There is just one rational cone-length

Octavian Cornea University of Rochester Rochester, NY 14627

November 20, 1993

Abstract

We show that the homotopic nilpotency of the algebra of piecewise polynomial forms on a simply-connected, finite type, CW -complex coincides with the strong L.S. category of the rationalization of that space. This is used to prove that, in the rational, simply-connected context all reasonable notions of cone-length agree. Both these two results are obtained as parts of a more general and functorial picture.

1 Introduction

Let S be the pointed category of pointed spaces of the homotopy type of a CW -complex and let F = F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ … ⊂ F_{n} ⊂ … be a filtered class of objects of S with the property that each F_{i} is closed with respect to wedges and suspensions and the zero dimensional sphere is in F_{0}.

Definition 1 For X ∈ Ob(S) the cone-length of X with respect to F, Cl_{F}(X), is the least natural number n for which there are cofibration sequences:

Z_{i} ⟶ X_{i} ⟶ X_{i+1}

for 0 ≤ i < n, with X_{0} = ∗, X_{n} ≃ X and Z_{i} ∈ F_{i} (if no such number exists, take Cl_{F}(X) = ∞).

Remark. Obviously, similar notions may be defined in more general categories where homotopy theory is possible.

Examples. 1. F_{i} = Ob(S) for all i. Then Cl_{F} = Cat, the strong Lusternik-Schnirelmann category of Ganea [6]. This invariant was introduced as an approximation for cat, the category of Lusternik and Schnirelmann. In its turn cat(M) + 1, [9], gives a lower bound for the minimal number of critical points of a smooth function on a smooth, closed manifold M; catX may be defined as the least natural number n such that there are n + 1 open subsets of X whose union covers X, each of them being contractible to the base point in X. A description of CatX, that does not involve the notion of cone-length, is the same as that of cat exept that the subsets covering X are required to be contractible in themselves. Obviously catX ≤CatX. Ganea and Takens [10] proved also that CatX ≤catX + 1. For further reference let’s mention here two alternative definitions for cat. The first one, due to Whitehead, asserts that catX ≤ n iff the diagonal X ⟶ X^{n+1} factors, up to homotopy, through the inclusion of the fat wedge T^{n+1}X ↪ X^{n+1}. The second one, given by Ganea, states that catX ≤ n iff the n-th Milnor fibration:

E_{n}ΩX ⟶ B_{n}ΩX ⟶ X

appearing in the classifying construction for ΩX admits a section up to homotopy.

2. F_{i} = the class of suspensions of order i. Denote Cl=Cl_{F} in this case. This notion was introduced in [2] where it is shown that Cl≤cat+1.

3. F_{i} = wedges of spheres. Let Cl_{S} be the resulting invariant. It was introduced (in the rational context) by Lemaire and Sigrist in [8]. Notice that, if we take F_{i}=wedges of spheres of dimension at least i, the invariant will not change.

In the context of the three examples above we have Cl_{S} ≥Cl≥Cat≥cat. In fact, Cl_{S} and Cat are the extreme examples of cone-length.

Let S_{0} be the category of rational simply-connected spaces of finite type. The definition 1 may be used to define similar notions in S_{0} with some obvious modifications : we start the cofibration sequences with i ≥ 1, define X_{1} to be of the homotopy type of the rationalization of a simply-connected suspension and require the spaces in F_{i} to be simply-connected (i ≥ 1). We preserve the same notations as in S.

Remark. In S_{0} we have Cl_{S} =Cl because rational suspensions are wedges of spheres.

In the above mentioned paper, Lemaire and Sigrist conjectured Cl_{S} =cat in S_{0}. This problem is the origin of the present work. An immediate consequence of our main result is that, in this rational setting, Cl_{S} =Cat.

The key to even formulate our result, and moreover to prove it, is a good understanding of the dictionary geometry⟷algebra specific to rational homotopy. It allows translations of different geometric notions (in particular, the various types of cone-length) in the category of rational, connected, graded, differential Lie algebras of finite type ,DGL, and in the category of rational, commutative, graded, differentiable, simply-connected algebras of finite type, DGA, (we refer to [7],[11] for the basics of rational homotopy).

To state our theorem we need a few more preliminaries.

Let h − S be the category of homotopy types of objects in S with morphisms given by homotopy classes of maps.

Each notion of cone-length induces a certain filtration on h − S. Denote by CL_{F} the filtration induced by Cl_{F}: (CL_{F})_{n} cosists of the homotopy types of objects X in S such that Cl_{F}X ≤ n and homotopy classes of maps which admit a level preserving representative in the sense that if Cl_{F}X ≤ n, Cl_{F}X′ ≤ n, then f : X ⟶ X′ is level preserving if they are objects of S: Z_{i}, Z′_{i} ∈ F_{i}, X_{i}, X′_{i} and morphisms g_{i}, f_{i} where 1 ≤ i < n such that the following diagrams are commutative:

[figure: Z_{i}; g_{i}; X_{i}; f_{i}; X_{i+1}; f_{i+1}; Z′_{i}; X′_{i}; X′_{i+1}]

the rows are cofibration sequences, X_{0} = X′_{0} = ∗, f_{0} = id, X_{n} ≃ X, X′_{n} ≃ X′, f_{n} ≃ f.

Let CAT and CL_{S} be the filtrations given by the previous construction using respectively Cat and Cl_{S}. We will use the same notations for the corresponding two filtrations obtained in S_{0} (here, of course, in the definition of a level preserving map we start with f_{1}) and for the filtrations obtained by translating everything in DGL language.

One of the main ingredients of our analysis of the rational cone-length is the approach introduced by Felix and Halperin in the study of the rational L.S. category.

Definition 2 Let A ∈ DGA. The nil-length of A, nilA, is the least natural number n (or ∞) for which there is a quasi-isomorphism A ⟶ B, with B ∈ DGA and \overline{B}^{n+1} = 0 (\overline{B} is the augmentation ideal of B). For X ∈ S_{0}, nilX =nil(M_{X}) where M_{X} is Sullivan’s minimal model of X.

We might think about nilX as the homotopic nilpotency of the algebra of P.L. forms on X.

In [4] Felix and Halperin give the following description for the L.S. category. If X ∈ S_{0} ,catX ≤ n iff X is a retract of some Y ∈ S_{0} such that nilY ≤ n. They also ask about a geometric interpretation for nil. As we shall see, another immediate consequence of our theorem is that Cat=nil.

Remark. It was shown in [2] that in the previous definition B does not have to be simply-connected, or finite type. If we require only that it is an augmented algebra we will still get the same invariant. In the same paper it was also shown that nil=Cl.

Let us denote by h − DGA the category of homotopy types of DGA’s taking as morphisms homotopy classes of morphisms in DGA.

The nil-length induces a filtration on h − DGA that we will denote by N IL: N IL_{n} consists of the homotopy types of algebras A with nilA ≤ n together with homotopy classes of morphisms that admit representatives g : A ⟶ A′ with \overline{A}^{n+1} = 0, \overline{A′}^{n+1} = 0.

Recall now the existence of a (classical) pair of functors L_{∗} : DGA ⟷ DGL : C^{∗} ,[11], which are homotopy adjoint.

Theorem 1.1 The functor L : (h − DGA) ⟶ (h − DGL) induced by L_{∗} preserves filtrations with respect to N IL and CL_{S}.

The functor C : (h − DGL) ⟶ (h − DGA) induced by C^{∗} preserves filtrations with respect to CAT and N IL.

Corollary 1.2 In h − DGL we have CL_{S} = CAT .

Remarks.1.This implies not only that Cl_{S} =Cat (everything is rational) but also that any homotopy class of maps that admits a level preserving representative in the sense of Cat admits also a level preserving representative in the sense of Cl_{S}; such a homotopy class of maps can be represented inside the DGA’s by a ”short” model: a morphism between two ”short” algebras.

2. The main cases when the Lemaire- Sigrist conjecture is known to hold are those of spaces which are formal or coformal or with cat≤ 2, [5]; in particular Cat=Cl_{S} was known to hold only in these cases.

The next two sections are devoted to the proof of the theorem. Below we would like to make a few comments on some closely related problems:

1. In the geometric (non-rational) context a natural problem is to distiguish among the different notions of cone-length. An approach to this question, along some quite classical lines, is to consider a cohomology theory h^{∗}(−) together with some natural transformations ψ^{i} : h^{∗} ⟶ h^{∗}, i ∈ I; suppose that the ψ^{i}’s vanish on ΣF (in the sense that ψ^{i}(u) = 0 for u ∈ h^{∗}(ΣX), X ∈ F, i ∈ I). If for some space Y there are i_{1}, i_{2} …, i_{n} ∈ I such that ψ^{i_{1}}ψ^{i_{2}} … ψ^{i_{n}}(u) ≠ 0 for some u ∈ h^{∗}(Y), then Cl_{F}Y ≥ n + 1. For example ordinary Z/2 cohomology together with the Steenrod squares may be used to show that Cl_{S}−Cl can reach arbitrarilly large values.

Two basic questions remain open:

2. Rationally are Cat and cat different ?

3. Geometrically are Cl and Cat different ? In this context it is worth mentioning that Ganea, [6], and more recently Clapp and Puppe, [1], have given some criteria that insure the equality of Cat and cat . The most efficient such condition (given in [1]) is: CatX =catX if X is p-connected, p ≥ 1, and dimX ≤ (2catX + 1)(p + 1) − 3.

The author would like to thank Joe Neisendorfer for his many helpful suggestions and for his advice, to John Moore for his sustained interest for these problems, to Steve Halperin for many useful discussions and for suggesting to us the statement (and the usefulness) of Lemma 2.2. and to Jean-Paul Doeraene for some helpful remarks.

2 L preserves filtrations

We want to show here that for any differential graded algebra morphism: f : A ⟶ B, with A, B ∈ DGA, \overline{A}^{n+1} = \overline{B}^{n+1} = 0 the induced morphism L_{∗}(f) : L_{∗}(B) ⟶ L_{∗}(A) fits into a homotopy commutative diagram :

[figure: L_{∗}(B); ψ_{1}; L_{∗}(A); ψ_{2}; L_{∗}(f); L_{1}; L_{2}; \overline{f}]

where ψ_{1}, ψ_{2} are quasi-isomorphisms, L_{1} = (L(X_{1}), d_{1}), L_{2} = (L(X_{2}), d_{2}) (L(X) being the free Lie algebra on X) and L_{1}, L_{2}, \overline{f} satisfy two conditions:

1. There are filtrations of X_{j} (j = 0, 1), 0 = F_{0}X_{j} ⊂ … ⊂ F_{n}X_{j} = X_{j} with d_{j}(F_{i+1}X_{j}) ⊂ L(F_{i}X_{j}).

Having these filtrations in place let F_{i}Z_{j} be a complement of F_{i}X_{j} in F_{i+1}X_{j}. Let L′_{ij} = (L(F_{i}Z_{j} ⊕ s^{−1}F_{i}Z_{j}), d_{ij}) where s^{−1} is the desuspension and d_{ij}(x) = s^{−1}x, for x ∈ F_{i}Z_{j}, d_{ij}(y) = 0 for y ∈ s^{−1}F_{i}Z_{j}. Let L_{ij} = (L(s^{−1}F_{i}Z_{j}), 0). There are unique morphisms u_{ij}, v_{ij} :

L_{ij} ⟶^{u_{ij}} (L(F_{i}X_{j}), d_{j}) ∗ L′_{ij} ⟶^{v_{ij}} (L(F_{i+1}X_{j}), d_{j})

such that v_{ij} restricts to inclusion on F_{i}X_{j} and on F_{i}Z_{j}, u_{ij}(y) = y − d(sy) for y ∈ L_{ij}. The existence of these morphisms is equivalent, geometrically, to the fact that the space represented by (L(F_{i+1}X_{j}), d_{j}) is the cofibre of a map from a wedge of spheres, represented by L_{ij}, into the space represented by (L(F_{i}X_{j}), d_{j}). The morphism \overline{f} has to satisfy:

2. There are morphisms f_{i}, g_{i}, 0 ≤ i < n, f_{n} = \overline{f} such that the following diagrams commute :

[figure: L_{i1}; g_{i}; L(F_{i}X_{1}) ∗ L′_{i1}; f′_{i}; L(F_{i+1}X_{1}); f_{i+1}; u_{i1}; v_{i1}; L_{i2}; L(F_{i}X_{2}) ∗ L′_{i2}; L(F_{i+1}X_{2}); u_{i2}; v_{i2}]

where f′_{i} = f_{i} ∗ g_{i} ∗ sg_{i} (notice that L′_{ij} is isomorphic, as Lie algebra, with L_{ij} ∗ sL_{ij}). This condition means, in Lie algebra language, that \overline{f} is ”level preserving”.

It is easy to see that to verify 2. it is enough to check that:

2’. The maps f_{i} as above exist and are compatible with the inclusions L(F_{i}X_{j}) ↪ L(F_{i+1}X_{j}) and, for some choice for F_{i}Z_{j}, the image of the restriction off_{i+1} to F_{i}Z_{1} is contained in F_{i}Z_{2}.

We will prove our claim by induction over n.

There is nothing to prove for n = 0. The induction step is provided by the next result.

Proposition 2.1 Let f : A ⟶ B be a DGA-morphism with \overline{A}^{n+1} = \overline{B}^{n+1} = 0. There is a DGL-morphism \overline{f} making the below diagram homotopy commutative:

[figure: L_{∗}(B); L_{∗}(A); L_{∗}(f); L_{1}; L_{2}; \overline{f}]

such that L_{1}, L_{2} are free as Lie algebras, the vertical arrows are quasiisomorphisms, L_{1} = (L′_{1} ∗ L′′_{1}, d_{1}), L_{2} = (L′_{2} ∗ L′′_{2}, d_{2}), if L_{i} = L(U_{i}), then d_{i}(U_{i}) ⊂ L′_{i} for i = 1, 2, \overline{f}(L′_{1}) ⊂ L′_{2}, \overline{f}(L′′_{1}) ⊂ L′′_{2} and:

∗) (L′_{1}, d_{1}) ≃ L_{∗}(B/\overline{B}^{n}) , (L′_{2}, d_{2}) ≃ L_{∗}(A/\overline{A}^{n})

Remark. This proposition implies that, in the given circumstances, A and B represent spaces which are cofibres of some maps from wedges of spheres to the spaces represented, repectively, by A/\overline{A}^{n} and B/\overline{B}^{n} and f represents a map that can be made level preserving with respect to these cofibration sequences.

The proposition follows from the next two lemmas.

Lemma 2.2 Let f : A ⟶ B be a DGA-morphism and let I ⊂ A, J ⊂ B be two ideals such that f^{−1}(J) = I, the restriction of f to I is injective and \overline{A} · I = 0, \overline{B} · J = 0. There is a DGL-morphism \overline{f} with all the properties in the statement of the proposition with the exeption of ∗) which is replaced by (L′_{1}, d_{1}) ≃ L_{∗}(B/J), (L′_{2}, d_{2}) ≃ L_{∗}(A/I).

Remark. The proof of this lemma is based on the ideas of [2]3, Proposition 3.3 . This satement and its utility in the present context was pointed out to us by Steve Halperin.

Proof.Recall the functor L_{∗} : DGA ⟶ DGL. Let CDGC be the category of cocommutative, differential, graded, simply-connected coalgebras of finite type, then L_{∗} = L ∘ ♯ where ♯ : DGA ⟶ CDGC is a dualization functor and L : CDCG ⟶ DGL verifies: L(C, δ) = (L(s^{−1}C), d′ + d′′), d′, d′′ induced by d′(s^{−1}x) = −s^{−1}(δx), d′′(s^{−1}x) = (−1/2)Σ(−1)^{|a_{i}|}[s^{−1}a_{i}, s^{−1}b_{i}] for \overline{Δ}x = Σa_{i} ⊗ b_{i}, x ∈ C, Δ being the coproduct in C (we will use ♯(A) = A^{∗}).

Let’s return to our lemma. We have vector space isomorphisms A ≃ (A/I) ⊕ I, B ≃ (B/J) ⊕ J. Choose vector space bases for A and B which respect these splittings. Dualize with respect to these bases. We get L_{∗}(B) = (L′_{1}∗L′′_{1}, d_{1}), L_{∗}(A) = (L′_{2}∗L′′_{2}, d_{2}) where L′_{1} ≃ L(s^{−1}(B/J)^{∗}), L′_{2} ≃ L(s^{−1}(A/I)^{∗}), L′′_{1} ≃ L(s^{−1}(J)^{∗}), L′′_{2} ≃ L(s^{−1}(I)^{∗}). It is immediate to check that L′_{1} and L′_{2} are closed to d_{1} respectively to d_{2} and that we have isomorphisms : (L′_{1}, d_{1}) ≃ L_{∗}(B/J), (L′_{2}, d_{2}) ≃ L_{∗}(A/I). Moreover, because f^{−1}(I) = J the map L_{∗}(f) respects the splittings on L_{∗}(B) and L_{∗}(A). The ideal J can be written, when viewed as a differential graded vector space, as a sum J = H_{1} ⊕ U_{1} such that each h ∈ H_{1}, is a cocycle and U_{1} is acyclic. A similar splitting exists on I and, because f is monomorphic on I, we can choose it I = H_{2} ⊕ U_{2} such that f^{−1}(H_{1}) ⊂ H_{2}. As a consequence of this we get splittings L′′_{i} = l′_{i} ∗ l′′_{i} where l′_{i} = L(s^{−1}H_{i}^{∗}) l′′_{i} = L(s^{−1}U_{i}^{∗}) (i = 1, 2). Remark that L_{∗}(f)(l′_{1}) ⊂ l′_{2}. Let’s denote L_{i} = L′_{i} ∗ l′_{i}.

We have Lie algebra maps : L_{1} ⟶^{j_{1}} L_{∗}(B) ⟶^{r_{1}} L_{1}, L_{2} ⟶^{j_{2}} L_{∗}(A) ⟶^{r_{2}} L_{2} with r_{i} ∘ j_{i} = id, j_{i} being obvious inclusions, r_{i} projections. Moreover we have the following commutative diagram:

[figure: L_{1}; j_{1}; L_{2}; j_{2}; L_{∗}(f); L_{∗}(A); L_{∗}(A); L_{∗}(f)]

Here the top arrow is the restriction of L_{∗}(f) to L_{1}. Let d_{i} = d′_{i} + d′′_{i} be the standard splitting of the differential in L_{∗}(B) for i = 1 and in L_{∗}(A)for i = 2. If x is a generator of l′_{i} then d′_{i}x = 0; because \overline{B} ·J = 0 and \overline{A}·I = 0 it is easy to see that d′′_{i}x ∈ L′_{i}. This means that d_{i} admits a restriction to L_{i}. With respect to this induced differential r_{i} becomes a differential map. To conclude the proof let’s just notice that both r_{i} and j_{i} are quasi-isomorphisms. Indeed l′′_{i} is d′_{i} acyclic, hence it is d_{i} acyclic.

We would like now to apply the previous lemma to the case of I = \overline{A}^{n}, J = \overline{B}^{n}. To do that we have to show that the conditions in the statement of the lemma may be satisfied.

Lemma 2.3 Let f : A ⟶ B be a DGA-morphism with \overline{A}^{n+1} = \overline{B}^{m+1} = 0, m ≤ n. It is possible to factor f: A ⟶^{f_{1}} A⊗_{d}E ⟶^{f_{2}} B, f_{1} being the inclusion x → x ⊗ 1, f_{2} is a quasi-isomorphism, \overline{(A ⊗_{d} E)}^{n+1} = 0 (clearly E is not free as an algebra).

Proof.Let A_{0} ⟶^{f′} A be a minimal model of A. It is a standard fact [3] that the product μ : A_{0} ⊗A_{0} ⟶ A_{0} admits an extension m : (A_{0} ⊗A_{0})⊗_{d} A′ ⟶ A_{0} where if A_{0} = (ΛZ, d), A′ = (Λs^{−1}Z, 0) and the twisted tensor product (A_{0} ⊗ A_{0}) ⊗_{d} A′ coresponds, geometrically, to the fibration ΩX ⟶ X ⟶^{Δ} X × X where X is a space represented by A_{0} and Δ is the diagonal (notice that A′ represents ΩX). It is easy to check that the same construction works for A. That means that we get a map t : (A ⊗ A) ⊗_{d} A′ ⟶ A extending the multiplication A ⊗ A ⟶ A; t (as m before) is easily seen to be a quasi-isomorphism.

Now f turns B into an A-module. Using this structure we get j_{1} = t ⊗_{A} B : (A ⊗ B) ⊗_{d} A′ ⟶ B which will still be a quasi-isomorphism. We have obtained a factorization of f:

A ⟶^{i_{1}} (A ⊗ B) ⊗_{d} A′ ⟶^{j_{1}} B

i_{1} is injective, j_{1} is a quasi-isomorphism. Let T_{1} = (A ⊗ B) ⊗_{d} A′. We also have a factorization of the identity 1_{B}:

B ⟶^{h_{1}} (A ⊗ B) ⊗_{d} A′ ⟶^{j_{1}} B

where h_{1}(x) = 1 ⊗ x ⊗ 1.

Suppose we have constructed T_{n} and factorizations of f and 1_{B} respectively: A ⟶^{i_{n}} T_{n} ⟶^{j_{n}} B, B ⟶^{h_{n}} T_{n} ⟶^{j_{n}} B. Build the next step by remarking, first, that j_{n} factors j_{n} : T_{n} ⟶ T_{n}/\overline{(T_{n})}^{n+1} ⟶^{i_{n}} B and construct T_{n+1} = (T_{n}/\overline{(T_{n})}^{n+1}) ⊗_{d} F_{n}, F_{n} free, together with morphisms h_{n+1} : B ⟶ T_{n+1} and j_{n+1} : T_{n+1} ⟶ B such that both are quasi-isomorphisms and j_{n+1} ∘ h_{n+1} = 1_{B}. We can now define i_{n+1} : A ⟶ T_{n+1} by the composition A ⟶^{i_{n}} T_{n} ⟶ T_{n+1}. Details on the construction of T_{n+1} together with the fact that passing to the limit (with respect to n) gives a splitting of 1_{B} : B ⟶^{i} T ⟶^{j} B with h, j quasi-isomorphisms, T of finite type, simplyconnected and \overline{T}^{n+1} = 0 can be found in [2] 3. At the limit we get also a map A ⟶^{f_{1}} T and A ⟶^{f_{1}} T ⟶^{j} B is our factorization of f.

Finally, to end the proof of the proposition, apply the above lemma to replace f : A ⟶ B by f_{1} : A ⟶ A ⊗_{d} E = B′ which is a monomorphism and satisfies f_{1}^{−1}(\overline{B′}^{n}) = \overline{A}^{n}. Now use Lemma 2.2. with I = \overline{A}^{n}, J = \overline{B′}^{n}.

3 C preserves filtrations

We recall that, if not mentioned otherwise, we are working inside the category of simply-connected, finite type rational spaces. It is immediate that our claim on C reduces to the next proposition:

Proposition 3.1 Suppose the following diagram is commutative:

[figure: Z; h; X; f; Y; g; i; p; Z′; X′; Y′; i′; p′]

where the rows are cofibration sequences and f admits a representative : \overline{f} : A′ ⟶ A; A, A′ ∈ DGA, \overline{A′}^{n+1} = \overline{A}^{n+1} = 0, A′ representing X′ and A representing X. Then g admits a representative \overline{g} : B′ ⟶ B; B, B′ ∈ DGA, \overline{B′}^{n+2} = \overline{B}^{n+2} = 0, B′ representing Y′ and B representing Y.

Proof.Consider the following homotopy commutative diagram:

[figure: F; ∗; F; ∗; Z; X; p; i; ∗; Y; Z/F; j_{1}; X; p_{1}; i_{1}; ΣF; Y; k_{1}]

Here F is the homotopy fibre of i (and it might not be simply-connected, however the fibration is clearly nilpotent), the vertical sequences are cofibrations and, as the top and middle square are push-out diagrams, so is the bottom square. Clearly a similar diagram exists in the ′ case. We denote F′ the homotopy fibre of i′. We obtain a map of push-out squares :

[figure: Z/F; j_{1}; h_{1}; X; p_{1}; f; i_{1}; ΣF; Σh; Y; g; k_{1}; Z′/F′; j′_{1}; X′; p′_{1}; i′_{1}; ΣF′; Y′; k′_{1}]

Recall that at the center of Sullivan’s point of view on rational homotopy there is the functor ”piecewise polynomial forms” A_{P L}(−). If we apply this functor to the previous diagram we get two short exact sequences of augmented algebras :

[figure: 0; K′; \overline{g′}; A_{P L}(ΣF′) ⊕ A_{P L}(X′); \overline{Σh}⊕\overline{f}; A_{P L}(Z′/F′); \overline{h}; 0; \overline{j′}−\overline{i′}; 0; K; A_{P L}(ΣF) ⊕ A_{P L}(X); A_{P L}(Z/F); 0; \overline{j}−\overline{i}]

Here \overline{j} = A_{P L}(j_{1}), \overline{j′} = A_{P L}(j′_{1}), \overline{i′} = A_{P L}(i′), \overline{i} = A_{P L}(i), \overline{Σh} = A_{P L}(Σh), \overline{f} = A_{P L}(f), \overline{h} = A_{P L}(h_{1}). By the elementary properties of the piecewise polynomial forms we get that \overline{g′} is a representative for g.

Remark. The maps \overline{j} − \overline{i} and \overline{j′} − \overline{i′} are not algebra maps. However K and K′ are clearly subalgebras of the respective algebras.

Next we plan to represent the previous exact sequences in terms of algebras as short as possible (in the sense that the nilpotency degree of the augmentation ideal of such an algebra should be as small as possible).

If we view the map i : Z ⟶ X as a fibration, by a standard tecnique of rational homotopy [7] we may represent i by an inclusion i_{2} : A ⟶ A ⊗_{d} F where A represents X (and appears in the statement of the proposition), F is the Sullivan model of F. Moreover the projection A⊗_{d} F ⟶ F represents the inclusion of the fibre F ↪ Z. Hence the map i_{1} : Z/F ⟶ X will be represented by i_{3} : A ⟶ Q ⊕ (\overline{A} ⊗_{d} F) = T ; i_{3} is uniquely defined by the property that when composed with the inclusion T ↪ A ⊗_{d} F it gives i_{2}. Everything we did till now can be done in the ′ case and even in a functorial way, hence we get a square :

[figure: A′; \overline{f}; T′; t; i′_{3}; A; T; i_{3}]

which represents the square (∗):

[figure: Z/F; h_{1}; X; f; i_{1}; Z′/F′; X′; i′_{1}]

Remark that \overline{T}^{n+1} = \overline{T′}^{n+1} = 0. We would like to show now that i_{3} and i′_{3} can be perturbed as to become surjective.

Lemma 3.2 Suppose f : U ⟶ V is a DGA-morphism with \overline{U}^{n+1} = 0, \overline{V}^{m+1} = 0, m ≤ n. Then it is possible to factor f in U ⟶^{f′} U′ ⟶^{f′′} V with f′′ surjective and f′ a quasi-isomorphism, \overline{U′}^{n+1} = 0.

Proof.By the basic properties of rational homotopy we can first factor f in U ⟶^{f_{1}} U_{1} ⟶^{f_{2}} V where U_{1} = U ⊗_{d} L, L being an acyclic, free, differential, graded algebra, f_{1} the obvious inclusion (in particular it is a quasiisomorphism) and f_{2} is surjective.

Let J = \overline{U_{1}}^{n+1}. As m ≤ n, f_{2} factors U_{1} ⟶ U_{1}/J ⟶ V. It is obvious that our claim is proved if we show that J is acyclic. Let {(E_{r}(U_{1}), d_{r})} be the spectral sequence obtained by filtering U_{1} by the powers of the augmentation ideal. To show that J is acyclic it is enough to notice that E_{1}^{≥n+1,∗}(U_{1}) = 0. But this follows from the below short exact sequence:

0 → \overline{U} ⊗_{d} L ⟶ U_{1} ⟶ L → 0

because E_{1}^{≥n+1,∗}(\overline{U} ⊗_{d} L) = 0, E_{1}(L) = Q where the involved spectral sequences are also induced by filtrations by the powers of the augmentation ideals.

Applying the previous lemma first to i_{3} and then to i′_{3} we will get a new square:

[figure: A′_{1}; \overline{f}_{1}; T′; t; i′_{4}; A_{1}; T; i_{4}]

representing the square (∗) and such that both i_{4} and i′_{4} are surjective and \overline{A_{1}}^{n+1} = \overline{A′_{1}}^{n+1} = 0.

The next step is to represent in a good way the diagram (∗∗):

[figure: Z/F; h_{1}; ΣF; Σh; j_{1}; Z′/F′; C; j′_{1}]

Standard rational homotopy theory allows us to represent this square by:

[figure: M′_{1}; u; T′; t; \overline{j′_{1}}; M_{1}; T; \overline{j_{1}}]

Where M_{1}, M′_{1} are free models of ΣF, ΣF′ respectively, both \overline{j_{1}}, \overline{j′_{1}} are surjective and u represents h_{1}. Moreover, because ΣF, ΣF′ are both coformal spaces, we may assume that the differentials in M_{1} and M′_{1} are at most quadratic.

Lemma 3.3 Let f : M ⟶ G be a DGA-morphism with \overline{G}^{n+1} = 0, M free with a differential at most quadratic. If nilM = m then there is a factorization of f: M ⟶^{f′} M′ ⟶^{f′′} G such that f′ is a quasi-isomorphism and \overline{M′}^{p+1} = 0 for any p ≥max\{m, n + 1\}.

Proof.Let {(E_{r}(M), d_{r})} be the spectral sequence induced by the filtration by the powers of the augmentation ideal of M. The differential of M being at most quadratic it results that d_{r} = 0 for r ≥ 2. Also, because nilM = m we get E_{2}^{>p,∗}(M) = 0. In particular if p ≥max{m, n + 1} we get E_{2}^{>p,∗} = 0. Let I = \overline{M}^{p+1}. It is clear that f factors: M ⟶ M/I ⟶^{f_{1}} G. The ideal I is not acyclic in general, however E_{2}^{>p,∗}(M) = 0 implies that we can find a minimal set of elements U = \{v_{i}\} such that v_{i} ∈ \overline{M}^{p}, v_{i} survives to E_{1} and d_{1}([v_{i}]) spans the image of d_{1} : E_{1}^{p,∗} ⟶ E_{1}^{p+1,∗}. Define now J = I ⊕ Q{U}. This is an acyclic ideal and, because p ≥ n + 1 we notice that f_{1}(U) = 0. Hence we get the wanted factorization of f : M ⟶ M/J ⟶ G.

Returning to the proposition, the spaces ΣF and ΣF′ are also formal, thus we have surjective quasi-isomorphisms: M_{1} ⟶^{ϕ} H = H^{∗}(ΣF), M′_{1} ⟶^{ϕ′} H′ = H^{∗}(ΣF′). Hence nilM′_{1} =nilM_{1} = 1. By using the lemma above we can represent the square (∗∗) by:

[figure: M′_{1}; u_{1}; T′; t; \overline{j′_{1}}; M_{2}; T; \overline{j_{2}}]

with \overline{M_{2}}^{n+2} = 0.

We cannot apply directly the same construction to M′_{1} because the construction of U in the proof of the lemma involves some choices that are not functorial. However the lemma gives us an ideal J′ ⊂ M′_{1} such that \overline{j′_{1}} factors: M′_{1} ⟶^{ψ} M′_{1}/J′ ⟶^{s} T′, \overline{(M′_{1}/J′)}^{n+2} = 0 with ψ a quasi-isomorphism. Now ϕ factors through M_{2}: M_{1} ⟶ M_{1}/J = M_{2} ⟶^{ϕ_{1}} H. Consider the composition u_{2} : M′_{1} ⟶^{u_{1}} M_{2} ⟶^{ϕ_{1}} H. Clearly u_{2} is null on J′ hence it factors through ψ.

We get the following commutative diagram (∗ ∗ ∗):

[figure: M′_{1}; u_{1}; M′_{1}/J′; ψ; M_{2}; H; ϕ_{1}]

where ψ and ϕ_{1} are surjective quasi-isomorphisms and \overline{M_{2}}^{n+2} = \overline{(M′_{1}/J′)}^{n+2} = 0.

Lemma 3.4 Suppose we have the following commutative diagram (of DGA’s):

[figure: M; r; G; v; f; M_{1}; G_{1}; h]

with r, v quasi-isomorphisms, v surjective and \overline{M_{1}}^{n+1} = \overline{G_{1}}^{n+1} = \overline{G}^{n+1} = 0, then there exists a DGA, M′ together with maps e : M ⟶ M′, k : M′ ⟶ G, l : M′ ⟶ M_{1} such that l ∘ e = r, k ∘ e = f and \overline{M′}^{n+1} = 0.

Proof.The construction we will use is very similar to that of Lemma 2.2 (and Lemma 3.7 of [2]).

It is clear that both f and r factor through M/\overline{M}^{n+1}. Let T_{0} = M, f_{0} = f, r_{0} = r, e_{0} = id; we have the commutative diagram (0):

[figure: M; r; e_{0}; G; v; f; T_{0}; t_{0}; f_{0}; r_{0}; T_{0}/\overline{T_{0}}^{n+1}; k_{0}; l_{0}; M_{1}; G_{1}; h]

Suppose that, by induction, we have constructed a diagram (i) with the property that it is the same as (0) but all indexes 0 are replaced by i, e_{i} is a quasi-isomorphism and t_{i} ∘ e_{i} is at least (i + 1)-connected. Define T_{i+1} = T_{i}/\overline{T_{i}}^{n+1} ⊗_{d} Z_{i} (with Z_{i} free) by succsesivelly killing the kernel of H^{∗}(l_{i}). Define e_{i+1} to be the composition M ⟶ T_{i} ⟶ T_{i}/\overline{T_{i}}^{n+1} ⟶ T_{i+1}; t_{i+1} is the obvious projection. It is easy to check that t_{i+1} ∘ e_{i+1} is at least i + 2connected. The map r_{i+1} appears by the construction of T_{i+1}. Let’s show that we can also construct f_{i+1}. We define it to be k_{i} on T_{i}/\overline{T_{i}}^{n+1}. We have to show that we can extend it also over Z_{i} (see Lemma 3.7 in [2]). The initial step is given by taking α ∈ T_{i}/\overline{T_{i}}^{n+1} such that [α] ∈ KerH^{∗}(l_{i}) and suppose β ∈ Z_{i} verifies dβ = α. Then h ∘ l_{i}(α) = dβ′. Also β′ = v(β′′) because v is surjective . We have v ∘ k_{i}(α) = h ∘ l_{i}(α), hence dβ′′ − k_{i}(α) ∈ Ker(v). But as v is a quasi-isomorphism, and d(dβ′′ − k_{i}(α)) = 0 there is a γ ∈ G such that dγ = dβ′′ − k_{i}(α). Define f_{i+1}(β) = β′′ − γ. The same type of argument may be applied at later stages in the construction of Z_{i}. Passing to the limit we get M′ as the limit of the T_{i}’s, l as that of the l_{i}’s and k as that of the k_{i}’s.

Returning now to the diagram (∗ ∗ ∗), by applying the previous lemma we get an algebra M′_{2} and maps making the next diagram commutative:

[figure: M′_{1}; e; u_{1}; ψ; M_{2}; M′_{2}; M′_{1}/J′; l; k]

with \overline{M′_{2}}^{n+2} = \overline{M_{2}}^{n+2} = 0, e, ψ, k quasi-isomorphisms, k surjective. Making use of this we can represent the square (∗∗) by :

[figure: M′_{2}; l; T′; t; \overline{j′_{2}}; M_{2}; T; \overline{j_{2}}]

where \overline{j′_{2}} = s ∘ k. Recall that \overline{M_{2}}^{n+2} = \overline{M′_{2}}^{n+2} = \overline{T}_{n+1} = \overline{T′}^{n+1} = 0. We may also suppose \overline{j_{2}}, \overline{j′_{2}} surjective (by Lemma 3.1.). Putting things together we get the following map of short exact sequences of augmented algebras:

[figure: 0; B′; \overline{g}; M′_{2} ⊕ A′_{1}; l⊕\overline{f_{1}}; T′; t; 0; \overline{j′_{2}}−i′_{4}; 0; B; M_{2} ⊕ A_{1}; T; 0; \overline{j_{2}}−i_{4}]

The map \overline{g} represents g and \overline{B′}^{n+2} = \overline{B}^{n+2} = 0.

Remark. In [2] it is proved that nil=Cl by showing that a map ΣF ⟶ X can be represented in terms of short algebras. In fact the ideal kind of proof for nil≤Cat would be along the same lines : perturb a map Z ⟶ Y without changing its cofibre such that it becomes representable in terms of short algebras.

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