2 Preliminaries

2.2 Sullivan models

In this subsection, we assume $\mathbb {K}$ of characteristic zero. The standard reference for this section is [Reference Félix, Halperin and Thomas7].

Let $V=\oplus _{i\geq 1}V^{i}$ be a graded vector space over $\mathbb {K}$ . The free commutative graded algebra over V, denoted $\Lambda V$ , is the quotient of the tensor graded algebra $TV$ by the graded ideal I generated by homogeneous elements of the form $x\otimes y-(-1)^{| x | | y |}y\otimes x$ ; $x, y \in V$ , where $|z|$ denotes the degree of $z\in V$ . Thus, $\Lambda V=\mathrm{Exterior}(V^{\text {odd}})\otimes \,\mathrm{Sym}(V^{\text {even}})=\bigoplus _{i\geq 0}\Lambda ^{i} V$ with $\Lambda ^{i} V$ denoting the linear span of the elements $v_{1}^{n_{1}}\ldots v_{r}^{n_{r}}$ ; ${n_{1}+\cdots +n_{r}=i}$ and $n_{i}=1$ if $v_{i}\in V^{\text {odd}}$ .

A Sullivan algebra is a commutative cochain algebra $(\Lambda V,d)$ generated by the graded vector space $V=\oplus _{i\geq 1}V^{i}$ that is the union of an increasing sequence of graded subspaces $\{V(k)\}_{k\geq 0}$ such that $d=0$ on $V(0)$ and $d: V(k)\rightarrow \Lambda V(k-1),\; k\geq 1$ . In other words, d preserves each $\Lambda V(k)$ and there exists a subspace $V_k\subseteq \Lambda V(k)$ such that $\Lambda V(k) = \Lambda V(k-1)\otimes V_k$ and $d: V_k\rightarrow \Lambda V(k-1),\; k\geq 1$ . Such an algebra is called minimal if moreover:

$$ \begin{align*}\mathrm{Im}(d) \subseteq \Lambda ^+V\otimes \Lambda ^+V.\end{align*} $$

If $(\Lambda V,d)$ is $1$ -connected, that is, $V^{1}=0$ , then $(\Lambda V,d)$ is minimal if and only if

$$ \begin{align*}\mathrm{Im}(d)\subset \Lambda^{\geq 2} V=\oplus_{i\geq 2}\Lambda ^{i} V.\end{align*} $$

A Sullivan model for a cdga $(A,d)$ is a quasi-isomorphism (i.e., a morphism inducing an isomorphism in cohomology):

$$ \begin{align*} m:\, (\Lambda V ,d)\stackrel{\simeq}{\rightarrow}(A,d) \end{align*} $$

from a Sullivan algebra $(\Lambda V ,d)$ . Often we simply say that $(\Lambda V,d)$ is a model of $(A,d)$ .

As shown in the foundational text by Félix, Halperin and Thomas [Reference Félix, Halperin and Thomas7, Proposition 12.2], any commutative cohomologically 1-connected cdga $(A,d)$ (i.e., $H^0(A)=\mathbb K$ and $H^1(A)=0$ ) has a minimal Sullivan model.

If X is a path-connected topological space, a Sullivan model

$$ \begin{align*} m:\, (\Lambda V ,d)\stackrel{\simeq}{\rightarrow} C^{*}(X,\mathbb{K}) \end{align*} $$

of the cochains $C^{*}(X,\mathbb {K})$ on X is called a Sullivan model for X. In particular, if X is $1$ -connected, then it admits a minimal Sullivan model.

A KS-extension of an augmented cdga $\epsilon :\, (A,d)\rightarrow (\mathbb {K},0)$ (i.e., a cdga with the algebra morphism $\epsilon $ sending the ideal of augmentation $A^+= A^{>0}$ to $0$ ) is an exact sequence of cdga morphisms

$$ \begin{align*}(A,d_A)\rightarrow (A\otimes \Lambda V,d_{A\otimes \Lambda V} )\rightarrow (\Lambda V,\overline{d}),\end{align*} $$

where $d_{A\otimes \Lambda V}$ restricts to $d_A$ on A, $dv_{i}\in A\otimes \Lambda V_{<i}$ , and $(\Lambda V,\bar {d})$ is a Sullivan algebra with differential $\bar {d}$ denoting the obvious quotient differential induced by $d_{A\otimes \Lambda V}$ on V.

2.3 Differential Ext

In what follows, we fix $(R,d)$ as a cdga over $\mathbb {K}$ .

A left $(R,d)$ -module $(M,d)$ is said to be semifree if M is the union of an increasing sequence $\{M(k)\}_{k\geq 0}$ of $(R,d)$ -submodules such that $M(0)$ and each of the quotients $M(k)/M(k-1)$ are R-free modules on a basis of cycles. Such an increasing sequence is called a semifree filtration of $(M,d)$ .

A semifree resolution of an $(R,d)$ -module $(A,d)$ is an $(R,d)$ -semifree module $(M,d)$ together with a quasi-isomorphism

$$ \begin{align*} m:\, (M,d)\stackrel{\simeq}{\rightarrow}(A,d) \end{align*} $$

of $(R,d)$ -modules. Recall from [Reference Félix, Halperin and Thomas7, Proposition 6.6] that:

1. (1) Every $(R,d)$ module $(A,d)$ has a semifree resolution $m:\, (M,d)\stackrel {\simeq }{\rightarrow }(A,d)$ .

2. (2) If $m':\, (M',d)\stackrel {\simeq }{\rightarrow }(A,d)$ is a second semifree resolution, then there is an equivalence of $(R,d)$ -modules $\alpha : (M',d)\rightarrow (M,d)$ such that $m \circ \alpha \sim m'$ .

Remark 2.1 Let $\{M(k)\}$ be a semifree filtration of $(M,d)$ . Then $M(0)$ and each $M(k)/M(k-1)$ have the form $(R,d)\otimes (Z(k),0)$ , where $Z(k)$ is a free $\mathbb {K}$ -module and so that each surjection $M(k) \rightarrow R\otimes Z(k)$ splits as follows:

$$ \begin{align*} M(k)=M(k-1)\oplus( R\otimes Z(k)), \text{ and } d:\,Z(k)\rightarrow M(k-1). \end{align*} $$

Thus, forgetting the differentials, we conclude that $M=R\otimes (\bigoplus _{k=0}^{\infty })$ is a free R-module.

Let $\eta :\, (P,d)\rightarrow (Q,d)$ be a morphism of $(R,d)$ -modules, and for any third $(R,d)$ -module $(M,d)$ , denote by

$$ \begin{align*}\begin{array}{lccl} \mathrm{Hom}_{R}(M,\eta): & \mathrm{Hom}_{R}(M,P) & \rightarrow & \mathrm{Hom}_{R}(M,Q)\\ & \varphi & \mapsto & \eta \circ\varphi \end{array} \end{align*} $$

the morphism of complexes induced by $\eta $ . Referring to [Reference Félix, Halperin and Thomas7, Proposition 6.4], if $(M,d)$ is a semifree $(R,d)$ -module and $\eta $ is a quasi-isomorphism, then

• • $\mathrm{Hom}_{R}(M,\eta )$ is a quasi-isomorphism.

• • Given a diagram of morphism of $(R,d)$ -modules,

  (2.1) $$ \begin{align} \begin{array}{c c c} & & (P,d)\\ & & \downarrow \simeq \eta\\ (M,d) & \stackrel{\psi}{\rightarrow} & (Q,d) \end{array} \end{align} $$
  there is a unique homotopy class of morphisms $\phi :\,(M,d) \rightarrow (P,d)$ such that $\eta \circ \phi \sim \psi $ .

• • A quasi-isomorphism between semifree $(R,d)$ modules is an equivalence.

The second property is called the lifting lemma.

Given $f:\,M\rightarrow M'$ and $g:\,P'\rightarrow P,$ two R-linear morphisms between left $(R,d)$ -modules and $h:\, Q\rightarrow Q'$ an R-linear morphism between right $(R,d)$ -modules. We define

$$ \begin{align*} \mathrm{Hom}_{R}(f,g):\, \mathrm{Hom}_{R}(M',P')\rightarrow \mathrm{Hom}_{R}(M,P) \text{ by } \phi \longmapsto (-1)^{\deg f(\deg g+\deg\phi)}g \circ \phi \circ f \end{align*} $$

and

$$ \begin{align*} h\otimes_{R}f:\, Q\otimes_{R}M\rightarrow Q'\otimes_{R}M'\text{ by } q\otimes_{R}m\longmapsto (-1)^{\deg f \deg q}h(q)\otimes_{R}f(m). \end{align*} $$

If moreover f, g, and h commute with the differentials, then $\mathrm{Hom}_{R}(f,g)$ and $h\otimes _{R}f$ are morphisms of graded chain complexes. Furthermore, if $(M,d)$ and $(M',d)$ are $(R,d)$ -semifree, then [Reference Félix, Halperin and Thomas7, Proposition 6.7] the following holds:

1. (1) If f and g are quasi-isomorphisms, then so is $\mathrm{Hom}_{R}(f,g)$ .

2. (2) If f and h are quasi-isomorphisms, then so is $h\otimes _{R}f$ .

Given two $(A,d)$ -modules $(M,d)$ and $(N,d)$ and $(P,d)$ a semifree resolution of $(M,d)$ . The Eilenberg–Moore generalized functor called here the Ext differential is defined as follows:

$$ \begin{align*}\mathrm{Ext}_{(A,d)}((M,d),(N,d))=H(\mathrm{Hom}_{A}(P,N), D). \end{align*} $$

It follows from the previous properties that this functor is independent of the choice of the semifree resolution of $(M,d)$ . More precisely,

$$ \begin{align*}\mathrm{Hom}_{A}(P,N)=\bigoplus_{p\geq 0}\mathrm{Hom}_{A}^{p,*}(P,N)=\bigoplus_{p\geq 0}\bigoplus_{q\geq 0}\mathrm{Hom}_{A}^{p,q}(P,N),\end{align*} $$

where $\mathrm{Hom}_{A}^{p,q}(P,N)=\mathrm{Hom}_{A}(P^{q},N^{p+q})$ has the differential D defined for any $f: P^q\rightarrow N^{p+q}$ , of degree p, by

(2.2) $$ \begin{align} D(f)=d_N\circ f+(-1)^{p+1}f\circ d_P. \end{align} $$

So, we obtain the graduation:

$$ \begin{align*}\mathrm{Ext}_{A}(M,N) &= \oplus_j[\mathrm{Ext}_{A}(M,N)]^j= \oplus_j [\oplus_{p} \mathrm{Ext}_{A}^{p,j-p}(M,N)]\\ & = \oplus_j [\oplus_{p}H^{p,j-p}(\mathrm{Hom}_{A}(P,N))],\end{align*} $$

where $H^{p,j-p}(\mathrm{Hom}_{A}(P,N))$ stands for the cohomology at the middle term in the following complex:

$$ \begin{align*}\dots \stackrel{D}{\rightarrow} \mathrm{Hom}_{A}(P^{j-p},N^{j-1})\stackrel{D}{\rightarrow} \mathrm{Hom}_{A}(P^{j-p},N^{j})\stackrel{D}{\rightarrow} \mathrm{Hom}_{A}(P^{j-p},N^{j+1})\stackrel{D}{\rightarrow} \cdots .\end{align*} $$

p is called the cohomological degree, $q=j-p$ the internal or the complementary degree, and j the total degree.

2.4 Gorenstein spaces and the evaluation map

In this subsection, we consider cohomologically $1$ -connected cdga’s and $1$ -connected spaces.

An augmented cdga $(A,d)$ is called a Gorenstein dga or simply a Gorenstein algebra if the graded $\mathbb {K}$ -vector space $Ext_{A}(\mathbb {K},A)$ is one-dimensional. The generating class $\Omega $ of $\mathrm{Ext}_{A}(\mathbb {K},A)$ and its degree $|\Omega |$ , denoted $fd(A,d)$ , are called the fundamental class and formal dimension of $(A,d), respectively$ .

A pointed topological space X is said to be a Gorenstein space over $\mathbb {K}$ (with formal dimension denoted $fd(X)$ ) if $C^{*}(X,\mathbb {K})$ is a Gorenstein algebra. Referring to [Reference Félix, Halperin and Thomas4], given a quasi-isomorphism $A\stackrel {\simeq }{\rightarrow }B$ of augmented cdga’s, we obtain the following chain of isomorphisms:

(2.3) $$ \begin{align} \mathrm{Ext}_{(A,d)}(\mathbb{K},(A,d))\stackrel{\cong}\rightarrow \mathrm{Ext}_{(A,d)}(\mathbb{K},(B,d))\stackrel{\cong}\longleftarrow \mathrm{Ext}_{(B,d)}(\mathbb{K},(B,d)). \end{align} $$

This composition will be viewed as an identification, that is,

$$ \begin{align*}\mathrm{Ext}_{(A,d)}(\mathbb{K},(A,d))= \mathrm{Ext}_{(B,d)}(\mathbb{K},(B,d)). \end{align*} $$

Thus, if $(\Lambda V,d)$ is a Sullivan model of $(A,d)$ (resp. of X), the latter is Gorenstein over $\mathbb {K}$ if and only if $(\Lambda V,d)$ is. This holds when V is finite dimensional [Reference Murillo17, Theorem A]. If, moreover, $\dim H^{*}(\Lambda V,d)<\infty $ , then $H^{*}(\Lambda V,d)$ satisfies Poincaré duality property over $\mathbb {K}$ and $fd(\Lambda V,d):=N$ is exactly the degree of its fundamental class $[\omega ]$ ; the unique generating element of $H^N(\Lambda V, d)$ [Reference Murillo17, Theorem A]. Recall that a graded algebra A of finite dimension is called of formal dimension m if $A^m\neq 0$ and $A^{>m} =0$ . Such an algebra is said a Poincaré duality algebra or it satisfies Poincaré duality property over $\mathbb {K}$ if it is commutative and satisfies the following conditions:

1. (1) $A^m \cong \mathbb {K}\omega $ .

2. (2) $\forall p, A^p\otimes A^{m-p} \rightarrow A^m \cong \mathbb {K}$ is a nondegenerate bilinear form.

The generating element $\omega $ of degree m is called the fundamental class of A.

Let $(A,d)$ be an augmented cdga and consider $\mathbb {K}$ as a cdga with differential $d_{\mathbb {K}}=0$ . The chain evaluation map

(2.4) $$ \begin{align} \begin{array}{cccc} cev: & \mathrm{Hom}_{(A,d)}((P,d),(A,d)) & \rightarrow & (A,d)\\ & f & \mapsto & f(p) \end{array} \end{align} $$

induces in homology, the evaluation map [Reference Félix, Halperin and Thomas4, Reference Murillo16] is the linear map

(2.5) $$ \begin{align} \begin{array}{cccc} ev_{(A,d)}: & \mathrm{Ext}_{(A,d)}(\mathbb{K},(A,d)) & \rightarrow & H(A,d)\\ & [f] & \mapsto & [f(p)]. \end{array} \end{align} $$

Here, $[f]$ is a cohomology class represented by a cocycle $f: (P,d) \rightarrow (A,d)$ , where $(P,d)\stackrel {\simeq }{\rightarrow } (\mathbb {K},0)$ is an $(A,d)$ -semifree resolution of $\mathbb {K}$ , and p is a cocycle in $(P,d)$ representing $1_{\mathbb {K}}$ . Again, from (2.3), we deduce that $ev_{A}$ is preserved by quasi-isomorphisms.

The evaluation map $ev_{X}$ of a pointed topological space X is by definition that of $C^{*}(X;\mathbb {K})$ . Thus, if $(\Lambda V,d)$ is a minimal Sullivan model of X, we will make the following identification in what follows:

$$ \begin{align*}ev_X = ev_{(\Lambda V,d)}: \mathrm{Ext}_{(\Lambda V,d)}(\mathbb{K},(\Lambda V,d)) \rightarrow H(\Lambda V,d).\end{align*} $$

A result of Murillo [Reference Murillo16, Theorem A] states that when $\dim V<\infty $ ,

$$ \begin{align*}\mathrm{dim}H^{*}(\Lambda V,d)<\infty \Longleftrightarrow ev_{(\Lambda V,d)}\neq0,\end{align*} $$

i.e., a Gorenstein algebra satisfies Poincaré duality if and only if its evaluation map is nonzero.

2.5 Spectral sequences

A cohomology spectral sequence (see [Reference Félix, Halperin and Thomas7, Reference McCleary14]) is a sequence of differential bi-graded complexes $(E_{r}^{*,*},d_{r},\sigma _{r})$ , $r\geq s$ (some integer s) over $\mathbb {K}$ with $E_{r}^{*,*}=\{E_{r}^{p,q}\}$ , $d_{r}$ is a differential of bi-degree $(r,-r+1),$ and $\sigma _{r}:\, H(E_{r}^{*,*})\stackrel {\cong }{\rightarrow } E_{r+1}^{*,*}$ is an isomorphism of bi-graded complexes.

The spectral sequence $(E_{r},d_{r})$ is called convergent if, for each $(p,q),$ there is an integer $r(p,q)$ such that

$$ \begin{align*}E^{p,q}_{n} = E^{p,q}_{r(p,q)},\;\; \forall n\geq r(p,q).\end{align*} $$

In this case, the $\infty $ term of the spectral sequence is the bi-graded complex $E^{*,*}_{\infty } =\oplus _{p,q} E^{p,q}_{\infty }$ , where

$$ \begin{align*}E^{p,q}_{\infty} = E^{p,q}_{r(p,q)},\;\; \forall (p,q).\end{align*} $$

Given a cdga $(A,d)$ . A filtration $FA$ on $(A,d)$ is a family of graded subalgebras $\{F^{p}A\}$ for $p\in \mathbb {Z}$ so that

$$ \begin{align*}\cdots \subseteq F^{p+1}A\subseteq F^{p}A \subseteq F^{p-1}A\subseteq \cdots \subseteq A\end{align*} $$

and $d(F^{p}A) \subseteq F^{p}A$ for all $p\in \mathbb {Z}$ . In particular, d induces a differential on each $F^pA$ so that $(F^pA,d_{F^pA})$ becomes also a differential graded algebra. The filtered cdga $(A, FA, d)$ determines the associated differential bi-graded algebra $(GA,Gd)$ , where

$$ \begin{align*}G^{p,q}A=(F^{p}A/F^{p+1}A)^{p+q}\end{align*} $$

and $Gd$ is the differential naturally induced by d. Here, p is the filtration degree, q is the complementary degree, and $p+q$ is the total degree.

Now, denote (for each integer $r\geq 0$ ) by $Z_{r}^{p}$ and $B_{r}^{p}$ the sub-algebras

$$ \begin{align*}Z_{r}^{p,q} = \{ x\in [F^pA]^{p+q} | dx\in [F^{p+r}(A)]^{p+q+1}\}\end{align*} $$

and

$$ \begin{align*}B_{r-1}^{p,q} = d([F^{p-r+1}A]^{p+q-1})\cap F^pA = d(Z_{r-1}^{p-r+1,q+r-2}).\end{align*} $$

Clearly, $B_{0}^{p}\subset B_{1}^{p}\subseteq \dots B_{r}^{p}\subseteq \cdots \subseteq Z_{r}^{p}\subseteq \cdots \subseteq Z_{0}^{p}.$ The r-term ( $r\geq 0$ ) of this spectral sequence is the bi-graded algebra $E_{r}=\{E_{r}^{p,q}\}$ , where

(2.6) $$ \begin{align} E_{r}^{p,q}=(Z_{r}^{p}/(Z_{r-1}^{p+1}+B_{r-1}^{p}))^{p+q}=Z_{r}^{p,q}/Z_{r-1}^{p+1,q-1} + B_{r-1}^{p,q}. \end{align} $$

It follows from the definition that $(E_{0},d_{0})=(GA,Gd)$ and d factors to define a differential $d_{r}$ in $E_{r}$ of bi-degree $(r,1-r)$ . Moreover, the inclusion $Z_{r+1}^{p}\hookrightarrow Z_{r}^{p}$ induces an isomorphism of bi-graded algebras $E_{r+1}\stackrel {\cong }{\rightarrow } H(E_{r})$ and the filtration $\{F^{p}A\}$ on A induces a filtration on $H(A,d)$ as follows:

$$ \begin{align*}F^{p}H(A,d)=\mathrm{Im}(H(F^{P}A)\rightarrow H(A)).\end{align*} $$

If the associated spectral sequence is convergent and there is an isomorphism $E_{\infty }\cong GH(A)$ of bi-graded algebras then, we say that it converges to $H(A)$ . This is expressed (quite often) by

(2.7) $$ \begin{align} E_2^{p,q} \Longrightarrow H^{p+q}(A,d). \end{align} $$

2.5.1 Milnor–Moore and Eilenberg–Moore spectral sequences

In this section, we consider again cohomologically $1$ -connected cdga algebras $1$ -connected spaces.

Given a minimal Sullivan algebra $(\Lambda V,d)$ with $d=d_{k}+d_{k+1}+\cdots $ (some $k\geq 2$ ), where $d_{i}(V)\subset \Lambda ^{\geq i} V$ ( $\forall i\geq k$ ). Notice that $d_k$ is indeed a differential. We endow $\Lambda V$ with the decreasing filtration:

(2.8) $$ \begin{align} F^{p}\Lambda V=\Lambda^{\geq p} V,\; p\geq 0. \end{align} $$

It is straightforward to verify that $(F^{p}\Lambda V)$ is a decreasing sequence and $dF^{p}\Lambda V \subset F^{p}\Lambda V$ . Consequently, we obtain the so-called (convergent) Milnor–Moore spectral sequence:

(2.9) $$ \begin{align} E_k^{p,q}(\Lambda V) = H^{p,q}(\Lambda V,d_{k})\Longrightarrow H^{p+q}(\Lambda V,d). \end{align} $$

Next (see [Reference Murillo17, Reference Rami19]), on the cdga $A=(\mathrm{Hom}_{\Lambda V}((\Lambda V\otimes \Lambda sV,d),\Lambda V),D)$ , where [Reference Félix, Halperin and Thomas7] $(\Lambda V\otimes \Lambda sV,d)$ stands for the $(\Lambda V, d)$ -semifree resolution of $\mathbb {K}$ with differential d extending that of $\Lambda V$ by $dsv=v-sdv$ , we consider the filtration

(2.10) $$ \begin{align} F^{p}A=\{f\in \mathrm{Hom}_{\Lambda V}((\Lambda V\otimes\Lambda sV,d),\Lambda V)\; |\; f(\Lambda sV)\subset\Lambda^{\geq p} V\},\; p\geq 0. \end{align} $$

Notice that since f is a $(\Lambda V, d)$ -morphism, then, $f\in F^pA$ if and only if $f(\Lambda V \otimes \Lambda sV)\subset \Lambda ^{\geq p} V$ , that is, $f(1)\subset \Lambda ^{\geq p} V$ and $f(\Lambda sV)\subset \Lambda ^{\geq p} V$ . Here again, it is straightforward to see that $(F^{p}A)$ is a decreasing sequence and $dF^{p}A \subset F^{p}A$ . Then, we get the Eilenberg–Moore spectral sequence [Reference Murillo17, Reference Rami19]:

(2.11) $$ \begin{align} E_k^{p,q}(A) = \mathrm{Ext}^{p,q}_{(\Lambda V,d_{k})}(\mathbb{K},(\Lambda V,d_{k})) \Longrightarrow \mathrm{Ext}^{p+q}_{(\Lambda V,d)}(\mathbb{K},(\Lambda V,d)). \end{align} $$

Recall that a Sullivan algebra is called elliptic if $\dim V$ and $\dim H(\Lambda V,d)$ are both finite dimensional. It results from the convergence of these spectral sequences that if $(\Lambda V, d_{k})$ is an elliptic (resp. a Gorenstein) Sullivan algebra, then $(\Lambda V, d)$ is.

2.5.2 Odd and odd-Eilenberg–Moore spectral sequences

To give a convenient characterization of ellipticity, S. Halperin associated [Reference Halperin9] to an arbitrary Sullivan algebra $(\Lambda V,d)$ whose generating space V is finite dimensional, another Sullivan algebra denoted $(\Lambda V,d_{\sigma })$ with differential $d_{\sigma }$ satisfying:

$$ \begin{align*}d_{\sigma}(V^{\text{even}})=0,\; \text{and}\; (d-d_{\sigma})(V^{\text{odd}})\subseteq \Lambda V^{\text{even}}\oplus \Lambda^+V^{\text{odd}} .\end{align*} $$

$(\Lambda V,d_{\sigma })$ is called the associated pure algebra of $(\Lambda V,d)$ . Notice that the second property is equivalent to $d_{\sigma }(V^{\text {odd}})\subseteq \Lambda V^{\text {even}}$ . He then introduced the following spectral sequence:

(2.12) $$ \begin{align} E_k^{p,q}(\Lambda V)(pure) = H^{p,q}(\Lambda V,d_{\sigma})\Longrightarrow H^{p+q}(\Lambda V,d), \end{align} $$

which connects $(\Lambda V,d)$ and $(\Lambda V,d_{\sigma })$ called the odd spectral sequence. Its main result states that $\dim H(\Lambda V,d) < \infty \Leftrightarrow \dim H(\Lambda V,d_{\sigma }) < \infty .$ Thus, $(\Lambda V,d)$ is elliptic if and only if $(\Lambda V,d_{\sigma })$ is.

Another spectral sequence, introduced by the fourth author in [Reference Rami18], is called the odd-Eilenberg–Moore or Ext-odd spectral sequence:

(2.13) $$ \begin{align} E_k^{p,q}(A)(pure) = \mathrm{Ext}^{p,q}_{(\Lambda V, d_{\sigma})}(\mathbb{K},(\Lambda V, d_{\sigma})) \Longrightarrow \mathrm{Ext}^{p+q}_{(\Lambda V, d)}(\mathbb{K},(\Lambda V, d)). \end{align} $$

More details about it will be discussed in §5.

3 A basis of $Ext_{(\Lambda V, d_{\sigma })}(\mathbb {K},(\Lambda V, d_{\sigma }))$

In this section, we construct a generating class of $\mathrm{Ext}^N_{(\Lambda V, d_{\sigma })}(\mathbb {Q},(\Lambda V, d_{\sigma }))$ for a pure Sullivan algebra. The algorithm proceeds in two steps:

1. (i) determination of f on $\Lambda V$ via the evaluation map, and

2. (ii) recursive construction on $\Lambda sV$ by solving successive differential obstructions.

In this section, we assume $\mathbb {K}=\mathbb {Q}$ . Recall that a Sullivan algebra $(\Lambda V,d)$ is said to be pure if $\dim V < \infty $ , $dV^{\text {even}}=0$ and $dV^{\text {odd}}\subseteq \Lambda V^{\text {even}}$ [Reference Félix, Halperin and Thomas7, Reference Halperin9]. If moreover $\dim H(\Lambda V, d)<\infty ,$ we say that $(\Lambda V, d)$ is an elliptic pure Sullivan algebra. Any space whose minimal model is a pure (elliptic) model is called a pure (elliptic) space.

We assume that $(\Lambda V, d)$ is a minimal pure Sullivan algebra, that is, $\dim V < \infty $ and its differential d satisfies $d= \sum _k d_{\geq k}$ (for some fixed $k\geq 2$ ), $dV^{\text {even}}=0$ and $dV^{\text {odd}}\subseteq \Lambda V^{\text {even}}$ .

Let $(x_{1},\ldots ,x_{n})$ be a basis of $V^{\text {even}}$ and $(y_{1}, \ldots ,y_{m})$ be a basis of $V^{\text {odd}}$ . Thus,

$$ \begin{align*}\Lambda V=\mathbb{Q}[x_{1}, \ldots ,x_{n}]\otimes \Lambda(y_{1}, \ldots ,y_{m}),\end{align*} $$

with $dx_{i}=0$ for all $1\leq i\leq n$ and $dy_{j}\in \mathbb {Q}[x_{1}, \ldots ,x_{n}]$ for all $1\leq j\leq m$ . Its formal dimension is given by the formula [Reference Félix, Halperin and Thomas4, Proposition 5.2]:

(3.1) $$ \begin{align} fd(\Lambda V,d) := N =\sum_{j=1}^{m}| y_{j}|+\sum_{i=1}^{n}(1-| x_{i}|). \end{align} $$

A $(\Lambda V, d)$ -semifree resolution of $(\mathbb {Q},0)$ , called an acyclic closure of $(\Lambda V, d)$ (see [Reference Félix, Halperin and Thomas7]), is quasi-isomorphic to the cdga $(\Lambda V\otimes (\mathbb {K}\oplus sV), d)$ with differential (denoted also d) extending that of $(\Lambda V,d)$ as follows:

(3.2) $$ \begin{align} d(sv)=v - s(dv),\; \text{for any} \; v\in V. \end{align} $$

Therefore, being $\dim V < \infty $ , $(\Lambda V, d)$ is a Gorenstein algebra so, there exists a unique generating class $\Omega = [f]$ of $\mathrm{Ext}_{(\Lambda V, d)}(\mathbb {Q},(\Lambda V, d))$ represented by a non-degenerate cocycle morphism $f\in \mathrm{Hom}_{(\Lambda V,d)}((\Lambda V\otimes (\mathbb {K}\oplus sV), d) , (\Lambda V, d))$ . In particular, we have $D(f) = d\circ f + (-1)^{|f|+1} f\circ d = 0$ (cf. (2.2)), that is,

(3.3) $$ \begin{align} d\circ f = (-1)^{|f|} f\circ d. \end{align} $$

The main result of this section is

We now apply this procedure for particular examples.

4 A basis of $\mathrm{Ext}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ from that of $\mathrm{Ext}_{(\Lambda V,d_{k})}(\mathbb {K},(\Lambda V,d_{k}))$

In this section, we transport generators through the Eilenberg–Moore spectral sequence from $(\Lambda V,d_{k})$ to $(\Lambda V,d)$ . We identify the $E_{\infty }$ -term with $\mathrm{Ext}_{(\Lambda V,d)}^N(\mathbb K,(\Lambda V,d))$ , recall the filtration, and apply the Lechuga–Murillo procedure to obtain the corrected generating class $[f_k^t]$ . This yields the explicit computation of the Ginsburg invariant $l_0(X)=t-1$ .

We work with a minimal Sullivan algebra $(\Lambda V,d)$ , where

$$ \begin{align*}d=\sum_{i\geq k}d_i\quad\text{and}\quad d_i(V)\subset\Lambda^{i}V\end{align*} $$

for a fixed $k\geq 2$ . We assume $\dim V<\infty $ and that $(\Lambda V,d)$ is elliptic, while $(\Lambda V,d_k)$ need not be. Both algebras are then Gorenstein and share the same dimension N (see (3.1)).

As recalled in §2, they are linked by the Eilenberg–Moore spectral sequence (2.11). This sequence arises from the filtration

$$ \begin{align*}F^p A=\{g\in A\mid g(\Lambda sV)\subset\Lambda^{\geq p}V\},\end{align*} $$

where $A=\mathrm{Hom}_{\Lambda V}(\Lambda V\otimes \Lambda sV,\Lambda V)$ . The differential on A is

$$ \begin{align*}D_A(f)=d\circ f+(-1)^{|f|+1}f\circ D_{\Lambda V\otimes\Lambda sV}.\end{align*} $$

We write $d_{\Lambda V}$ simply as d, and denote both $D_A$ and $D_{\Lambda V\otimes \Lambda sV}$ by D.

Degree reasons ensure that $d_k$ induces a differential $D_k$ on A. The first term of the spectral sequence is then

$$ \begin{align*}E_k^{p,q}=H^{p,q}\big(\mathrm{Hom}_{\Lambda V}(\Lambda V\otimes\Lambda sV,\Lambda V),D_k\big),\end{align*} $$

where p is the filtration degree and q its complement (so $p+q$ is the total degree).

Conceptual step: Transport via Eilenberg–Moore spectral sequence

We identify the $E_{\infty }$ -term of the convergent Eilenberg–Moore spectral sequence with $\mathrm{Ext}_{(\Lambda V,d)}^N(\mathbb K,(\Lambda V,d))$ . The Lechuga–Murillo procedure then corrects the generator $[f_k]$ of the quadratic level step by step.

Technical verification:

The Gorenstein property of $(\Lambda V,d_k)$ implies a unique bi-degree $(p,q)$ with $p+q=N$ such that $\operatorname {Ext}_{(\Lambda V,d_k)}(\mathbb {K},(\Lambda V,d_k))$ is one-dimensional. Let $[f]$ be its generator.

The Gorenstein property of $(\Lambda V,d_{k})$ implies a unique bi-degree $(p,q)$ with ${p+q=N}$ such that $\mathrm{Ext}_{(\Lambda V,d_{k})}(\mathbb {K},(\Lambda V,d_{k}))$ is one-dimensional. Let $[f]$ be its generator. Using notation of §2, we have $f\in Z^{p,q}_{k-1}$ is the unique cocycle, up to a coboundary. Thus,

(4.1) $$ \begin{align} f(\Lambda sV)\subset (\Lambda^{\geq p} V)^N\; \text{and}\; Df(\Lambda sV)\subset (\Lambda^{\geq p+k-1} V)^{N+1}. \end{align} $$

1. (1) Suppose that $Df=0$ .

   If moreover $f=Dg$ , for some $g\in A^{N-1}$ , then $f=D_{k}g+\sum _{i\geq k+1}D_{i}g$ . So, $f-D_{k}g$ is a $D_k$ cocycle such that $(f-D_{k}g)(\Lambda sV)\subset \Lambda ^{\geq p+1}V$ . Necessarily, $[f-D_{k}g]=0$ in A. Otherwise, it should be a generating class of $\mathrm{Ext}_{(\Lambda V,d_{k})}(\mathbb {K},(\Lambda V,d_{k}))$ . But, being of bi-degree $(p+1, q-1)$ , we get a contradiction to the uniqueness of $(p,q)$ . As a consequence, f is not a coboundary and its class is the desired generator (in this case) of A or equivalently, the spectral sequence (2.11) degenerates at the first term $E_{k}$ .

   The same reasoning will be applied in the spectral sequence of §5, with the differential $d_k$ replaced by $d_{\sigma }$ .

2. (2) Assume that $Df\neq 0$ . Since $Df$ is a $(\Lambda V,d)$ -morphism, it is determined by its restriction to $\Lambda sV$ . Let $h^{0}_{i}$ denote the component mapping $(\Lambda V\otimes \Lambda sV)^{r,s}$ to $(\Lambda V)^{p+N+s+r+i+1,*}$ . For degree reasons, there exists a fixed integer $t\geq 0$ such that

   $$ \begin{align*}Df_{|sV}=h^{0}_{0}+\cdots+h^{0}_{t}\end{align*} $$
   (with $h^0_t\neq 0$ ). Indeed, if $r= max\{|v_i|,\; v_i\in V\}$ , then [Reference Félix, Halperin and Thomas7, Corollary 1, p. 441] gives $r\leq 2N-1$ and one obtains $t\leq {(N+2-2p-2k+r)}/{2}$ .

   This is equivalent to

   (4.2) $$ \begin{align} Df(\Lambda sV)\subseteq \Lambda^{p+k-1}V\oplus\cdots\oplus\Lambda^{p+k-1+t}V. \end{align} $$

   Until the last of section, we consider only the restrictions to $\Lambda sV$ .

   Now, since $D^{2}f=Dh^{0}_{0}+\cdots +Dh^{0}_{t}=0$ and, by word-length argument, $D_{k}h^{0}_{0}=0$ (in fact, $D_{k}h^{0}_{0}(\Lambda sV)\subset \Lambda ^{2(k-1)+p}V$ is the unique term which has this least word-length). But, $|h^{0}_{0}|=N+1>N$ implies that there exist some $\xi _{1}\in \mathrm{Hom}_{\Lambda V}(\Lambda V\otimes \Lambda sV;\Lambda V)$ such that $h^{0}_{0}=D_{k}\xi _{1}$ . Moreover, as $h^{0}_{0}(\Lambda sV)=D_{k}\xi _{1}(\Lambda sV)\subset \Lambda ^{p+k-1}V$ , we see that $\xi _{1}(\Lambda sV)\subset \Lambda ^{p}V$ with total degree $N={p+q}$ .

   We now introduce $f^{1}=f-\xi _{1}$ .

   On one hand, since, $Df^{1} = Df - D\xi _1 = h^{0}_{0}+\cdots +h^{0}_{t} - D_k\xi _1 - (D-D_{k})\xi _1 = (h^{0}_{1}- D_{k+1}\xi _1) + h^{0}_{2} + \cdots + h^{0}_{t} - D_{\geq k+2}\xi _1$ and $D^2f^1 = 0$ , we have, for degree reason, $D_k(h^{0}_{1}- D_{k+1}\xi _1) =0$ . So, being $|h^{0}_{1}- D_{k+1}\xi _1| = N+1$ , there is some $\xi _2\in A^{N}$ such that $h^{0}_{1}- D_{k+1}\xi _1 = D_k\xi _2$ . This can be resumed (for the same t due again for analogous degree reason) as:

   $$ \begin{align*}Df^{1} = h_1^1 + h^{1}_{2}+\cdots+ h^{1}_{t},\end{align*} $$
   where $h_1^1 = h^{0}_{1}- D_{k+1}\xi _1 = D_k\xi _2$ .

   We continue by introducing $f^2 =f^1 - \xi _2 = f - \xi _1 - \xi _2$ and repeat the process until we reach

   $$ \begin{align*}\left\{ \begin{array}{@{}l} f^t = f^{t-1} - \xi_t = f -\xi_1 - \cdots - \xi_t\\ Df^t=h_t^t = h_t^0 - D_{k+t}\xi_1 - D_{k+t-1}\xi_2 - \cdots - D_{k+1}\xi_t. \end{array} \right. \end{align*} $$

   If $Df^t \neq 0$ , we continue the above process to get $Df^t =h_t^t = D_k\xi _{t+1}$ (some ${\xi _{t+1}\in A^N}$ ). But, by definition of $t,$ we have $D_{\geq k+1}(\xi _{t+1})=0$ . Therefore, $Df^t = D\xi _{t+1}$ so that $D(f^t -\xi _{t+1})=0$ . We (should) then take $f^t -\xi _{t+1}$ instead of $f^t$ .

   Then, to simplify, we assume that $Df^t= 0$ and show that $f^t$ is not a coboundary.

   Assume that $f^t =Dg$ (some $g\in A^{N-1})$ . On one hand, since all terms of the spectral sequence (2.11) are one-dimensional (as graded vector spaces), we still have

   $$ \begin{align*}f^t(\Lambda sV)\subseteq \Lambda^{p}V\oplus \cdots \oplus \Lambda^{p+t}V.\end{align*} $$

   Thus, we should have $g(\Lambda sV)\subseteq \Lambda ^{p-k+1}V \oplus \cdots \oplus \Lambda ^{p+t-k+1}V$ so that

   $$ \begin{align*}g =g^0_0 + g_1^0 + \cdots + g_t^0,\end{align*} $$
   where $g_i^0(\Lambda (sV)\subseteq \Lambda ^{p-k+1+i}V$ . On the other hand, by (4.2), we should have
   $$ \begin{align*}f(\Lambda sV)\subseteq \Lambda^{p}V\oplus\cdots\oplus\Lambda^{p+t}V.\end{align*} $$

   Thus, we may decompose f as

   $$ \begin{align*}f =f^0_0 + f_1^0 + \cdots + f_t^0,\end{align*} $$
   where $f_i^0(\Lambda (sV))\subseteq \Lambda ^{p+i}V$ . It follows, using word-length argument, that $f_0^0 - \xi _1 = D_kg^0_0$ , Hence, $D_kf_0^0 - D_k\xi _1 = D_k^2(g_0^0)=0$ and since $D_kf_0^0=0$ (by word length), we get $h_0^0 =D_k\xi _1= 0$ . This contradicts the fact that $f\in F^p(\Lambda V)$ . As a conclusion, $[f^{t}]$ is a generating class of $Ext^*_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ .

Thus, the corrected generating class $[f_k^t]$ of $\mathrm{Ext}_{(\Lambda V,d)}^N(\mathbb {K},(\Lambda V,d))$ has been obtained. By Theorem 1.2, this immediately gives the Ginsburg invariant $l_0(X)=t-1$ , and by Theorem 1.3, we have $\mathrm {R}_0(X)=e_0(X)=\mathrm{cat}_0(X)$ .

Thus, we have established the following

Remark 4.2 From the algorithm inducing $f^t$ , the following formulas arise:

$$ \begin{align*}\left\{ \begin{array}{@{}l} h_0^0 = D_k\xi_1\\ h_1^1 = h_1^0 - D_{k+1}\xi_1= D_k\xi_2\\ h_2^2 = h_2^0 - D_{k+2}\xi_1 - D_{k+1}\xi_2= D_k\xi_3\\ \vdots\\ h_t^t = h_t^0 - D_{k+t}\xi_1 - D_{k+t-1}\xi_2 - \cdots - D_{k+1}\xi_t= D_k\xi_{t+1}. \end{array} \right. \end{align*} $$

We now describe how this is used explicitly to determine $f^t$ (cf. Example 4.2 below).

From the equation $Df(1) = d(f(1)) = h_0^0(1) + h_1^0(1) + \cdots h_t^0(1)$ with $f(1)$ represents the generating class $\omega _{0}$ of $H(\Lambda V,d_k)$ , we deduce the components $h^0_i(1)$ ( $0\leq i\leq t$ ). These are essential in obtaining iteratively the $f^i$ ’s ( $1\leq i\leq t$ ) as follows: we first determine $\xi _{1}(1)$ from the first equation, evaluated at $1\in \mathbb {Q}$ . This determines $\xi _{1}$ on $\Lambda V$ . Then, we determine $\xi _{1}$ on $\Lambda sV$ and therefore, $f^1 = f - \xi _{1}$ on $\Lambda V \otimes \Lambda sV$ . We proceed inductively using the $(i+1)$ -th equation to determine $\xi _{i}$ , hence $f^i = f^{i-1}-\xi _i$ on all $\Lambda V \otimes \Lambda sV$ . As it is shown previously, the process terminates while we reach $f^t$ .

Alternatively, we can apply the algorithm of Lechuga and Murillo [Reference Lechuga and Murillo13, Theorem 5] (or its extension [Reference Boutahir and Rami2]) to determine the $\xi _{i}(1)$ ; the only difference is that the first term is $h_0^0(1)$ instead of $h_1^0(1)$ .

Consequently, from $f^t = f - \xi _1 - \xi _{2} - \cdots - \xi _t$ , we obtain $f^t(1) = f(1) - \xi _1(1) - \xi _{2}(1) - \cdots - \xi _t(1),$ which indeed yields the fundamental class $[f^t(1)] = ev_{(\Lambda V,d)}([f^t])$ of $(\Lambda V,d)$ , that is, the generating class of $H^N(\Lambda V,d)$ .

4.1 Ext-Ginsburg and $\mathrm {R}_0(\Lambda V,d)$ invariants

4.1.1 Ext-Ginsburg invariant

Recall that M. Ginsburg defined [Reference Ginsburg8], in terms of the following topological Milnor–Moore spectral sequence:

(4.3) $$ \begin{align} E_2^{p,q} = \mathrm{Ext}_{H_*(\Omega X, \mathbb{Q})}^{p,q}(\mathbb{Q} , \mathbb{Q}))\Longrightarrow \mathrm{Ext}_{C_*(\Omega X, \mathbb{Q})}^{p+q}(\mathbb{Q} , \mathbb{Q}) \end{align} $$

a homotopy invariant, denoted $l_0(X)$ , to be the largest integer j such that the differential $d_j$ of the j-th term $E_j$ is non-zero. He then showed that $l_0(X) \leq \mathrm{cat}_0(X)$ . For our purpose, we use the algebraic isomorphic spectral sequence (2.9) to give the following equivalent definition of $l_0(X)$ :

$$ \begin{align*}l_0(\Lambda V,d) = \mathrm{sup}\{ j\geq 0\; | \; d_j\neq 0 \}\end{align*} $$

or equivalently:

$$ \begin{align*}l_0(\Lambda V,d) = \mathrm{sup}\{m\; | \; E_{m+1}^{*,*}(\Lambda V) = E_{\infty}^{*,*}(\Lambda V) \}.\end{align*} $$

In other words, $l_0(\Lambda V,d)+1$ designates the stage where the spectral sequence (2.9) degenerate.

Notice that the two definitions agree, since, the first terms from $E_2$ up to $E_{k-2}$ in (2.9) are all identical to the terms $E_0 = (\Lambda V,0)$ and $E_{k-1}\cong (\Lambda V,d_k)$ so that, if $E_{\infty }^{p,q}=E_{s+1}^{p,q}$ in (4.3) and $E_{\infty }^{p,q}(\Lambda V)=E_{t+1}^{p,q}(\Lambda V)$ in (2.9) (where the first term is $E_{k}^{p,q}(\Lambda V)$ ), then $t+1$ should be equal to $(s+1-k +1) +(k-1) =s+1$ .

In the same spirit, we introduced [Reference Acharqy and Rami1] in terms of (2.11) an Ext-version of $l_0(\Lambda V,d)$ as follows:

$$ \begin{align*}\mathrm{L}_0(\Lambda V,d) = \mathrm{sup}\{j\; | \; \delta_j\neq 0\}.\end{align*} $$

(see (4.5) below for the definition of $\delta _j$ ). Clearly, an equivalent definition is

$$ \begin{align*}\mathrm{L}_0(\Lambda V,d) = \mathrm{sup}\{m\; | \; E_{m+1}^{*,*}(A) = E_{\infty}^{*,*}(A)\}.\end{align*} $$

Now, referring to [Reference Rami18, Theorem 1.1], we know that, if X is a simply connected finite-type CW-complex, the spectral sequence (2.11) is isomorphic to the Eilenberg–Moore spectral sequence:

(4.4) $$ \begin{align} E_2^{p,q} = \mathrm{Ext}_{H_*(\Omega X, \mathbb{Q})}^{p,q}(\mathbb{K} , H_*(\Omega X, \mathbb{Q}))\Longrightarrow \mathrm{Ext}_{C_*(\Omega X, \mathbb{Q})}^{p+q}(\mathbb{K} , C_*(\Omega X, \mathbb{Q})). \end{align} $$

Let $d_* =(d_j)_j$ (resp. $\delta _* =(\delta _j)_j$ ) denote the differentials in (4.4) (resp. (2.11)). We then introduce the following invariant:

$$ \begin{align*}\mathrm{L}_0(X) = \mathrm{max}\{j\; | \; d_j\neq 0\} = \mathrm{max}\{m\; | \; E_{m+1}^{*,*} = E_{\infty}^{*,*}\}.\end{align*} $$

Here, once again, $\mathrm {L}_0(\Lambda V,d) $ (resp. $\mathrm {L}_0(X)$ ) is the order minus one at which the spectral sequence (2.11) (resp.(4.4)) degenerate. We call each of these the Ext-Ginsburg invariant.

Next, we present an equivalent algorithm, based on spectral sequence argument, to obtain the generating class $[f^t]$ of $\mathrm{Ext}^{p+q}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ from that of $\mathrm{Ext}^{p,q}_{(\Lambda V,d_{k})}(\mathbb {K};(\Lambda V,d_{k}))$ denoted here again by $[f]$ .

Recall from §2 that $f\in Z^{p,q}_{k-1}$ . That is, $f(\Lambda sV)\subset \Lambda ^{\geq p} V$ and $Df(\Lambda sV)\subset (\Lambda ^{\geq p+k-1}V)^{N+1}$ . Clearly, if $D(f)=0,$ then, as in the above algorithm, $[f]$ generates $\mathrm{Ext}^{p+q}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ , that is, (2.11) collapses at the first term. We then assume that $Df\neq 0$ .

The decomposition $Df = h_0^0 + h_1^0 + \cdots + h_t^0$ and the condition $D^2f=0$ lead us to introduce $f^1 = f - \xi _{1}$ , where $\xi _1$ is the solution of $h_0^0 =D_k(\xi _1)$ . This implies that $\xi _{1}(sV)\subseteq \Lambda ^{p}V$ , hence,

1. (1) $f^{1}(sV) = (f-\xi _{1})(sV)\subseteq \Lambda ^{\geq p}V$ .

2. (2) $Df^{1}(sv) = D(f-\xi _1)(sV) = [h^{0}_{1}+ \cdots + h^{0}_{t}-(D-D_{k})\xi _{_{1}}](sV)\subseteq \Lambda ^{\geq p+k}V$ .

Therefore, $f^{1}\in Z^{p,q}_{k}.$

To continue, recall from §2 that the differential D of $A=\mathrm{Hom}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ induces on $E_k^{*,*}$ the differential

(4.5) $$ \begin{align} \begin{array}{llll} \delta _k : & E_k^{p,q} & \rightarrow & E_k^{p+k,q-k+1}\\ & [h]_k & \mapsto & [Dh]_k. \end{array} \end{align} $$

Applying this to $h= f^{1}$ , we obtain

$$ \begin{align*}\delta _k[f^{1}]_k = [Df^{1}]_k = [h^{0}_{1}- D_{k+1}\xi_1]_k.\end{align*} $$

Now, we show that $[f^{1}]_k$ is a cocycle representing a non zero class in $(E_{k+1}^{*,*}, \delta _k)$ .

On one hand, since,

(4.6) $$ \begin{align} Df^{1} = h^{0}_{0}+\cdots+h^{0}_{t} - D_k\xi_1 - (D-D_{k})\xi_1 = (h^{0}_{1}- D_{k+1}\xi_1) + h^{0}_{2} + \cdots + h^{0}_{t} - D_{\geq k+2}\xi_1 \end{align} $$

and $D^2f^1 = 0,$ we have, by degree reason, $D_k(h^{0}_{1}- D_{k+1}\xi _1) =0$ . So, being $|h^{0}_{1}- D_{k+1}\xi _1| = N+1$ , there is some $\xi _2\in A$ such that $h^{0}_{1}- D_{k+1}\xi _1 = D_k\xi _2$ . Therefore, using the isomorphism $E_k^{p+k,q-k+1} \cong H^{p+q+1}(A,D_k)$ , we deduce that $\delta _k[f^{1}]_k=0$ and $[f^{1}]_k$ represents a cohomology class in $E_{k+1}^{p,q}$ .

On the other hand, if $[f^1]_k = \delta _k[g]_k= [Dg]_k$ , for some $g\in Z_k^{p-k+1,q+k-2}$ , then ${[f^1 - Dg]_k =0}$ in $E_k^{p,q}$ or equivalently, using (2.6), the component $pr_{-k+1}\circ (f^1 - Dg)$ in $\Lambda ^p V$ of $f^1 - Dg$ , which we denote by $(f^1 - Dg)_{\Lambda ^p V}$ , decomposes as $(h_1 + Dh_2)_{\Lambda ^p V}$ in $Z^{p+1,q-1}_{k-1} + B_{k-1}^{p,q}$ . Clearly, only the first term $D_k$ of D acts and then we have ${(f^1 - D_kg)_{\Lambda ^p V}= (D_k{h_2})_{\Lambda ^p V}}$ . Hence, applying $D_k$ , we get $D_kf^1_{\Lambda ^pV} = D_kf_{\Lambda ^pV} - D_k{\xi _1}_{\Lambda ^pV}= - D_k{\xi _1}_{\Lambda ^pV}=0,$ that is, $h_0^0 =0$ . This contradicts the fact that $f\in Z_{k-1}^{p,q}$ , hence $[f^{1}]_k$ is indeed a cocycle representing a non zero class in $(E_{k+1}^{*,*}, \delta _k)$ .

We continue by reconsidering the above expression of $Df^1$ which actually is as follows:

$$ \begin{align*}Df^{1} = D_k\xi_2 + h^{0}_{2}+\cdots+ h^{0}_{t} - D_{\geq k+2}\xi_1.\end{align*} $$

This can be resumed (for the same t due again to degree reason) as

$$ \begin{align*}Df^{1} = h_1^1 + h^{1}_{2}+\cdots+ h^{1}_{t},\end{align*} $$

where $h_1^1 = h^{0}_{1}- D_{k+1}\xi _1 = D_k\xi _2$ .

We then introduce $f^2 =f^1 - \xi _2 = f - \xi _1 - \xi _2$ and show, using the same argument, that $[f^{2}]_k$ is a cocycle representing a non zero class in $E_{k+2}^{p,q}$ . Moreover, we may write

$$ \begin{align*}Df^{2} = h_2^2 + h^{2}_{3}+\cdots+ h^{2}_{t},\end{align*} $$

where $h_2^2 = h^{2}_{0}- D_{k+2}\xi _1 - D_{k+1}\xi _2$ .

We continue this process until we reach the stage s where the spectral sequence degenerates. This induces the generating class of $E_{\infty }^{p,q} =E_{s}^{p,q}$ represented by $f^s = f - \xi _1 - \xi _2 - \cdots - \xi _s$ . Thus, using the aforementioned identification $E_{\infty }^{p,q} = \mathrm{Ext}_{(\Lambda V,d)}^{p+q}(\mathbb {K},(\Lambda V,d))$ , we take $[f^s]$ as the required generating class of $\mathrm{Ext}_{(\Lambda V,d)}^{p+q}(\mathbb {K},(\Lambda V,d))$ .

As an application, we present below an immediate proof of [Reference Acharqy and Rami1, Theorem 2] which we recall here for convenience.

Theorem 4.3 Let X be a rationally elliptic space. Then, the spectral sequence (4.4) degenerate at t, hence, $L_0(X) = l_0(X) = t-1$ .

Proof 2 The above algorithm shows that the spectral sequence (2.11) degenerates exactly at $s=t$ , hence $L_0(X) = t-1$ . Because the evaluation map $ev_{(\Lambda V,d)}([f^t]) = [f^t(1)]$ coincides with the fundamental class obtained from the Milnor–Moore spectral sequence (2.9) via the Lechuga–Murillo algorithm [Reference Boutahir and Rami2, Theorem 2] (see Remark 4.2), the same index t governs the degeneration of (2.9). Consequently, $l_0(X)=t-1$ as well.

4.1.2 The invariant $\mathrm {R}_0(\Lambda V,d)$

Recall from [Reference Rami18] that the second author introduced, in terms of (2.11), the invariant

$$ \begin{align*}\mathrm{R}_0(\Lambda V,d) = \mathrm{sup}\{m\; | \; E_{\infty}^{m,*} \neq 0\}.\end{align*} $$

This is given in spirit of Toomer’s invariant which in turn is given in terms of (2.9) by

$$ \begin{align*}e_0(\Lambda V,d) := \mathrm{sup}\{m\; | \; E_{\infty}^{m,*} \neq 0\}.\end{align*} $$

An equivalent definition of $\mathrm {R}_0(\Lambda V,d)$ given in [Reference Rami18, Remark 3.4], in terms of the generating class $\Omega $ of $\mathrm{Ext}^{*}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ , is as follows:

$$ \begin{align*}\mathrm{R}_0(\Lambda V,d) = \mathrm{sup}\{m\; | \; \Omega \; \text{can be represented by a cocycle in}\; F^m(A)\}.\end{align*} $$

Our first result in this section gives another proof of [Reference Acharqy and Rami1, Theorem 1] which relies on the algorithm defining $f^t$ .

Theorem 4.4 Let X be a rationally elliptic CW-complex. Then, $\mathrm {R}_0(X)=e_0(X)=\mathrm{cat}_0(X).$

Proof 3 By hypothesis, X being elliptic, its Sullivan minimal model $(\Lambda V,d)$ is elliptic hence $\dim V < \infty $ and $\dim H(\Lambda V,d)<\infty $ . In particular, $(\Lambda V,d_k)$ is a Gorenstein algebras, that is,

$$ \begin{align*}\mathrm{Ext}^{*,*}_{(\Lambda V,d_k)}(\mathbb{K},(\Lambda V,d_k)) = \mathrm{Ext}^{p',q'}_{(\Lambda V,d_k)}(\mathbb{K},(\Lambda V,d_k)),\end{align*} $$

where the unique bi-degree $(p',q')$ derived from the filtration (2.10) satisfies $p'+q'=N$ , the formal dimension of $(\Lambda V,d)$ . By the convergent spectral sequence (2.11), we have also

$$ \begin{align*}\mathrm{Ext}^{*,*}_{(\Lambda V,d)}(\mathbb{K},(\Lambda V,d)) = \mathrm{Ext}^{p',q'}_{(\Lambda V,d)}(\mathbb{K},(\Lambda V,d)).\end{align*} $$

This implies $\mathrm {R}_0(\Lambda V,d) = p' = \mathrm {R}_0(X)$ (cf. §6 for more explanation).

Next, since the chain evaluation map (2.4):

$$ \begin{align*}\begin{array}{cccc} cev: & A=\mathrm{Hom}_{(\Lambda V,d)}((\Lambda V \otimes \Lambda sV, D) , (\Lambda V,d)) & \rightarrow & (\Lambda V,d)\\ & f & \mapsto & f(1) \end{array} \end{align*} $$

preserves the filtrations (2.10) and (2.8), in homology, it induces a morphism between the convergent spectral sequences (2.11) and (2.9). Hence, the $E_\infty $ morphism

$$ \begin{align*}E_{\infty}(ev): E_{\infty}^{p',q'}(A)\cong \mathrm{Ext}^{p',q'}_{(\Lambda V,d)}(\mathbb{K},(\Lambda V,d)) \rightarrow E_{\infty}^{p',q'}(\Lambda V)\cong H^{p'+q'}(\Lambda V,d)\end{align*} $$

is indeed an isomorphism (recall that $\dim H^{N}(\Lambda V,d)=1$ ). By the above theorem, $L_0(\Lambda V,d) = l_0(\Lambda V,d)$ therefore, by (1.1), we obtain $e_0{(\Lambda V,d)} = p'=e_0(X)$ and thus, $\mathrm {R}_0(X)=e_0(X)=\mathrm{cat}_0(X).$

To illustrate, we apply the algorithm to a specific Sullivan model.

Example 4.5 Let us reconsider the Sullivan model from example 3.1: $(\Lambda x_{2},x_{4},y_{5},y_{7}, d)$ , with differentials

$$ \begin{align*}dx_{2}=dx_{4}=0,\quad dy_{5}=x_{2}^{3}-2x_{2}x_{4},\quad dy_{7}=x_{4}^{2}-x_{2}^{2}x_{4}.\end{align*} $$

Its formal dimension is $N=8$ . The quadratic associated model $(\Lambda V,d_{2})$ is a pure but non-elliptic, with top class $\omega =[x_{2}^{2}x_{4}]$ and $H^8(\Lambda V,d_{2})=\mathbb Q[x_2^4]$ . We apply the algorithm of §4 to the Eilenberg–Moore spectral sequence

$$ \begin{align*}\mathrm{Ext}^{(2,6)}_{(\Lambda V,d_{2})}(\mathbb{K};\,(\Lambda V,d_{2}))\;\Longrightarrow\; \mathrm{Ext}^{8}_{(\Lambda V,d)}(\mathbb{K};\,(\Lambda V,d)).\end{align*} $$

Starting from the generating class $\Omega _2=[f]$ of $\mathrm{Ext}^{(2,6)}_{(\Lambda V,d_{2})}(\mathbb {K};\,(\Lambda V,d_{2}))$ obtained in Example 3.2 (with $f(1)=2x_4^2$ ), the Lechuga–Murillo procedure yields, after three successive corrections $\xi _1,\xi _2,\xi _3$ , a generating class $[f^3]$ of $\mathrm{Ext}^{8}_{(\Lambda V,d)}(\mathbb {K};\,(\Lambda V,d))$ such that $Df^3=0$ and $[f^3]$ is not a D-coboundary. Consequently, $t=3$ , $L_0(X)=t-1=2,$ and

$$ \begin{align*}e_0(\Lambda V,d)=e_0\!\left([x_2^4]\right)=4=cat_0(X).\end{align*} $$

5 A basis of $\mathrm{Ext}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ from that of $Ext_{(\Lambda V,d_{\sigma })}(\mathbb {K};(\Lambda V,d_{\sigma }))$

In this section, we lift the generating class from the pure model $(\Lambda V,d_{\sigma })$ to the full model $(\Lambda V,d)$ using the odd-Eilenberg–Moore spectral sequence. The proof of Theorem 1.1 is completed here, showing that $[(f_{k,\sigma }^t)^l(1)]$ is precisely the fundamental class of $(\Lambda V,d)$ .

Let $(\Lambda V,d)$ be a minimal Sullivan model with finite-dimensional V. Following Halperin [Reference Félix, Halperin and Thomas7, Reference Halperin9], we introduce a bi-grading on $\Lambda V$ by setting $Q= V^{\text {even}}$ and $P = V^{\text {odd}}$ . Specifically,

$$ \begin{align*}(\Lambda V)^{q+n,-q} = (\Lambda Q \otimes \Lambda^q P)^n,\end{align*} $$

where n is the total degree and $p=n+q$ the filtering degree. The filtration is

$$ \begin{align*}F^{p}(\Lambda V) = (\Lambda V)^{\geq p, *}.\end{align*} $$

This filtration induces a convergent spectral sequence whose $E_0$ -term is the pure Sullivan model $(\Lambda V,d_{\sigma })$ . The differential $d_{\sigma }$ satisfies

(5.1) $$ \begin{align} dv=d_{\sigma}(v)+\phi(v),\; \phi(v) \in \Lambda^{+} V^{\text{odd}}\otimes \Lambda V^{\text{even}} \end{align} $$

with $d_{\sigma }(V^{\text {even}})=0$ and $d_{\sigma }(V^{\text {odd}})\subseteq \Lambda V^{\text {even}}$ . The associated spectral sequence

(5.2) $$ \begin{align} E_2^{p,-q} = H^{p,-q} (\Lambda V,d_{\sigma}) \Longrightarrow H^{p-q}(\Lambda V,d) \end{align} $$

is called the odd spectral sequence. Its main consequence is that $(\Lambda V,d)$ is elliptic if and only if $(\Lambda V,d_{\sigma })$ is elliptic. Thus, as already noted in §2, any Sullivan model satisfying the above condition is called a pure Sullivan model.

We now consider the acyclic closure $(\Lambda V\otimes \Lambda sV, d)\stackrel {\simeq }{\rightarrow } (\mathbb {Q},0)$ . The differential D on $A=\mathrm{Hom}_{\Lambda V}(\Lambda V\otimes \Lambda sV;\Lambda V)$ is

$$ \begin{align*}D(f)= f\circ d_{\Lambda V \otimes \Lambda sV} + (-1)^{|f|+1}d_{\Lambda V}\circ f.\end{align*} $$

The pure differential $d_{\sigma }$ induces $D_{\sigma }$ on A defined by

$$ \begin{align*}D_{\sigma}(f)= f\circ {d_{\sigma}}_{\Lambda V \otimes \Lambda sV} + (-1)^{|f|+1}{d_{\sigma}}_{\Lambda V}\circ f.\end{align*} $$

Therefore, we obtain the relation $D(f)=D_{\sigma }(f)+\varphi (f),\; \forall f\in A$ with $\varphi (f) = (D-D_{\sigma })(f)$ .

In [Reference Rami18], denoting

$$ \begin{align*}(\Lambda V\otimes\Lambda sV)^{r,s}=\bigoplus_{r=a_{1}+a_2+2a_{3}}(\Lambda Q\otimes \Lambda^{a_{1}}sQ \otimes \Lambda ^{a_2}P\otimes \Lambda^{a_{3}}sP)^{s},\end{align*} $$

the second author introduced on $A^n=\mathrm{Hom}_{\Lambda V}^n(\Lambda V\otimes \Lambda sV,\Lambda V)$ ( $n\geq 0$ ) the following filtration:

(5.3) $$ \begin{align} F^{p}A^{n}=\bigoplus_{r,s}\mathrm{Hom}_{\Lambda V}((\Lambda V\otimes\Lambda sV)^{r,s},(F^{p+n+s+r}(\Lambda V))^{n+s}). \end{align} $$

That is, $f\in F^{p}A^{n}$ if and only if $f((\Lambda V\otimes \Lambda sV)^{r,s}) \subseteq (\Lambda ^{\geq p+n+s+r, *}V)^{n+s}$ . Note that the differential on $(\Lambda V\otimes \Lambda sV)^{r,s}$ preserves both s (total degree) and r (filtering degree). Thus, he obtains the following convergent spectral sequence, called the Ext-odd spectral sequence of $(\Lambda V,d)$ :

(5.4) $$ \begin{align} \mathrm{Ext}^{p,-q}_{(\Lambda V,d_{\sigma})}(\mathbb{K},(\Lambda V,d_{\sigma}))\Longrightarrow \mathrm{Ext}^{p-q}_{(\Lambda V,d)}(\mathbb{K},(\Lambda V,d)). \end{align} $$

Since V is finite-dimensional, both $(\Lambda V,d)$ and $(\Lambda V,d_{\sigma })$ are Gorenstein dga’s with the same formal dimension N. Consequently, there exists a unique bi-degree $(p,q)$ such that $\mathrm{Ext}_{(\Lambda V,d_{\sigma })}^{*,*}(\mathbb {K},(\Lambda V,d_{\sigma }))= \mathrm{Ext}^{(p,-q)}_{(\Lambda V,d_{\sigma })}(\mathbb {K},(\Lambda V,d_{\sigma }))$ is one-dimensional. Recall also that $A:= \mathrm{Hom}_{\Lambda V}(\Lambda V\otimes \Lambda sV,\Lambda V).$

Following the approach for the spectral sequence (2.11), we proceed to determine a generating class $[h]$ of $\mathrm{Ext}^{p-q}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ from that, say $[f]$ , of $\mathrm{Ext}^{p,-q}_{(\Lambda V,d_{\sigma })}(\mathbb {K},(\Lambda V,d_{\sigma }))$ . Using notation of §§1.4, we have $f\in Z_{1}^{p,-q}$ and $|f|=N= p-q$ , that is, $f(\Lambda V\otimes \Lambda sV)^{r,s}\in F^{p+N+r+s}(\Lambda V)$ and $Df(\Lambda V\otimes \Lambda sV)^{r,s}\subset F^{p+N+r+s+1}(\Lambda V).$

• • First, assume $Df=0$ . If, moreover, $f = Dg$ , for some $g\in A^{N-1}$ , then $f-D_{\sigma }(g) = (D-D_{\sigma })(g)$ is a $D_{\sigma }$ -cocycle. Now, as $|g|=N-1$ and $d_{| {\Lambda V \otimes \Lambda sV}}$ sends $(\Lambda V\otimes \Lambda sV)^{r,s}$ to $(\Lambda V\otimes \Lambda sV)^{r-1,s+1}\oplus (\Lambda V\otimes \Lambda sV)^{\geq r,s+1}$ , g should send ${(\Lambda V\otimes \Lambda sV)^{r-1,s+1}}$ to $F^{p+N+r+s}(\Lambda V)=F^{(p+1)+(N-1)+(r-1)+(s+1)}(\Lambda V)$ . This yields the following diagram:

  (5.5) $$ \begin{align} \begin{array}{ccc} (\Lambda V\otimes \Lambda sV)^{r,s} & \stackrel{{d_{\sigma}}_{|{\Lambda V \otimes \Lambda sV}}}{\longrightarrow} & (\Lambda V\otimes \Lambda sV)^{r-1,s+1} \\ \downarrow g & & g \downarrow \\ F^{(p+1)+(N-1)+r+s}(\Lambda V ) & \stackrel{{d_{\sigma}}_{|{\Lambda V}}}{\longrightarrow} & F^{p+N+r+s}(\Lambda V). \end{array} \end{align} $$

  Therefore, $D_{\sigma }(g) = g\circ {d_{\sigma }}_{|{\Lambda V \otimes \Lambda sV}} + (-1)^N {d_{\sigma }}_{|{\Lambda V}}\circ g$ sends $(\Lambda V\otimes \Lambda sV)^{r,s}$ to $F^{p+N+r+s}(\Lambda V)$ . The same analysis give us the following diagram:

  (5.6) $$ \begin{align} \begin{array}{ccc} (\Lambda V\otimes \Lambda sV)^{r,s} & \stackrel{{(d-d_{\sigma})}_{|{\Lambda V \otimes \Lambda sV}}}{\longrightarrow} & (\Lambda V\otimes \Lambda sV)^{\geq r,s+1} \\ \downarrow g & & g \downarrow \\ F^{(p+1)+(N-1)+r+s}(\Lambda V) & \stackrel{{(d-d_{\sigma})}_{|{\Lambda V}}}{\longrightarrow} & F^{>p+N+r+s}(\Lambda V). \end{array} \end{align} $$

  It follows that $f-D_{\sigma }(g) = (D-D_{\sigma })(g)$ sends $(\Lambda V\otimes \Lambda sV)^{r,s}$ to $F^{(p+1)+N+r+s}(\Lambda V)$ . So, if $f-D_{\sigma }(g)$ is not a $D_{\sigma }$ -coboundary, then $[f-D_{\sigma }(g)]$ becomes a generating class of $ \mathrm{Ext}^{*,*}_{(\Lambda V,d_{\sigma })}(\mathbb {K},(\Lambda V,d_{\sigma }))$ of bi-degree $(p+1,q-1)$ , which contradicts uniqueness of $(p,q)$ . We then conclude that if $Df=0,$ then $[f]$ is a generating class of $ \mathrm{Ext}^{*,*}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d))$ .

• • Second, assume that $Df\neq 0$ . Decompose $Df$ as in the previous section:

  $$ \begin{align*} Df_{|(\Lambda V\otimes \Lambda sV)^{r,s}} = g_{0}^{0} + \cdots + g_{l}^{0}, \end{align*} $$
  where l is the greatest integer with $g_{l}^{0}\neq 0$ and each $g_i^0$ sends $(\Lambda V\otimes \Lambda sV)^{r,s}$ to $(\Lambda V )^{p+N+s+r+i+1,*}$ . Let $m= \mathrm{max}\{|v_i|,\; v_i\in V\}$ ; by [Reference Félix, Halperin and Thomas7, Corollary 1, p. 441], we have $2\leq m\leq 2N-1$ . The degree condition yields $l\leq \frac {-N+m}{2}-(r+s+p+1)$ .

  This decomposition extends to $Df$ so that:

  (5.7) $$ \begin{align} Df=h_{0}^{0}+\cdots+h_{l}^{0}. \end{align} $$

  As in the above section, this is equivalent to:

  (5.8) $$ \begin{align} Df((\Lambda V\otimes\Lambda sV)^{r,s})\subseteq (\Lambda V)^{p+N+s+r+1,*} \oplus\cdots\oplus (\Lambda V)^{p+N+s+r+l+1,*}. \end{align} $$

  Now, using the fact $D^{2}f=0$ and by considering word length, we get ${D_{\sigma }(h_{0}^{0})=0}$ . Moreover, as $|h_0^0| = |Df| = N+1$ , we have $[h_0^0] =0,$ hence, $h_{0}^{0}=D_{\sigma }\beta _{1}$ for some $\beta _{1} \in A^N$ .

  The iterative construction of the correctors $\beta _j$ is analogous to the Lechuga–Murillo procedure described in §4, with the filtration (5.3) playing the role of the filtration (2.10). We therefore only sketch the main steps.

  To continue, we consider $f^{1}=f-\beta _{1}$ . Since, $|\beta _1|=N$ , $h_{0}^{0}=D_{\sigma }\beta _{1}$ , and $h_0^0(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+1}(\Lambda V )$ , we have $\beta _1(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+1}(\Lambda V )$ and ${(D-D_{\sigma })(\beta _1)(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+2}(\Lambda V )}$ (compare with g and $(D-D_{\sigma })(g)$ in the diagrams (5.5) and (5.6), respectively). It follows that $f^{1}(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+1}(\Lambda V )$ . Afterwards, since

  $$ \begin{align*}Df_1 =Df - D\beta_1 = (D-D_{\sigma})(f) - h_0^0 - (D-D_{\sigma})(\beta_1) = h_1^0 + \cdots + h_l^0 -(D-D_{\sigma})(\beta_1),\end{align*} $$
  we deduce that $Df^{1}(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+2}(\Lambda V)$ .

  Next, expressing $F^{p+N+r+s+2}(\Lambda V ) = (\Lambda V) ^{\geq p+N+r+s+2, *}$ as the sum:

  $$ \begin{align*}F^{p+N+r+s+2} = (\Lambda V) ^{p+N+r+s+2, *} \oplus (\Lambda V) ^{p+N+r+s+3, *} \oplus \cdots \oplus (\Lambda V) ^{ p+N+r+s+l+1, *}(\Lambda V )\end{align*} $$
  we see that $(D-D_{\sigma })(\beta _1)$ may be decomposed as follows:
  $$ \begin{align*}(D-D_{\sigma})(\beta_1) = \beta_1^{(1)} + \beta_1^{(2)} + \cdots + \beta_1^{(l)},\end{align*} $$
  where $\beta _1^{(i)}$ , ( $1\leq i \leq l$ ), is the component of $(D-D_{\sigma })(\beta _1),$ which sends ${(\Lambda V\otimes \Lambda sV)^{r,s}}$ to $(\Lambda V)^{p+N+r+s+i+1, *}$ . Therefore, we may write
  $$ \begin{align*}Df^1 = h_1^1 + h_2^1 = \cdots + h_l^1\end{align*} $$
  with $h_1^1 = h_1^0 - \beta _1^{(1)}$ . We then continue by analyzing the behaviors of $f^1$ .

  Again, using the equation $D^2f^1=0$ , we obtain some $\beta _2\in A^{N}$ such that $h_1^1 = D_{\sigma }\beta _2$ . This leads to $f^2 = f^1-\beta _2 = f -\beta _1 - \beta _2$ . Similarly, we may show that $\beta _2(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+2}(\Lambda V )$ and $(D-D_{\sigma })(\beta _2)(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+N+r+s+3}(\Lambda V )$ , so that $(D-D_{\sigma })(\beta _2)$ can be expressed as follows:

  $$ \begin{align*}(D-D_{\sigma})(\beta_2) = \beta_2^{(2)} + \beta_2^{(3)} + \cdots + \beta_2^{(l)},\end{align*} $$
  where $\beta _2^{(i)}$ , ( $2\leq i \leq l$ ) is the component of $(D-D_{\sigma })(\beta _2),$ which sends ${(\Lambda V\otimes \Lambda sV)^{r,s}}$ to $(\Lambda V) ^{p+N+r+s+i+1, *}$ .

  Continuing this process, we inductively define a sequence $(f^{j}, \beta _{j})\in A^N \times A^N$ for $1\leq j \leq l$ with $f^0 := f$ :

  $$ \begin{align*}\left \{ \begin{array}{l}\!\! f^{j}=f^{j-1}-\beta_{j}= f -\beta_1 -\beta_2 - \cdots - \beta_j\\ \!\! Df^{j}(\Lambda V\otimes \Lambda sV)^{r,s}\subseteq F^{p+r+s+N+j+1,*}\\ \!\! Df^j = h_j^j + h_{j+1}^j = \cdots + h_l^j. \end{array} \right. \end{align*} $$

  It remains to show that $f^{l} = f -\beta _1 -\beta _2 - \cdots - \beta _l$ is a cocycle representing the generating class of $\mathrm{Ext}^{p-q}_{(\Lambda V,d)}(\mathbb {K},(\Lambda V,d)).$

  If $Df^l \neq 0$ , we proceed using the same procedure as in the previous section to obtain $Df^l =h_l^l = D_\sigma \beta _{l+1}$ (for some $\beta _{l+1}\in A^N$ ). Additionally, based on the definition of l, we find that $(D-D_{\sigma })(\beta _{l+1})=0$ . Consequently, $Df^l =h_l^l = D\beta _{l+1}$ which implies $D(f^l -\beta _{l+1})=0$ . Subsequently, we may substitute $f^l -\beta _{l+1}$ for $f^l$ .

  Next, we assume for simplicity that $Df^l= 0$ and show that $f^l$ is not a coboundary.

  Assuming the contrary, i.e., there is some $g\in A^{N-1}$ such that $f^l =Dg$ .

  On one hand, using (5.8), we see that

  $$ \begin{align*}f((\Lambda V\otimes\Lambda sV)^{r,s})\subseteq (\Lambda V)^{p+N+s+r,*} \oplus \cdots \oplus (\Lambda V)^{p+N+s+r+l,*}.\end{align*} $$

  Therefore, f decomposes as

  $$ \begin{align*}f =f^0_0 + f_1^0 + \cdots + f_l^0,\end{align*} $$
  where $f_i^0(\Lambda V\otimes \Lambda sV)^{r,s})\subseteq (\Lambda V)^{p+N+s+r+i,*}$ .

  On the other hand, since all terms of the spectral sequence (2.13) are one-dimensional (as graded vector spaces), we still have

  $$ \begin{align*}f^l((\Lambda V\otimes\Lambda sV)^{r,s})\subseteq (\Lambda V)^{p+N+s+r,*} \oplus \cdots \oplus (\Lambda V)^{p+N+s+r+l,*}.\end{align*} $$

  Thus, we should have

  $$ \begin{align*}g((\Lambda V\otimes\Lambda sV)^{r,s})\subseteq (\Lambda V)^{p+N+s+r-1,*} \oplus \cdots \oplus (\Lambda V)^{p+N+s+r+l-1,*}\end{align*} $$
  so g also decomposes as
  $$ \begin{align*}g =g^0_0 + g_1^0 + \cdots + g_l^0\end{align*} $$
  with $g_i^0((\Lambda V\otimes \Lambda sV)^{r,s})\subseteq (\Lambda V)^{p+N+s+r+i-1,*}$ .

  It results, due to degree constraints, that $f_0^0 - \beta _1 = D_\sigma (g_0^0)$ . Hence, applying $D_{\sigma }$ , we get $D_\sigma \beta _1 = 0$ . That is, $h_0^0 =0$ . This contradicts (5.8) or equivalently the fact that $f\in F^p(A^N)$ and concludes the algorithm.

Remark 5.1 A similar description given in Remark 4.2 may be used in terms of the following system of equations to determine $f^l$ :

$$ \begin{align*}\left \{ \begin{array}{@{}l} h_0^0 = D_{\sigma}\beta_1\\ h_1^1 = h_1^0 - \beta_1^{(1)}\\ h_2^2 = h_2^0 - \beta_1^{(2)} - \beta_2^{(2)}\\ \vdots\\ h_l^l = h_l^0 - \beta_1^{(l)} - \beta_2^{(l)} - \cdots - \beta_l^{(l)}. \end{array} \right. \end{align*} $$

Alternatively, Lechuga–Murillo’s algorithm [Reference Lechuga and Murillo13, Reference Murillo15] can be applied to determine the $\beta _i(1)$ , starting with $h_0^0(1)$ instead of $h_1^0(1)$ .

Here again, from $f^l = f - \beta _1 - \beta _{2} - \cdots - \beta _l$ , we obtain $f^l(1) = f(1) - \beta _1(1) - \beta _{2}(1) - \cdots - \beta _l(1),$ which indeed give us the fundamental class $[f^l(1)] = ev_{(\Lambda V,d)}([f^l(1)])$ of $(\Lambda V,d)$ from that of $(\Lambda V,d_{\sigma })$ .

We now prove Theorem 1.1

Example 5.3 Consider the Sullivan model $(\Lambda (a,b,x,u,v,w), d)$ with degrees

$$ \begin{align*}|a|=2,\; |b|=4,\; |x|=|u|=3,\; |v|=5,\; |w|=7,\end{align*} $$

and differentials

$$ \begin{align*}da=dx=0, \quad du=a^{2},\quad db=ax,\quad dv=ab-ux,\quad dw=b^{2}-2vx.\end{align*} $$

The pure model $(\Lambda V,\,d_{\sigma })$ has fundamental class $\omega _{0}=abxv-b^{2}xu$ and formal dimension $N=14$ :

$$ \begin{align*}d_{\sigma}a=d_{\sigma}b=d_{\sigma}x=0,\; d_{\sigma}u=a^{2},\; d_{\sigma}v=ab,\; d_{\sigma}w=b^{2},\end{align*} $$

with fundamental class represented by $\omega _{0}=abxv-b^{2}xu.$

Applying the algorithm of §3, one obtains the generating class $[f]$ of $\mathrm{Ext}_{(\Lambda V,\,d_{\sigma })}(\mathbb Q,\,(\Lambda V,d_{\sigma }))$ . The algorithm of §5 then lifts $[f]$ to a generating class of $Ext_{(\Lambda V,\,d)}(\mathbb Q,\,(\Lambda V,d))$ via the Ext-odd spectral sequence

$$ \begin{align*}\mathrm{Ext}^{p,-q}_{(\Lambda V,\,d_{\sigma})}(\mathbb K;\,(\Lambda V,d_{\sigma}))\;\Longrightarrow\;\mathrm{Ext}^{p-q}_{(\Lambda V,\,d)}(\mathbb K;\,(\Lambda V,d)).\end{align*} $$

The iterative correction process converges at $l=1$ : one corrector $\beta _1$ suffices, so that $f^1=f-\beta _1$ satisfies $Df^1=0$ and $[f^1]$ is nonzero. Hence, $[f^1]$ generates $\mathrm{Ext}_{(\Lambda V,\,d)}(\mathbb Q,\,(\Lambda V,d))$ , and $[f^1(1)]$ represents the fundamental class of $(\Lambda V,d)$ .

6 Concluding remarks

1. (1) It is worth noting that if $(\Lambda V,d_{k})$ , hence $(\Lambda V,(d_{k})_{\sigma })$ , is not elliptic, then $(f_{k,\sigma }^t)^l(1)$ is not necessarily the representative that gives the exact value of $e_0(X)$ . Indeed, as shown in the proof of Theorem 1.1, when the representative $f_{k,\sigma }(1)$ of the top class of $(\Lambda V,(d_{\sigma })_{k})$ is a coboundary, since $(\Lambda V,d_{\sigma })$ is itself elliptic, $(f_{k,\sigma }^t)(1)$ is not a coboundary as is $(f_{k,\sigma }^t)^l(1)$ by the convergence of the spectral sequence (2.12). Thus, $[(f_{k,\sigma }^t)^l(1)]$ is the fundamental class of $(\Lambda V,d).$

   It follows from the algorithm that, if $(p,q)$ designates the bi-degree of $f_{k,\sigma }$ , then $(f_{k,\sigma }^t)(1)\in \Lambda ^{\geq p}V$ and also $(f_{k,\sigma }^t)^l(1)\in \Lambda ^{\geq p}V$ . By [Reference Boutahir and Rami2, Theorem 2], the fundamental class $\omega \in H^N(\Lambda V,d)$ that results from some class $\omega _{0} \in H^N(\Lambda V,(d_{\sigma })_{k})$ should be represented by $\alpha \in \Lambda ^{\geq p'}V$ with $p'\geq p$ . Thus, $e_0(\Lambda V,d) =p'\geq p$ .

2. (2) In the case, where $(\Lambda V,(d_{k})_{\sigma })$ is elliptic, $[f_{k,\sigma }(1)]\in H^N(\Lambda V,(d_{\sigma })_{k})$ is the fundamental class of $H^N(\Lambda V,(d_{\sigma })_{k})$ . Using the algorithm in §4, this persists to yield the fundamental class $[f_{k,\sigma }^t(1)]\in H^N(\Lambda V,d_{\sigma })$ , and using the algorithm in §3, we get the desired fundamental class $[(f_{k,\sigma }^t)^l]$ of $(\Lambda V,d)$ . It follows that $e_0(\Lambda V,(d_{\sigma })_{k}) = e_0(\Lambda V,d_{\sigma }) = e_0(\Lambda V,d)$ . This is determined by the explicit formula [Reference Lechuga and Murillo13, Theorem 6]:

   $$ \begin{align*}e_0(\Lambda V,(d_{\sigma})_{k}) = \dim{V^{odd}} + (k-2)\dim{V^{even}}.\end{align*} $$

   This demonstrates the extensibility of our method to the general case.

3. (3) For details on the requirements imposed to accomplish the algorithm of §4, see Example 4.5. From this example, we can see the NP-hard complexity proven in [Reference Lechuga and Murillo11, Theorem 2] when determining this type of invariant.