. For this class of spectra the condition (S) is obviously satisfied.

Corollary 5.17 For an S-algebra E satisfying the condition (S) any multiplicative operation MU → E can be lifted (perhaps in a non-unique way) to an S-algebra map.

Proof. The condition (S) guarantees that the map Mult(MU, MU) → Mult(MU, E) induced by the given map f : MU → E is surjective. In other words any multiplicative operation g : MU → E can be represented as a composition h ∘ f where h ∈ Mult(MU, MU). Since h can be lifted to an S-algebra self-map of MU our claim follows.

As another consequence of Theorem 5.4 we will show that the p-local Brown-Peterson spectrum BP is an A_{∞}-retract of MU_{(p)}, the spectrum MU localized at p. Recall from e.g. [13], 4.1 that there exists a multiplicative cohomology operation g : MU_{(p)} → MU_{(p)} which is idempotent and whose image is the p-local spectrum BP.

Theorem 5.18 There exists an S-algebra map f : MU_{(p)} → BP which has a right inverse S-algebra map h : BP → MU_{(p)}.

Proof. According to Theorem 5.4 the multiplicative operation g : MU_{(p)} → MU_{(p)} determines a map of S-algebras which we will denote by the same letter. Without loss of generality we can assume g to be a q-cofibration of S-algebras. Consider the diagram in the category of S-algebras:

MU_{(p)} →^{g} MU_{(p)} →^{g} …
g ↓  g ↓
MU_{(p)} →^{id} MU_{(p)} →^{id} …

Each square in this diagram is commutative since the operation g is idempotent. The colimit of the upper row taken in the category of S-algebras coincides with the colimit taken in the category of spectra by Cofibration Hypothesis ([8],VII.4) and both are equivalent to BP. (That shows that BP is an S-algebra). Now the map f : MU_{(p)} → BP is just the canonical map to the colimit. Next the colimit of the lower row is obviously MU_{(p)} and therefore there exists an S-algebra map h : BP → MU_{(p)}. It follows that f ∘ h : BP → BP is homotopic to the identity and Theorem 5.18 is proved.

Remark 5.19 It can be shown (cf. [10], [1], [7]) that BP actually supports a structure of an MU-algebra.

Remark 5.20 It seems natural to conjecture that any Q-preferred S-algebra self-map of MU lifts to a commutative S-algebra self-map. This conjecture, if true, would imply the existence of a canonical E_{∞} ring structure on BP, a long-standing problem posed by P.May. The first (to author’s knowledge) serious attack on this problem was undertaken by I. Kriz in his 1993 preprint [9]. This paper inspired much activity in the area, however it is still regarded as a program for further work rather than a definitive solution.

Even though we don’t know whether BP is a commutative S-algebra we can use Theorem 5.18 to compute homotopy classes of A_{∞}-maps out of BP.

Corollary 5.21 For an S-algebra E satisfying the condition (S) every multiplicative operation BP → E lifts to an A_{∞} ring map BP → E (perhaps in a non-unique way).

Proof. The composition of the multiplicative operation BP → E with the canonical projection MU → BP determines a multiplicative operation MU → E. This operation lifts to an S-algebra map. Composing this S-algebra map with the splitting map BP → MU (which we know is an Salgebra map by Theorem 5.18) we find the desired S-algebra map BP → E.

Acknowledgement. The author would like to thank P.Goerss and N.Kuhn for many stimulating discussions.

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