VIRTUAL OPERAD ALGEBRAS AND REALIZATION OF HOMOTOPY TYPES

VLADIMIR HINICH

1. Introduction

1.1. Let k be a base commutative ring, C(k) be the category of complexes of k-modules. The category of operads Op(k) in C(k) admits a closed model category (CMC) structure with quasiisomorphisms as weak equivalences and surjective maps as fibrations (see [H], Sect. 6 and also Section 2 below).

Let now O be a cofibrant operad. The main result of this note (see Theorem 3.1) claims that the category of O-algebras admits as well a CMC structure with quasi-isomorphisms as weak equivalences and surjective maps as fibrations. This allows one, following the pattern of [H], 5.4, to construct the homotopy category of virtual O-algebras for any operad O over C(k) as the homotopy category of P-algebras for a cofibrant resolution P → O of the operad O.

The main motivation of the note was to understand the following main result of Mandell’s recent paper [Man].

1.2. Theorem. The singular cochain functor with coefficients in \overline{𝔽}_{p} induces a contravariant equivalence from the homotopy category of connected p-complete nilpotent spaces of finite p-type to a full subcategory of the homotopy category of E_{∞} \overline{𝔽}_{p}-algebras.

In his approach, Mandell realizes the homotopy category of E_{∞}-algebras as a localization of the category of algebras over a “particular but unspecified” operad E. One of major technical problems was that the category of E-algebras did not seem to admit a CMC structure.

We suggest to choose E to be a cofibrant resolution of the Eilenberg-Zilber operad. Then according to Theorem 3.1, the category of E-algebras admits a CMC structure. This considerably simplifies the proof of Theorem 1.2.

1.3. Content of Sections. The main body of the note (Sections 2 – 4) can be considered as a complement to [H] where some general homology theory of operad algebras is presented.

In Section 2 we recall some results of [H] we need in the sequel. In Section 3 we prove the Main theorem 3.1. In Section 4 we present, using Theorem 3.1, a construction of the homotopy category Viral(O) of virtual O-algebras.

In Section 5 we review the proof of Mandell’s theorem [Man], stressing the simplifications due to our Theorem 3.1.

1.4. Acknowledgement. This work was made during my stay at the Max-Planck Institut für Mathematik at Bonn. I express my gratitude to the Institute for the hospitality. I am also grateful to P. Salvatore for a useful discussion.

2. Homotopical algebra of operads: a digest of [H]

In this Section we recall some results from [H] and give some definitions we will be using in the sequel.

2.1. Category of operads. Let k be a commutative ring and let C(k) denote the category of complexes of k-modules.

Recall ([H], 6.1.1) that the category Op(k) of operads in C(k) admits a closed model category (CMC) structure in which weak equivalences are componenwise quasi-isomorphisms and fibrations are componentwise surjective maps.

Cofibrations in Op(k) are retractions of standard cofibrations; a map O → O′ is a standard cofibration if O′ = lim_{→}_{s∈ℕ}O_{s} with O_{0} = O and each O_{s+1} is obtained from O_{s} by adding a set of free generators g_{i} with prescribed values of d(g_{i}) ∈ O_{s}.

2.2. Algebras over an operad. Let O ∈ Op(k).

The category of O-algebras is denoted by Alg(O). For X ∈ C(k) we denote by F (O, X) the free O-algebra generated by X.

For any d ∈ ℤ denote by W_{d} ∈ C(k) the contractible complex

0 → k = k → 0

concentrated in degrees d, d + 1.

2.2.1. Definition. An operad O ∈ Op(k) is called H_{1}-operad if for any A ∈ Alg(O) the natural map

A → A ⊔ F (O, W_{d})

is a quasi-isomorphism.

2.2.2. Proposition. (see [H], Thm. 2.2.1) Let O be an H_{1}-operad. Then the category of Oalgebras admits a CMC structure with quasi-isomorphisms as weak equivalences and surjective maps as fibrations.

2.3. Examples.

2.3.1. First of all, not all operads are H_{1}-operads. In fact, let k = 𝔽_{p}, O = COM (the operad of commutative algebras). Then the symmetric algebra of W_{d} fails to be contractible in degree p.

2.3.2. Proposition. (see [H], Thm. 4.1.1) Any Σ-split operad (see [H], 4.2) is H_{1}-operad.

In particular, all operads over k ⊇ ℚ are H_{1}-operads. Also, all operads of form T^{Σ} where T is an asymmetric operad, in particular, ASS (see [H], 4.2.5), are H_{1}-operads.

2.3.3. The main result of this note claims that any cofibrant operad is an H_{1}-operad.

2.4. Base change and equivalence. Let f : O → O′ be a map of operads. Then a pair of adjoint functors

f^{∗} : Alg(O) → Alg(O′) : f_{∗}  (1)

is defined in a standard way.

2.4.1. Proposition. (see [H], 4.6.4.) Let f : O → O′ be a map of H_{1}-operads. The inverse and direct image functors (1) induce the adjoint functors

Lf^{∗} : Hoalg(O) → Hoalg(O′) : Rf_{∗} = f_{∗}  (2)

between the corresponding homotopy categories.

2.4.2. Definition. A map f : O → O′ of operads is called strong equivalence if for each d = (d_{1}, …, d_{n}) ∈ ℕ^{n} the induced map

O(|d|) ⊗_{Σ_{d}} k → O′(|d|) ⊗_{Σ_{d}} k

is a quasi-isomorphism.

Here |d| = ∑ d_{i} and Σ_{d} = Σ_{d_{1}} × … × Σ_{d_{n}} ⊆ Σ_{|d|}.

2.4.3. Proposition. Let f : O → O′ be a strong equivalence of H_{1}-operads. Then the functors Lf^{∗}, f_{∗} are equivalences.

In Section 5 we will be using the following version of Proposition 2.4.3.

2.4.4. Proposition. Let f : O → O′ be a strong equivalence of operads. Suppose O is H_{1}-operad. Then for each cofibrant O-algebra A the natural map

A → f_{∗}(f^{∗}(A))

is an equivalence.

2.4.5. Remark. A quasi-isomorphism of Σ-split operads compatible with the Σ-splittings is necessarily a strong equivalence.

Theorem 4.7.4 of [H] actually proves Proposition 2.4.4 and Proposition 2.4.3 together with the last Remark.

3. Main theorem

3.1. Theorem. Any cofibrant operad O ∈ Op(k) is an H_{1}-operad.

In particular, the category of algebras Alg(O) over a cofibrant operad O admits a CMC structure with quasi-isomorphisms as weak equivalences and epimorphisms as fibrations.

3.2. Proof of the theorem. First of all, we can easily reduce the claim to the case O is standard cofibrant. In fact, since O is cofibrant, it is a retraction of a standard cofibrant operad O′. Let

O ⟶^{α} O′ ⟶^{π} O

be a retraction. Let A be a O-algebra. We can consider A as a O′-algebra via π. Then the map A → A ⊔ F (O, M) is a retraction of the map A → A ⊔ F (O′, M). This reduces the theorem to the case O is standard cofibrant.

3.3. Standard cofibrant case. Let O = lim_{→}_{s∈ℕ}O_{s} (see notation of 2.1, O_{0} = 0) be a standard cofibrant operad. Let {g_{i}}, i ∈ I be a set of free (homogeneous) generators of O.

Let a function s : I → ℕ be given so that O_{s} is freely generated as a graded operad by g_{i} with s(i) ≤ s and, of course, dg_{i} ∈ O_{s(i)−1}.

Let, finally, val : I → ℕ and d : I → ℤ be the valency and the degree functions defined by the condition g_{i} ∈ O(val(i))^{d(i)}.

The collection I = (I, s, val, d) will be called a type of O.

Since we deal with free operads and free algebras, it is worthwhile to have an appropriate notion of tree. Fix a type I = (I, s, val, d).

Put I^{+} = I ∪ {a, m} (a and m will be special marks on some terminal vertices of our trees) and extend the functions val : I → ℕ and d : I → ℤ to I^{+} by setting val(a) = val(m) = d(a) = d(m) = 0.

3.3.1. Definition. A I-tree is a finite connected directed graph such that any vertex has ≤ 1 ingoing arrows; each vertex is marked by an element i ∈ I^{+} so that val(i) equals the number of outgoing arrows which are numbered by 1, …, val(i).

The set of vertices of a tree T will be denoted by V (T). Terminal vertices of a I-tree are the ones having no outgoing arrows. In particular, all vertices marked by a or by m are terminal.

3.3.2. Definition. A I-tree T is called proper if the following property (P) is satisfied.

(P) For any vertex v of T one of the possibilities (a)–(c) below occurs:

(a) v is terminal;

(b) v admits an outgoing arrow to a non-terminal vertex;

(c) v admits an outgoing arrow to a vertex marked by m.

We denote by P(I) the set of isomorphism classes of proper I-trees. The following obvious result justifies the notion of proper tree.

3.3.3. Proposition. Let O be a standard cofibrant operad of type I = (I, s, val, d), A be a O-algebra and M ∈ C(k). Then the coproduct B := A ⊔ F (M) is given, as a graded k-module, by the formula

B = ⨁_{T ∈P(I)} A^{⊗a(T)} ⊗ M^{⊗m(T)}[d(T)]  (3)

where a(T) (resp., m(T)) is the number of vertices of type a (resp., of type m) in T and d(T) = ∑_{v∈V (T)} d(v).

3.3.4. Let W be the set of maps ℕ → ℕ having finite support. Endow W with the following lexicographic order. For f, g ∈ W we will say that f > g if there exists a s ∈ ℕ such that f(s) > g(s) and f(t) = g(t) for all t > s. The set W well-ordered. Our next step is to define a filtration of B = A ⊔ F (M) indexed by W.

3.3.5. Definition. Let T ∈ P(I). The weight of T, w(T) ∈ W is the function ℕ → ℕ which assigns to any s ∈ ℕ the number of vertices v of T whose mark i ∈ I satisfies s(i) = s. Now we are able to define a filtration on B.

3.3.6. Let A, M, B = A ⊔ F (M) be as above. For each f ∈ W define

F_{f}(B) = ⨁_{T:w(T)≤f} A^{⊗a(T)} ⊗ M^{⊗m(T)}[d(T)].

The homogeneous components of the associated graded complex are defined as

gr_{f}^{F}(B) = F_{f}(B)/ ∑_{g<f} F_{g}(B).

3.3.7. Proposition. 1. For each f ∈ W the graded submodule F_{f} is a subcomplex of B.

2. One has F_{0} = A.

3. Suppose M is a contractible complex. Then for each f > 0 the homogeneous components gr_{f}^{F} are contractible.

Proof. Obvious.

3.3.8. Corollary. The natural map A → B = A ⊔ F (O, M) is a quasi-isomorphism of complexes. This implies Main Theorem 3.1.

Proof. Obvious.

4. Virtual algebras

4.1. Theorem 3.1 suggests the following definition. Let O ∈ Op(k). The homotopy category of virtual O-algebras Viral(O) is defined as Hoalg(P) where P → O is a cofibrant resolution of O in the category of operads. One should, however, do some work, to ensure the definition above makes sense.

4.2. Base change. Any morphism f : P → Q of operads induces a pair of adjoint functors

f^{∗} : Alg(P) ⟷ Alg(Q) : f_{∗}.  (4)

Theorem 3.1 together with 2.4.1 give immediately the following

4.2.1. Proposition. For any morphism f : P → Q of cofibrant operads the adjoint functors (4) induce a pair of adjoint functors

Lf^{∗} : Hoalg(P) ⟷ Hoalg(Q) : Rf_{∗} = f_{∗}  (5)

between the homotopy categories.

4.2.2. Proposition. 1. Let f : P → Q be a weak equivalence of cofibrant operads. Then f is a strong equivalence. In paticular, the derived functors of inverse and direct image (5) establish an equivalence of the homotopy categories.

2. Let f, g : P → Q be homotopic maps between cofibrant operads. Then there is an isomorphism of functors

f_{∗}, g_{∗} : Hoalg(Q) → Hoalg(P).

This isomorphism depends only on the homotopy class of the homotopy connecting f with g.

Proof. 1. Let d = (d_{1}, …, d_{n}), |d| = ∑ d_{i} and let Σ_{d} = ∏ Σ_{d_{i}} ⊆ Σ_{|d|}.

We have to check that the map

P(|d|) ⊗_{Σ_{d}} k → Q(|d|) ⊗_{Σ_{d}} k,

induced by f, is a quasi-isomorphism.

Since P and Q are cofibrant operads, P(|d|) and Q(|d|) are cofibrant as complexes of k(Σ_{|d|})- modules. Therefore, their quasi-isomorphism is a homotopy equivalence of k(Σ_{|d|})-modules and therefore is preserved after tensoring by k.

2. We present here a proof which is identical to the proof of Lemma 5.4.3(2) of [H].

Let Q ⟶^{α} Q^{I} ⟶ ⟶^{p_{0},p_{1}} Q be a path diagram for Q (see [Q], ch. 1) so that α is an acyclic cofibration. Since the functors p_{0∗} and p_{1∗} are both quasi-inverse to an equivalence α_{∗} : Hoalg(Q^{I}) → Hoalg(Q), they are naturally isomorphic. Therefore, any homotopy F : P → Q^{I} between f and g defines an isomorphism θ_{F} between f_{∗} and g_{∗}. Let now F_{0}, F_{1} : P → Q^{I} be homotopic. The homotopy can be realized by a map h : P → R where R is taken from a path diagram

Q^{I} ⟶^{β} R ⟶^{q_{0}×q_{1}} Q^{I} ×_{Q×Q} Q^{I}  (6)

where β is an acyclic cofibration, q_{0} × q_{1} is a fibration, q_{i} ∘ h = F_{i}, i = 0, 1. Passing to the corresponding homotopy categories we get the functors q_{i∗} ∘ p_{j∗} : Hoalg(Q) → Hoalg(R) which are quasi-inverse to α_{∗} ∘ β_{∗} : Hoalg(R) → Hoalg(Q). This implies that θ_{F_{0}} = θ_{F_{1}}.

4.3. Virtual operad algebras. Our construction of the category of virtual O-algebras follows the construction of virtual modules in [H], 5.4.

Let Op^{c}(k) denote the category of cofibrant operads in C(k). For each P ∈ Op^{c}(k) let Hoalg(P) be the homotopy category of P-algebras. These categories form a fibred category Hoalg over Op^{c}(k), with the functors Rf_{∗} = f_{∗} playing the role of “inverse image functors”.

Let O ∈ Op(k). Let Op^{c}(k)/O be the category of maps P → O of operads with cofibrant P. The obvious functor

c_{O} : Op^{c}(k)/O → Op^{c}(k)

assigns the cofibrant operad P to an arrow P → O.

4.3.1. Definition. The (homotopy) category Viral(O) of virtual O-algebras is the fibre of Hoalg at c_{O}. In other words, an object of Viral(O) consists of a collection A_{a} ∈ Hoalg(P_{a}) for each a : P_{a} → O in Op^{c}(k)/P and of compatible collection of isomorphisms ϕ_{f} : A_{a} → f_{∗}(A_{b}) given for every f : P_{a} → P_{b} in Op^{c}(k)/O.

4.3.2. Corollary. Let α : P → O be a weak equivalence of operads with cofibrant P. Then the obvious functor

                            q_{α} : Viral(O) → Hoalg(P)

is an equivalence of categories.

Proof. We will construct a quasi-inverse functor q^{α} : Hoalg(P) → Viral(O). For this choose for any map β : Q → O a map f_{β} : Q → P making the corresponding triangle homotopy commutative. Then, for any A ∈ Hoalg(P) we define q^{α}(A) to be the collection of f_{β∗}(A) ∈ Hoalg(Q). According to Proposition 4.2.2, the definition does not depend on the choice of f′_{β}s.

The corollary means that the homotopy category of virtual O-algebras is really the category of algebras over a cofibrant resolution of O.

4.3.3. Any map f : O → O′ defines an obvious functor Op^{c}(k)/O → Op^{c}(k)/O′. This induces a direct image functor

f_{∗} : Viral(O′) → Viral(O).

According to Corollary 4.3.2, this functor admits a left adjoint inverse image functor f^{∗} which can be calculated using cofibrant resolutions for O and O′.

4.4. Comparing Viral(O) with Hoalg(O).

4.4.1. Suppose k ⊇ ℚ. Let O ∈ Op(k) and let f : P → O be a cofibrant resolution of O. Both O and P admit a Σ-splitting (see [H], 4.2.4 and 4.2.5.2.) Moreover, the quasi-isomorphism f preserves the Σ-splittings. Therefore, the categories Viral(O) = Hoalg(P) and Hoalg(O) are equivalent by [H], 4.7.4.

Thus, in the case k ⊇ ℚ virtual operad algebras give nothing new.

4.4.2. Let T be an “asymmetric operad” i.e. a collection of complexes T (n) ∈ C(k) (with no action of the symmetric group), associative multiplication

T (n) ⊗ T (m_{1}) ⊗ … ⊗ T (m_{n}) → T (∑ m_{i})

and unit element 1 ∈ T (1) satisfying the standard properties.

Let O = T^{Σ} be the operad induced by T (see [H], 4.2.1).

Lemma. Suppose T (n) are cofibrant in C(k) (for example, T (n) ∈ C^{−}(k) and consist of projective k-modules). Then the natural functor

                         Viral(O) = Hoalg(P) → Hoalg(O)

induced by a(ny) resolution P → O, is an equivalence of categories.

Proof. It is enough to check that the map P(n) → O(n) is a homotopy equivalence of k(Σ_{n})- complexes for each n.

But P(n) is cofibrant over k(Σ_{n}) since P is a cofibrant operad; O(n) = T (n)⊗k(Σ_{n}) is cofibrant over k(Σ_{n}) since T (n) is cofibrant over k. This proves the claim.

4.4.3. Although the categories Viral(O) and Hoalg(O) turn out to be equivalent in all examples of Σ-split operads we know ([H], 4.2.5), we do not see any reason why this should always be the case. No doubt, the category Viral(O) should always be used when it differs from Hoalg(O).

5. Application: realization of homotopy p-types

Mandell’s theorem [Man] on the realization of homotopy p-types can be reformulated in terms of virtual commutative algebras. The advantage of this approach is that we can work with the category of operad algebras which has a CMC structure. This makes unnecessary a big part of [Man].

In this Section we review the proof Mandell’s theorem 1.2.

5.1. Adjoint functors C^{∗} and U.

5.1.1. Recall [HS] that the cochain complex C^{∗}(X) of an arbitrary simplicial set X ∈ Δ^{op}Ens admits a canonical structure of algebra over the Eilenberg-Zilber operad Z which is weakly equivalent to the operad COM of commutative algebras. Choose any cofibrant resolution E of Z. The category of virtual commutative algebras Viral(COM) is canonically equivalent to Hoalg(E).

5.1.2. For each commutative ring k define

C^{∗}( , k) : (Δ^{op}Ens)^{op} → Alg(k ⊗ E)  (7)

(here and below ⊗ means tensoring over ℤ) to be the functor of normalized k-valued cochains.

This functor admits an obvious left adjoint functor

U_{k} : Alg(k ⊗ E) → (Δ^{op}Ens)^{op}  (8)

given by the formula

U_{k}(A)_{n} = Hom(A, C^{∗}(Δ^{n}, k))  (9)

The pair of functors C^{∗}( , k) and U_{k} satisfies the requirements of Quillen’s theorem [Q], §4, Theorem 3.

Since the functor C^{∗}( , k) preserves weak equivalences, one therefore obtains a pair of derived adjoint functors

𝕌_{k} : Viral(COM) = Hoalg(k ⊗ E) ⟷ Ho : C^{∗}( , k),  (10)

Ho being the homotopy category of simplicial sets.

5.2. Following [Man], we call X ∈ Δ^{op}Ens k-resolvable if the natural map

u_{X} : X → 𝕌_{k}C^{∗}(X, k)

is a weak equivalence.

The following two lemmas allow one to construct resolvable spaces.

5.2.1. Lemma. ([Man], Thm. 1.1) Let X be the limit of a tower of Kan fibrations

… → X_{n} → … → X_{0}.

Assume that the canonical map from H^{∗}X to colim H^{∗}X_{n} is an isomorphism. If each X_{n} is k-resolvable, then X is k-resolvable.

5.2.2. Lemma. ([Man], Thm. 1.2) Let X, Y and Z be connected simplicial sets of finite type, and assume that Z is simply connected. Let X → Z and Y → Z be given, so that Y → Z is a Kan fibration. Then, if X, Y and Z are k-resolvable then so is the fibre product X ×_{Z} Y.

Lemma 5.2.1 follows form the fact that the functor 𝕌 carries homotopy colimits in Alg(E) into homotopy limits in Δ^{op}Ens. The proof of Lemma 5.2.2 is similar, but needs in addition Proposition 5.2.3 below which can be also easily deduced from Theorem 3.1.

Using the CMC structure on Op(k), one can embed the obvious map of operads ASS → COM into the following commutative diagram

ASS_{∞} ⟶^{α} E
↓  ^{τ}↓

                             ASS  ⟶^{\overline{α}} \overline{E} ⟶^{π} COM

where ASS_{∞} is the operad of A_{∞}-algebras, α is a cofibration, π is a weak equivalence and the square is cocartesian.

5.2.3. Proposition. (compare to [Man], Lemma 5.2). Let A → B and A → C be cofibrations of cofibrant E-algebras. Let \overline{A} = τ^{∗}(A), and similarly for \overline{B}, \overline{C}. Then the natural maps

B ⊔^{A} C ⟶^{t} \overline{B} ⊔\overline{A} \overline{C} ⟵^{r} \overline{B} ⊗\overline{A} \overline{C}

are quasi-isomorphisms in C(k). Here t is induced by τ and r is induced by the composition

\overline{B} ⊗ \overline{C} → (\overline{B} ⊔\overline{A} \overline{C}) ⊗ (\overline{B} ⊔\overline{A} \overline{C}) ⟶^{mult.} \overline{B} ⊔\overline{A} \overline{C}.

Proof. 1. t is a quasi-isomorphism. The functor τ^{∗} commutes with colimits. Therefore, it is enough to prove that the natural map A → τ_{∗}τ^{∗}(A) is a weak equivalence for a cofibrant algebra A. According to 2.4.4, it is enough to check that τ : E → \overline{E} is a strong equivalence of operads.

Since α is a cofibration, \overline{α} is a cofibration as well. Therefore, both E(n) and \overline{E}(n) are cofibrant over kΣ_{n}. Then the strong equivalence of E and \overline{E} follows from their weak equivalence.

2. r is a quasi-isomorphism.

Suppose A is standard cofibrant and the maps A → B, A → C are standard cofibrations. Let {e_{i}, i ∈ I}, {e_{j}, j ∈ I ∪ J}, {e_{k}, k ∈ I ∪ K}, be graded free bases of A, B and C respectively (the index sets I, J, K are disjoint).

The sets I, J and K are well-ordered and the differential of e_{i} is expressed through e_{i′} with i′ < i.

Put S = I ∪ J ∪ K with the order given by i < j < k for i ∈ I, j ∈ J, k ∈ K. Let S̃ be the set of maps S → ℕ with finite support and with the lexicographic order as in 3.3.4.

For f ∈ S̃ denote |f| = ∑_{s∈S} f(s).

The algebra \overline{B}⊔\overline{A} \overline{C} has an obvious increasing filtration by subcomplexes \{F_{f}\} indexed by f ∈ S̃.

The homogeneous component of the associated graded complex for f ∈ S̃ takes form

gr_{f}(F) = \overline{E}(|f|) ⊗_{Σ_{f}} e^{f}

where e^{f} = ∏_{s∈S} e_{s}^{f(s)} and Σ_{f} = ∏_{s∈S} Σ_{f(s)}.

Define a filtration \{F′_{f}\} of \overline{B} ⊗\overline{A} \overline{C} indexed by the same set S̃. It is given by the formula

F′_{f} = ⨁_{g<f} \overline{E}(|g|_{1}) ⊗ \overline{E}(|g|_{2}) ⊗_{Σ_{g}} e^{g}

where |g|_{1} = ∑_{s∈I∪J} g(s) and |g|_{2} = ∑_{s∈K} g(s). The homogeneous component for f ∈ S̃ is given by

gr_{f}(F′) = \overline{E}(|f|_{1}) ⊗ \overline{E}(|f|_{2}) ⊗_{Σ_{f}} e^{f}.

The map r : \overline{B} ⊗\overline{A} \overline{C} → \overline{B} ⊔\overline{A} \overline{C} is compatible with the filtrations. The corresponding map of the homogeneous components

gr_{f}(r) : \overline{E}(|f|) ⊗_{Σ_{f}} e^{f} → \overline{E}(|f|_{1}) ⊗ \overline{E}(|f|_{2}) ⊗_{Σ_{f}} e^{f}

is induced by the map

\overline{E}(|f|_{1}) ⊗ \overline{E}(|f|_{2}) → \overline{E}(|f|)  (11)

which is obviously quasi-isomorphism. The assertion then follows from the observation that both the left and the right hand side of (11) are cofibrant over k(Σ_{f}).

5.3. To construct k-resolvable spaces using 5.2.1 and 5.2.2 one needs a space “to start with”. This is the Eilenberg-Maclane space K(ℤ/p, n). The key step in [Man] is the following

5.3.1. Theorem. (cf. [Man], Prop. A.7). The space K(ℤ/p, n) is k-resolvable iff k ⊇ 𝔽_{p} and the frobenius F : k → k gives rise to a short exact sequence of abelian groups

0 → 𝔽_{p} → k ⟶^{1−F} k → 0.  (12)

Proof. This is the most important part of Mandell’s result and we cannot simplify the original Mandell’s proof.

1. The main step is to construct an explicit cofibrant resolution of C := C^{∗}(K(ℤ/p, n), 𝔽_{p}) over k := 𝔽_{p}.

Let k = 𝔽_{p}. Let E be a cofibrant resolution of the operad COM over ℤ. Recall that E-algebra structure on A gives rise to the action of the generalized Steenrod algebra 𝔅 on H(A ⊗ 𝔽_{p}) — see [May].

Let A be a chain complex of a topological space and let the operad E be endowed with a weak equivalence E → Z to the Eilenberg-Zilber operad of [HS], so that A becomes an E-algebra. Then the action of fB on H(A ⊗ 𝔽_{p} induces an action of the (conventional) Steenrod algebra 𝔄 which is a quotient of 𝔅 by the ideal generated by P^{0} − 1, P^{0} being the degree zero generalized Steenrod operation.

Choose a fundamental cycle e ∈ C^{n}. This cycle defines a map ϕ : E_{𝔽_{p}}⟨x⟩ → C from the free 𝔽_{p} ⊗ E-algebra with a generator x to C sending x to e. Since P^{0} acts trivially on H(C), the cohomology class P^{0}([x]) − [x] of E_{𝔽_{p}}⟨x⟩ (here [x] is the cohomology class of x), belongs to the kernel of H(ϕ). Choose a representative z of the cohomology class P^{0}([x]) − [x] of E_{𝔽_{p}}⟨x⟩.

Finally, define B = E_{𝔽_{p}}⟨x, y; dy = z⟩. This is the 𝔽_{p} ⊗ E-algebra obtained from the free algebra E_{𝔽_{p}}⟨x⟩ by adding a variable to kill the cycle z see also haha 2.2.2.

The map ϕ can be obviously extended to a map ψ : B → C.

Theorem. (see [Man], Thm. 6.2). The map ψ is a quasi-isomorphism.

The proof of the theorem given in [Man], Sect. 12, is based on a study of free unstable modules over 𝔅 and 𝔄.

2. Once we have found a cofibrant resolution B of the algebra C of cochains of K(ℤ/p, n), the life becomes very easy.

We have to study the map u_{X} : X → 𝕌_{k}(C^{∗}(X, k)) for X = K(ℤ, n).

One has 𝕌_{k}(C^{∗}(X, k)) = U_{k}(B_{k}) where B_{k} = k ⊗_{𝔽_{p}} B = E_{k}⟨x, y; dy = z⟩. Since the functor U_{k} carries cofibrations to Kan fibrations and colimits to limits, one has a cartesian diagram of spaces

U_{k}(B_{k}) ⟶ U_{k}(E_{k}⟨z, y; dy = z⟩)
↓  ↓
U_{k}(E_{k}⟨x⟩) ⟶^{p}  U_{k}(E_{k}⟨t⟩)

The vertical maps are Kan fibrations and U_{k}(E_{k}⟨z, y; dy = z⟩) is contractible since E_{k}⟨z, y; dy = z⟩ is a contractible k ⊗ E-algebra.

Furthermore, U_{k}(E_{k}⟨t⟩) identifies easily with the Eilenberg-Mac Lane space K(k, n) and the map p is induced by 1 − F : k → k where F : k → k is the frobenius ([Man], prop. 6.4, 6.5).

Then the long exact sequence of the homotopy groups for the fibration U_{k}(B_{k}) → U_{k}(E_{k}⟨x⟩) gives the long exact sequence

… → π_{i}(K(k, n + 1)) → π_{i}(U_{k}(B_{k})) → π_{i}(K(k, n)) ⟶^{p} π_{i+1}(K(k, n + 1)) → …

where the map p is induced by 1 − F.

Now, if the condition on k is not fulfilled, U_{k}(B_{k}) is not an Eilenberg-Mac Lane space. If the sequence (12) is exact, the natural map K(ℤ/p, n) → U_{k}(B_{k}) induces isomorphism of homotopy groups and this proves the assertion.

References

[H] V. Hinich, Homological algebra of homotopy algebras, Comm. in algebra, 25(10)(1997), 3291–3323.

[HS] V. Hinich, V. Schechtman, Homotopy limit of homotopy algebras, Lecture notes in math., 1289(1987), 240–264.

[Man] M. Mandell, E_{∞}-algebras and p-adic homotopy theory, Hopf preprint server, October, 1998.

[May] P. May, A general algebraic approach to Steenrod operations, Lecture notes in math., 168(1970), 153–231.

[Q] D. Quillen, Homotopical algebra, Lecture notes in math., 43 (1967).

Dept. of Mathematics, University of Haifa, Mount Carmel, Haifa 31905 Israel