# An upper bound for the Lusternik-Schnirelmann category of relative Sullivan
algebras

Jiawei Zhou

(April 24, 2024)

###### Abstract

This paper addresses a question posed by Félix, Halperin and Thomas. We prove
that the Lusternik-Schnirelmann category of a relative Sullivan algebra is
finite if such invariants of the base algebra and fiber algebra are both
finite. Furthermore, we provide a similar estimation for the Toomer invariant.

##  1 Introduction

The Lusternik-Schnirelmann category (LS category), denoted as $cat(X)$, of a
topological space $X$, is defined as the smallest non-negative integer $m$
such that $X$ can be covered by $m+1$ contractible open subsets. This concept
was originally introduced in [6] with the aim of estimating the number of
critical points of smooth functions on a manifold.

While the definition is straightforward, computing the LS category is
generally not easy. To estimate it, several other invariants are introduced,
such as the cone length, the Toomer invariant, and the cup length. In rational
homotopy theory, these invariants also have their rational versions, which are
defined as the smallest among all equivalent spaces. This equivalence is
established by continuous maps that induce isomorphisms on rational
homologies.

Sullivan [7] employed commutative differential graded algebras (CDGA) as
models to represent these equivalence classes. Every equivalence class
contains a special type of CDGAs, which are called the Sullivan algebras,
along with the Sullivan models of the corresponding spaces. For simply
connected spaces, the rational LS category and other rational invariants can
be computed from the algebraic structures of their Sullivan models. This
approach extends to defining analogous invariants of quasi-isomorphisms for
all CDGAs.

Many fibrations can be represented by relative Sullivan algebras. Consider a
fibration $F\to E\to B$. If $\Lambda Z$ and $\Lambda Z\otimes\Lambda W$ are
Sullivan models of $B$ and $E$ respectively, then $\Lambda W$ is a Sullivan
model of $F$ under certain conditions, for example, that $F$ and $B$ are
simply connected with either $H^{*}(F,\mathbb{Q})$ or $H^{*}(B,\mathbb{Q})$
being of finite type. However, when these conditions are not met, the Sullivan
model of $F$ can differ significantly from $\Lambda W$.

It is easy to see that $cat(E)\leq(cat(F)+1)(cat(B)+1)-1$. Consequently, in
their book [2], Félix, Halperin and Thomas posed the question of whether the
LS category of $\Lambda Z\otimes\Lambda W$ can also be bounded by those of
$\Lambda Z$ and $\Lambda W$. This paper will give a positive answer.

######  Theorem 1.1.

Let $\Lambda Z\otimes\Lambda W$ be a relative Sullivan algebra. Suppose that
$cat(\Lambda Z)=m$ and $cat(\Lambda W)=n$, then $cat(\Lambda Z\otimes\Lambda
W)\leq(m+1)(n+2)-2$. Moreover, if $\Lambda Z\otimes\Lambda W$ itself is a
minimal Sullivan algebra, then $cat(\Lambda Z\otimes\Lambda
W)\leq(m+1)(n+1)-1$.

The proof pertains to the module category $mcat$, initially introduced by
Halperin and Lemaire [3], which offers an analogy to the LS category for
differential graded modules (DG modules). Hess’ theorem [5] shows that the LS
category and the module category of a minimal Sullivan algebra are same. So
Theorem 1.1 is a special case of Theorem 3.1 and Theorem 3.2, which are
statements concerning module categories.

A minimal Sullivan algebra $\Lambda V$ can be interpreted as a relative
Sullivan algebra, where its base $\Lambda V^{1}$ is generated by degree-1
generators, and its fiber $\Lambda V^{\geq 2}$ is generated by the generators
of higher degrees. Thus, the corollary below follows from Theorem 1.1,
addressing the first part of the question by Félix, Halperin and Thomas.

######  Corollary 1.2.

Let $\Lambda V$ be a minimal Sullivan algebra. Then $cat(\Lambda V)<\infty$ if
and only if both $cat(\Lambda V^{1})$ and $cat(\Lambda V^{\geq 2})$ are
finite.

Using the same idea, we can also give an estimation for the Toomer invariant.
Theorem 3.8 is its generalization to module categories.

######  Theorem 1.3.

Let $\Lambda V=\Lambda Z\otimes\Lambda W$ be a relative Sullivan algebra.
Suppose that $e_{\Lambda Z}(\Lambda V)=m$ and $e_{\Lambda W}(\Lambda^{\geq
q}W)=n$ for all $q\geq 0$, then $e_{\Lambda V}(\Lambda V)\leq(m+1)(n+2)-2$. If
$\Lambda V$ itself is a minimal Sullivan algebra, then $e_{\Lambda V}(\Lambda
V)\leq(m+1)(n+1)-1$.

This paper is organized as follows. In Section 2, we review the basic
definitions and former results. Trying to make this paper self-contained, we
provide proofs of certain consequences that are not covered in [1] and [2].
The fact that DG modules form a model category is not directly used, although
some statements follow from it immediately. In Section 3, we prove the main
Theorems: Theorem 3.1, Theorem 3.2 and Theorem 3.8, which lead to the
conclusions outlined in Theorem 1.1, Corollary 1.2 and Theorem 1.3.

The author would like to thank Ruizhi Huang for helpful discussions. This
research is supported by the National Key Research and Development Program of
China No. 2020YFA0713000.

##  2 Preliminary

###  2.1 Notations and Conventions

In this paper, Notations and conventions generally follow from [4], [1] and
[2].

As in the introduction, we will abbreviate commutative differential graded
algebra as CDGA, differential graded modules as DG modules, and Lusternik-
Schnirelmann category as LS category.

The ground ring is assumed to be a field $\mathbbm{k}$ of characteristic 0.
CDGAs are assumed to be non-negatively graded, while DG modules is allowed to
be $\mathbb{Z}$-graded. When referring to a morphism over a CDGA $A$, we may
simply use the terms $A$-module morphism or morphism. If the map is only a
morphism as graded modules, we will specify this.

We will use $\cong$ to denote isomorphisms, $\simeq$ to denote quasi-
isomorphisms, and $\sim$ to denote homotopic DG module morphisms. In this
section, we will also provide the definition of the latter.

Let $(\Lambda V,d)$ be a CDGA which is also a free graded algebra generated by
a graded vector space $V$ of non-negative degree. We call it a KS complex,
short for Koszul-Sullivan complex, if it satisfies the Sullivan condition,
i.e. $V$ is the union of an increasing family of subspaces

$0=V(-1)\subset V(0)\subset V(1)\subset\ldots\subset V(k)\subset\ldots$

such that $dV(k)\subset\Lambda V(k-1)$. We will simply write it as $\Lambda V$
when there is no ambiguity about the differential.

For $v\in V$ with homogeneous degree, we denote its degree as $|v|$. The
subspace of all $v\in V$ with $|v|=n$ will be written as $V^{n}$. We also use
$\Lambda^{m}V$ to represent the subspace of $\Lambda V$ consisting of elements
with wordlength $m$, and use $\Lambda V^{n}$ to denote the graded algebra
generated by $V^{n}$. Note that the latter is distinct from the subspace of
$\Lambda V$ concentrated in degree $n$, which is usually written as $(\Lambda
V)^{n}$. For the subspace of $V$ spanned by elements with degree smaller than
$n$, we use $V^{<n}$. Similar notations, such as $\Lambda^{\leq m}V,\Lambda
V^{\geq n}$ and so on, will also be employed.

When $V^{0}$ is non-trivial, we assume that there is an augmentation
$\epsilon:V^{0}\to\mathbbm{k}$. This augmentation, which is necessary for
Theorem 2.2 of [4], is only used to prove Theorem 2.18.

We mainly focus on the case that $V=V^{\geq 1}$. A KS complex generated by
such $V$ is referred to as a Sullivan algebra.

A KS complex $\Lambda V$ is termed minimal if it also satisfies
$dV^{n}\subset\Lambda V^{\leq n}$. If $\Lambda V$ is a Sullivan algebra, this
is equivalent to $dV\subset\Lambda^{\geq 2}V$. It is a well-known result that
given any connected CDGA $A$ (i.e. $H^{0}(A)=\mathbbm{k}$), there exists a
quasi-isomorphism from a Sullivan algebra $\Lambda V$ to $A$, which is called
the Sullivan model of $A$. Furthermore, $\Lambda V$ can be chosen to be
minimal up to isomorphism, and in this case, it is called the minimal Sullivan
model or minimal model of $A$.

Let $B$ be a connected CDGA. The inclusion $B\to B\otimes\Lambda W$ is termed
a $\Lambda$-extension if it is a CDGA morphism satisfying the following
condition. $W$ is the union of an increasing family of subspaces

$0=W(-1)\subset W(0)\subset W(1)\subset\ldots\subset W(k)\subset\ldots$

such that $d(1\otimes W(k))\subset(\mathbbm{k}\oplus B^{\geq 1})\otimes\Lambda
W(k-1)$. This $\Lambda$-extension induces a KS complex $(\Lambda W,\bar{d})$
where $\bar{d}$ is the $\mathbbm{k}\otimes\Lambda W$ component of $d$. The KS
complex $(\Lambda W,\bar{d})$ is referred to as the fiber of the
$\Lambda$-extension, and the CDGA $B$ is called the base.

If in addition $d(1\otimes W^{n})\subset B\otimes W^{\leq n}$, the
$\Lambda$-extension is referred to as Sullivan extension.

When $W$ is concentrated in positive degrees, the $\Lambda$-extension is
called a Sullivan extension, and the CDGA $B\otimes\Lambda W$ is termed a
relative Sullivan algebra.

###  2.2 Semifree Resolutions

######  Definition 2.1.

Let $P$ be a DG module over a CDGA $A$. We call $P$ semifree if it can be
written as the union of an increasing sequence

$0=P(-1)\subset P(0)\subset P(1)\subset\ldots\subset P(k)\subset\ldots$

of sub $A$-modules, such that $P(k)/P(k-1)\simeq A\otimes W(k)$ with
$dW(k)=0$, i.e. $P(k)/P(k-1)$ is a free DG module over $A$.

As the ground ring is assumed to be a field, the exact sequence

$0\to P(k-1)\to P(k)\to P(k)/P(k-1)\to 0$

splits. So by induction $P$ can be written as $A\otimes V$ where
$V=\bigcup_{k}V(k)$ for an increasing sequence of vector spaces $V(k)$ such
that $dV(k)\subset A\otimes V(k-1)$.

######  Definition 2.2.

Let $M$ be a DG module over a CDGA $A$. An $A$-semifree resolution is a
semifree $A$-module $P$ together with a quasi-isomorphism
$P\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}M$.

According to Proposition 6.6 of [1], every DG module has a semifree
resolution. Actually, the quasi-isomorphic of the semifree resolution can be
made surjective.

######  Proposition 2.3.

Let $A$ be a CDGA. Every $A$-module $M$ has a semifree resolution
$f:P\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}M$ such that $f$ is
surjective.

###### Proof.

Proposition 6.6 of [1] gives a semifree resolution
$f:P\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}M$, and $P(0)$ is
identified as $A\otimes V(0)$ where $V(0)$ is the space of cocycles in $M$. So
for each $x\in M$, $dx\in V(0)$ and there exists a cocycle in $\alpha\in P$
such that $f(\alpha)=dx$. As $f$ is a quasi-isomorphism, $\alpha=d\beta$ for
some $\beta\in P$. Then $d(x-f(\beta))=dx-f(d\beta)=0$. Hence, $x-f(\beta)\in
V(0)\subset\operatorname{im}f$. It follows that $x\in\operatorname{im}f$. ∎

######  Definition 2.4.

Two DG module morphisms $f$ and $g$ from $M$ to $N$ over a CDGA $A$ are called
homotopic, if $f-g=d\theta+\theta d$ for some graded module morphism
$\theta:M\to N$ over $A$. We will write $f\sim g$.

######  Proposition 2.5.

Suppose $A$ is a CDGA, and $P$ is a semifree $A$-module, Let $\phi:P\to N$ and
$f:M\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}N$ be $A$-module
morphisms with $f$ being quasi-isomorphic.

(i) (Proposition 6.4(ii) of [1]) There is a unique homotopy class of morphisms
$\psi:P\to M$ such that $\phi\sim f\circ\psi$.

(ii) (Exercise 4 of Section 6 of [1]) If $f$ is surjective, we can choose
$\psi$ such that $\phi=f\circ\psi$. Such $\psi$ is called a lift of $\phi$
through $f$.

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\simeq}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\textstyle{N}$

###### Proof.

(i) The proof is provided in [1].

(ii) Write $P$ as the union of $P(k)=A\otimes V(k)$ with $dV(k)\subset
P(k-1)$. Suppose we have constructed $\psi$ on $P(k-1)$ such that $\psi\circ
f=\phi$ when restricted to $P(k-1)$, we will extend it to $P(k)$.

Choose a linearly independent set $\\{v_{\alpha}\\}$ such that $V(k)$ is the
direct sum of $V(k-1)$ and the space spanned by $\\{v_{\alpha}\\}$. By
hypothesis $\psi(dv_{\alpha})$ is defined as a cocycle in $M$, and satisfies

$f(\psi(dv_{\alpha}))=\phi(dv_{\alpha})=d(\phi(v_{\alpha})).$

As $f$ is a quasi-isomorphism, $\psi(dv_{\alpha})$ is a coboundary in $M$.
Choose some $m_{\alpha}\in M$ such that $dm_{\alpha}=\psi(dv_{\alpha})$. Set
$\psi(v_{\alpha})=m_{\alpha}$. Then $\psi$ can be extended to $P(k)$ as an
$A$-module morphism.

By induction, we obtain a morphism $\psi$ defined on $P$ satisfying
$\phi=f\circ\psi$. ∎

###  2.3 Lusternik-Schnirelmann category

######  Definition 2.6.

Let $\Lambda V$ be a minimal Sullivan algebra. Its LS category $cat(\Lambda
V)$ is the least integer $m$ such that there is a minimal relative
$\Lambda$-algebra $\Lambda V\otimes\Lambda W$ with base $\Lambda V$, together
with CDGA morphisms $f:\Lambda V\to\Lambda V\otimes\Lambda W$, $g:\Lambda
V\otimes\Lambda Z\to\Lambda V$ and $\phi:\Lambda V\otimes\Lambda Z\to\Lambda
V/\Lambda^{>m}V$ making the following diagram commutative.

---
$\textstyle{\Lambda
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{id_{\Lambda
V}}$$\textstyle{\Lambda V}$$\textstyle{\Lambda V\otimes\Lambda
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\phi}$$\scriptstyle{\simeq}$$\textstyle{\Lambda
V/\Lambda^{>m}V}$

Here $f$ needs to be the natural inclusion, $\phi$ needs to be quasi-
isomorphic, and $q$ is the natural projection.

######  Remark 2.7.

As mentioned in [2], the condition $gf=id_{\Lambda V}$ can be weakened as that
$gf$ and $id_{\Lambda V}$ are homotopic as CDGA morphisms from Sullivan
algebras.

Every connected CDGA has a unique minimal Sullivan model up to isomorphism. So
its LS category can be defined by the minimal Sullivan model.

######  Definition 2.8.

Let $A$ be a connected CDGA and $\Lambda V$ be its minimal Sullivan model.
Then $cat(A)$ is defined as $cat(\Lambda V)$.

######  Definition 2.9.

Let $\Lambda V$ be a minimal Sullivan algebra and $P$ be a $\Lambda
V$-semifree module. The module category of $P$, $mcat_{\Lambda V}(P)$, is the
least number $m$ such that there is a semifree $\Lambda V$-module $Q$ together
with DG module morphisms $f:P\to Q$, $g:Q\to P$, and $\phi:Q\to
P/(\Lambda^{>m}V\cdot P)$ satisfying the following conditions. $\phi$ is
quasi-isomorphic, $g\circ f\sim id_{P}$, and $\phi\circ f\sim q$ where $q:P\to
P/(\Lambda^{>m}V\cdot P)$ is the natural projection. In other words, the
following diagram is homotopy commutative.

(2.7)

We will simply write $mcat_{\Lambda V}(P)$ as $mcat(P)$ when there is no
ambiguity.

As shown in the appendix of [3], the homotopy commutative diagram above can be
made commutative. This also follows from Proposition 2.14 below, which will be
frequently used to prove our main Theorem.

The advantage of the definition by homotopy commutative is that $P$ can be
replaced by any quasi-isomorphic $\Lambda V$-semifree modules. Applying
Proposition 2.5, it is easy to see that all the quasi-isomorphic semifree
modules have the same module category. Thus, we can define the module category
on general DG modules by their semifree resolutions.

######  Definition 2.10.

Let $\Lambda V$ be its minimal Sullivan model, $M$ be a $\Lambda V$-module and
$P$ be a $\Lambda V$-semifree resolution of $M$. Then $mcat_{\Lambda V}(M)$ is
defined as $mcat_{\Lambda V}(P)$.

######  Definition 2.11.

Let $\Lambda V$ be a minimal Sullivan algebra and $P$ be a $\Lambda
V$-semifree module. The Toomer invariant, $e_{\Lambda V}(P)$ or simply written
as $e(P)$ when there is no ambiguity, is the least integer $m$ such that the
projection $P\to P/(\Lambda^{>m}V\cdot P)$ is injective on cohomology.

In particular, $e(\Lambda V)$ is the least integer $m$ such that the morphism
$H(\Lambda V)\to H(\Lambda V/\Lambda^{>m}V)$ induced by the projection is
injective.

For a general $\Lambda V$-module $M$, $e(M)$ is also defined as the Toomer
invariant of its semifree resolution.

######  Proposition 2.12 (Proposition 9.3 of [2]).

Let $\Lambda V$ be a minimal Sullivan algebra and $M$ be a $\Lambda V$-module.
Then $e(M)\leq mcat(M)\leq mcat(\Lambda V)$.

######  Theorem 2.13 (Hess, [5]; Theorem 9.4 of [2]).

Let $\Lambda V$ be a minimal Sullivan algebra. Then $cat(\Lambda
V)=mcat_{\Lambda V}(\Lambda V)$.

######  Proposition 2.14.

Let $A$ be a CDGA and $P=A\otimes V$ be a semifree $A$-module.

(i) Suppose that $f:P\to Q$ is an $A$-module morphism. Then it factors as
$f=p\,\circ F$, where
$p:\widetilde{Q}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Q$ is a
surjective quasi-isomorphism and $F:P\to\widetilde{Q}$ is injective.

(ii) Suppose in addition that there exist $A$-module morphisms $g:Q\to M$ and
$h:P\to M$ such that $g\circ f\sim h$. Then there is an $A$-module quasi-
isomorphism
$p:\widetilde{Q}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Q$, together
with morphisms $F:P\to\widetilde{Q}$ and $G:\widetilde{Q}\to M$ such that
$p\circ F=f$ and $G\circ F=h$.

(iii) With the hypothesis of (i), let $I\subset A$ be an ideal and $q:P\to
P/(I\cdot P)$ be the projection. Suppose that there exists an $A$-module
morphism $g:Q\to P$ such that $gf\sim id_{P}$, and a quasi-isomorphism
$\phi:Q\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(I\cdot P)$ such
that $q\sim\phi\circ f$ and $gf\sim id_{P}$. Then $Q,f,g,\phi$ can be replaced
by $\widetilde{Q}$, $F:P\to\widetilde{Q}$, $G:\widetilde{Q}\to P$, and
$\Phi:\widetilde{Q}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(I\cdot
P)$ respectively, such that $\widetilde{Q}$ is quasi-isomorphic to $Q$,
$q=\Phi\circ F$ and $G\circ F=id_{P}$.

---
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{q}$$\scriptstyle{id_{P}}$$\textstyle{P}$$\textstyle{\widetilde{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\scriptstyle{\Phi}$$\scriptstyle{\simeq}$$\textstyle{P/(I\cdot
P).}$

.

(iv) All the $\widetilde{Q}$ in the above statements can be assumed to be
semifree.

###### Proof.

(i) Set $\widetilde{Q}=P\oplus Q\oplus(A\otimes sV)$, where
$(sV)^{n}=V^{n+1}$. Then $s:V\to sV$ induces a graded module morphism $S:P\to
A\otimes sV$ over $A$, i.e. $S(a\cdot v)=(-1)^{|a|}a\otimes sv$ for $a\in
A,v\in V$. The differential $D$ on $\widetilde{Q}$ is given as follows.
$Dx=dx$ for $x\in P$, $D\alpha=d\alpha$ for $\alpha\in Q$, and
$D(sv)=v+f(v)-S(dv)$. A straightforward calculation shows that $D$ is well
defined.

Set $F:P\to\widetilde{Q}$ as the natural inclusion. Let $p:\widetilde{Q}\to Q$
be $f$ when restricted to $P$, be $-id_{Q}$ when restricted to $Q$, and be $0$
when restricted to $A\otimes sV$. For $sv\in SV$ we have

$p\circ D(sv)=f(v)-id_{Q}\circ f(v)=0=D\circ p(sv).$

It follows that $p$ is a surjective DG module morphism over $A$. We will show
that it is a quasi-isomorphism.

Let $x+\alpha+\sum a_{i}\otimes sv_{i}$ be a cocycle in $\widetilde{Q}$ with
$x\in P,\alpha\in Q,a_{i}\in A$ and $sv_{i}\in sV$. Suppose $p(x+\alpha+\sum
a_{i}\otimes sv_{i})=d\beta$ for some $\beta\in Q$. Then
$f(x)-\alpha=p(x+\alpha+\sum a_{i}\otimes sv_{i})=d\beta$. On the other hand,
$D(x+\alpha+\sum a_{i}\otimes sv_{i})=0$ implies that

$dx+\sum(-1)^{|a_{i}|}a_{i}\otimes v_{i}=0,\quad
d\alpha+\sum(-1)^{|a_{i}|}a_{i}\otimes f(v_{i})=0,\quad\sum da_{i}\otimes
v_{i}-(-1)^{|a_{i}|}a_{i}\cdot S(dv_{i})=0.$

Write $x$ as $\sum b_{j}\otimes w_{j}$ for $b_{j}\in A,w_{j}\in V$. Then
$S(x)=\sum(-1)^{|b_{j}|}b_{j}\otimes sw_{j}$. Hence,

$\displaystyle D(Sx)$ $\displaystyle=\sum b_{j}\otimes w_{j}+\sum b_{j}\otimes
f(w_{j})+\sum[(-1)^{|b_{j}|}db_{j}\otimes sw_{j}-b_{j}\cdot S(dw_{j})]$
$\displaystyle=x+f\left(\sum b_{j}\otimes w_{j}\right)+\sum[-S(db_{j}\otimes
w_{j})-(-1)^{|b_{j}|}S(b_{j}\cdot dw_{j})]$ $\displaystyle=x+f(x)-S\left(\sum
d(b_{j}\otimes w_{j})\right)$ $\displaystyle=x+f(x)-S(dx).$

As $dx=-\sum(-1)^{|a_{i}|}a_{i}\otimes v_{i}$, $S(dx)=-\sum a_{i}\otimes
sv_{i}$. Thus,

$x+\alpha+\sum a_{i}\otimes sv_{i}=[\alpha-f(x)]+[x+f(x)-S(dx)]=D(-\beta+Sx).$

This proves that $p$ injective is on cohomology.

For each cocycle $\alpha\in Q$, $-\alpha$ is a cocycle in $\widetilde{Q}$ and
$p(-\alpha)=\alpha$. So $p$ surjective is on cohomology.

(ii) Construct
$p:\widetilde{Q}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Q$, and
$F:P\to\widetilde{Q}$ as in (i). Then $p\circ F=f$. Since $gf\sim h$, there
exists a graded module morphism $\theta:P\to P$ over $A$ such that
$gf-h=d\theta+\theta d$. Set $G(x)=h(x)$ for $x\in P$, $G(\alpha)=-g(\alpha)$
for $\alpha\in Q$, and $G(sv)=-\theta(v)$ for $sv\in sV$. Then

$G\circ S(a\otimes v)=(-1)^{|a|}G(a\otimes
sv)=-(-1)^{|a|}a\otimes\theta(v)=-\theta(a\otimes v)$

for any $a\in A$. So $G\circ S=-\theta$ on $P$, and we have

$G\circ D(sv)=G(v+f(v)-S(dv))=h(v)-gf(v)+\theta(dv)=-d\circ\theta(v)=d\circ
G(sv).$

It follows that $G$ is a DG module morphism over $A$. By construction clearly
$G\circ F=h$.

(iii) First apply (i) to construct $F:P\to\widetilde{Q}$. This process is only
depending on $f$. So we can apply the remaining step in (ii) twice on the
homotopies $q\sim\phi\circ f$ and $id_{P}\sim g\circ f$. Then we obtain two
morphisms $\Phi:\widetilde{Q}\to P/(I\cdot P)$ and $G:\widetilde{Q}\to P$ such
that $q=\Phi\circ F$ and $id_{P}=G\circ F$.

(iv) By Proposition 2.3 there exists a surjective semifree resolution
$p^{\prime}:\widetilde{Q}^{\prime}\to\widetilde{Q}$, and by Proposition
2.5(ii) there exists a lift $F^{\prime}:P\to\widetilde{Q}^{\prime}$ of $f$
through $p^{\prime}$ such that $p^{\prime}\circ F^{\prime}=F$. Then
$\widetilde{Q},p,F,G,\Phi$ can be replaced by $\widetilde{Q}^{\prime},p\circ
p^{\prime},F^{\prime},G\circ p^{\prime},\Phi\circ p^{\prime}$ respectively.
Since $F$ is injective, it follows that $F^{\prime}$ is also injective. As
both $p$ and $p^{\prime}$ are surjective, so is $p\circ p^{\prime}$. ∎

######  Corollary 2.15.

Let $\Lambda V$ be a minimal Sullivan algebra and $P$ be a semifree $\Lambda
V$-module. If $mcat(P)=m$, then there is a $\Lambda V$-module $Q$, together
with $\Lambda V$-module morphisms $f:P\to Q$, $g:Q\to P$, and a quasi-
isomorphism
$\phi:Q\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(\Lambda^{>m}V\cdot
P)$ making the graph (2.7) commutative, i.e. $\phi\circ f$ is the natural
projection $q:P\to P/(\Lambda^{>m}V\cdot P)$ and $g\circ f=id_{P}$.

###  2.4 Module Categories over general KS complex

In [1], for a general Sullivan algebra $\Lambda V$, $cat(\Lambda V)$ can also
be defined following the format of Definition 2.6. Its can be shown that this
definition matches the definition by its minimal model. Similarly, $e(\Lambda
V)$ can be defined following the format of Definition 2.11. In this
subsection, we will prove that the module category and Toomer invariant of DG
modules over non-minimal Sullivan algebras can also be defined in a similar
way. Such invariants will be used to prove the main theorems.

######  Definition 2.16.

Let $\Lambda V$ be a connected KS complex (not necessarily being Sullivan or
minimal), and $P$ be a $\Lambda V$-module. $mcat_{\Lambda V}(P)$ is defined as
the least number $m$ such that there is a semifree $\Lambda V$-module $Q$
together with DG module morphisms $f:P\to Q$, $g:Q\to P$, and $\phi:Q\to
P/(\Lambda^{>m}V\cdot P)$ making the diagram (2.7) commutative (or just
homotopy commutative according to Proposition 2.14).

$e_{\Lambda V}(P)$ is defined as the least number $m$ such that the projection
$q:P\to P/(\Lambda^{>m}V\cdot P)$ is injective on cohomology.

For a general $\Lambda V$-module $M$, we also define $mcat_{\Lambda V}(M)$ as
the module category of its semifree resolution, since it is independent of the
choice of the resolution. Similarly, $e_{\Lambda V}(M)$ is defined as the
Toomer invariant of its semifree resolution.

######  Remark 2.17.

We may also give such definitions even if $\Lambda V$ is not connected. In
this case there exists a cocycle $v\in V^{0}$. Then the cohomology classes of
all $v^{m}$ are non-trivial and $q^{*}:H^{*}(\Lambda V)\to H^{*}(\Lambda
V/\Lambda^{>m}V)$ can never be injective. So $e_{\Lambda V}(\Lambda
V)=mcat_{\Lambda V}(\Lambda V)=\infty$.

When $\Lambda V$ is connected, it has a minimal model $\Lambda Z$. Let
$\psi:\Lambda Z\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Lambda V$ be
the quasi-isomorphism. The $\Lambda V$-semifree module $P$ naturally has a
$\Lambda Z$-module structure, defined as $z\cdot x:=\psi(z)\cdot x$ for $z\in
Z,x\in P$.

######  Theorem 2.18.

With the hypotheses and notations above, $mcat_{\Lambda V}(P)=mcat_{\Lambda
Z}(P)$.

###### Proof.

We will use the fact that $\Lambda V$ can be written as a tensor product of
two KS complexes $(\Lambda Z,d)\otimes(\Lambda(U\oplus dU),d)$, where $\Lambda
Z$ is its minimal model and $\Lambda(U\oplus dU)$ is contractible (i.e.
$d:U\to dU$ is an isomorphism) [4, Theorem 2.2]. When $\Lambda V$ is a
Sullivan algebra, a proof can also be found in Theorem 14.9 of [1]. Without
loss of generalization, we may assume that $\psi:\Lambda
Z\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Lambda V$ is just the
inclusion.

Observe that $P$ is also $\Lambda Z$-semifree. Indeed, write it as $\Lambda
Z\otimes\Lambda(U\oplus dU)\otimes W$ such that $W$ is the union of an
increasing sequence of vector space satisfying $dW(k)\subset\Lambda
Z\otimes\Lambda(U\oplus dU)\otimes W(k-1)$. As $\Lambda(U\oplus dU)$ is a KS
complex, we can give $\Lambda(U\oplus dU)\otimes W$ a filtration
$\widetilde{W}(k)$ such that $d\widetilde{W}(k)\subset\Lambda
Z\otimes\widetilde{W}(k-1)$. Namely, let $\widetilde{W}(-1)=0$ and inductively
set $\widetilde{W}(k)$ as the subspace of elements whose image under $d$ is in
$\Lambda Z\otimes\widetilde{W}(k-1)$. Then use induction on the filtrations of
$\Lambda(U\oplus dU)$ and $W$ to show $\Lambda(U\oplus dU)\otimes
W\subset\bigcup\widetilde{W}(k)$.

So when $mcat_{\Lambda V}(P)=m$, the commutative diagram (2.7) of $\Lambda
V$-modules is also of of $\Lambda Z$-modules. The projection $q$ factors as
$P\stackrel{{\scriptstyle q_{1}}}{{\longrightarrow}}P/(\Lambda^{>m}Z\cdot
P)\stackrel{{\scriptstyle q_{2}}}{{\longrightarrow}}P/(\Lambda^{>m}V\cdot P)$.
We will show that $q_{2}$ is a quasi-isomorphism, so that
$P/(\Lambda^{>m}V\cdot P)$ in (2.7) can be replaced by $P/(\Lambda^{>m}Z\cdot
P)$. This proof does not use the condition $mcat_{\Lambda V}(P)=m$, so $q_{2}$
is a quasi-isomorphism for any $m$.

Equip $P$ with a filtration by setting $P(m)=\Lambda^{\geq m}V\cdot P$, which
is isomorphic to $\Lambda^{\geq m}(Z\oplus U\oplus dU)\otimes W$. This also
gives filtrations on the quotient spaces $P/(\Lambda^{>m}Z\cdot P)$ and
$P/(\Lambda^{>m}V\cdot P)$, which are preserved by $q_{2}$. The induced
$E_{0}$ pages of spectral sequences are

$(\Lambda^{\leq m}Z,0)\otimes(\Lambda(U\oplus dU),d)\otimes(W,\bar{d})$

and

$\bigoplus_{0\leq i\leq m}(\Lambda^{m-i}Z,0)\otimes(\Lambda(U\oplus
dU)/\Lambda^{>i}(U\oplus dU),d)\otimes(W,\bar{d})$

respectively. Here $\bar{d}:W(k)\to W(k-1)$ is induced by the differential on
$W$.

Note that the projection $\Lambda(U\oplus dU)\to\Lambda(U\oplus
dU)/\Lambda^{>i}(U\oplus dU)$ is quasi-isomorphic for all $i\geq 0$, as
$\Lambda^{>i}(U\oplus dU)$ has a trivial cohomology. Actually, since
$\Lambda(U\oplus dU)$ is contractible, for any cocycle
$x\in\Lambda^{>i}(U\oplus dU)$, there exist a constant $c\in\mathbbm{k}$ and
some $y\in\Lambda(U\oplus dU)$ such that $x-c=dy$. As $d$ does not change the
wordlength, $x$ is equal to $d$ acting on the part of $y$ whose wordlength is
greater than $i$. It follows that $x$ is a coboundary in $\Lambda^{>i}(U\oplus
dU)$ (and $c$ has to be 0).

Therefore, $q_{2}$ induces a quasi-isomorphism between the $E_{0}$ pages, then
isomorphisms on higher $E_{r}$ pages. As $P(m+1)=\Lambda^{>m}V\cdot P$, the
spectral sequence for $P/(\Lambda^{>m}V\cdot P)$ collapses at the page
$E_{m+1}$. Then so is the the spectral sequence for $P/(\Lambda^{>m}Z\cdot
P)$. Thus, $q_{2}$ induces an isomorphism between cohomologies.

Lift
$\phi:Q\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(\Lambda^{>m}V\cdot
P)$ through the surjective quasi-isomorphism $q_{2}$. We obtain a quasi-
isomorphism
$\Phi:Q\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(\Lambda^{>m}Z\cdot
P)$ such that $q_{2}\circ\Phi=\phi$. Then $q_{2}\circ q_{1}=q=\phi\circ
f=q_{2}\circ\Phi\circ f$, where $f:P\to Q$ is the morphism in (2.7). By
Proposition 2.5 (i) we have $q_{1}\sim\Phi\circ f$, and by Proposition 2.14 we
can assume $q_{1}=\Phi\circ f$, i.e. the following diagram commutes.

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q_{1}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\simeq}$$\scriptstyle{\Phi}$$\scriptstyle{\simeq}$$\textstyle{P/(\Lambda^{>m}Z\cdot
P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\scriptstyle{\simeq}$$\textstyle{P/(\Lambda^{>m}V\cdot
P)}$

Therefore, we can replace $P/(\Lambda^{>m}V\cdot P)$, $q$, $\phi$ by
$P/(\Lambda^{>m}Z\cdot P)$, $q_{1}$, $\Phi$ respectively. It follows that
$mcat_{\Lambda Z}(P)\leq m=mcat_{\Lambda V}(P)$.

Conversely, suppose $mcat_{\Lambda Z}(P)=m$. We have a diagram of the form
(2.7) of $\Lambda Z$-modules, where $P/(\Lambda^{>m}V\cdot P)$ is replaced by
$P/(\Lambda^{>m}Z\cdot P)$. Let $\Lambda V\otimes_{\Lambda Z}-$ acting on all
the modules and morphisms. Then we obtain the following commutative diagram of
$\Lambda V$-modules.

---
$\textstyle{\Lambda V\otimes_{\Lambda
Z}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{\Lambda
V\otimes_{\Lambda Z}P}$$\textstyle{\Lambda V\otimes_{\Lambda
Z}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Lambda
V\otimes_{\Lambda Z}\left(P/(\Lambda^{>m}Z\cdot P)\right)}$

Observe that $\Lambda V\otimes_{\Lambda Z}P\cong(\Lambda(U\oplus
dU),d)\otimes(P,d)$ is $\Lambda V$-free and quasi-isomorphic to $P$. Besides,

$\Lambda V\otimes_{\Lambda Z}\left(P/(\Lambda^{>m}Z\cdot
P)\right)\cong(\Lambda(U\oplus dU)\otimes
P)/(\Lambda^{>m}Z\cdot(\Lambda(U\oplus dU)\otimes P))$

Same as the discussion about $q_{2}$ above, we have a quasi-isomorphism

$(\Lambda(U\oplus dU)\otimes P)/(\Lambda^{>m}Z\cdot(\Lambda(U\oplus dU)\otimes
P))\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\Lambda(U\oplus
dU)\otimes P)/(\Lambda^{>m}V\cdot(\Lambda(U\oplus dU)\otimes P)).$

Therefore,

$mcat_{\Lambda V}(P)=mcat_{\Lambda V}(\Lambda(U\oplus dU)\otimes P)\leq
m=mcat_{\Lambda Z}(P).$

This proves that $mcat_{\Lambda V}(P)=mcat_{\Lambda Z}(P)$.

Finally, as shown above, $q_{2}:P/(\Lambda^{>m}Z\cdot
P)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(\Lambda^{>m}V\cdot P)$
is a quasi-isomorphism for all $m$. So $q_{1}:P\to P/(\Lambda^{>m}Z\cdot P)$
is injective on cohomology if and only if $q_{2}\circ q_{1}$ is. Thus,
$e_{\Lambda V}(P)=e_{\Lambda Z}(P)$. ∎

######  Corollary 2.19.

For any connected Sullivan algebra $\Lambda V$, $mcat_{\Lambda V}(\Lambda
V)=cat(\Lambda V)$.

###### Proof.

Let $\Lambda Z$ be a minimal Sullivan model of $\Lambda V$, then

$mcat_{\Lambda V}(\Lambda V)=mcat_{\Lambda Z}(\Lambda V)=mcat_{\Lambda
Z}(\Lambda Z)=cat(\Lambda Z)=cat(\Lambda V).$

∎

##  3 Main Theorems

###  3.1 Estimate the Module Category

######  Theorem 3.1.

Let $\Lambda V=\Lambda Z\otimes\Lambda W$ be a relative Sullivan algebra.
Suppose that $P$ is a $\Lambda V$-semifree module of the form $\Lambda
V\otimes U$, where $U$ is a vector space of the form $\bigcup U(k)$ satisfying
$dU(k)\subset\Lambda Z\otimes U(k-1)$. Then the subspace $\overline{P}=\Lambda
Z\otimes U$ is a $\Lambda Z$-semifree module. If $mcat_{\Lambda
Z}(\overline{P})=m$ and $cat(\Lambda W)=n$, then $mcat_{\Lambda
V}(P)\leq(m+1)(n+2)-2$.

###### Proof.

By hypothesis, there exists a $\Lambda Z$-semifree module $\overline{Q}$
quasi-isomorphic to $\overline{P}/(\Lambda^{>m}Z\cdot\overline{P})$ such that
the following diagram of $\Lambda Z$-modules commutes.

---
$\textstyle{\overline{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\scriptstyle{id}$$\textstyle{\overline{P}}$$\textstyle{\overline{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\overline{P}/(\Lambda^{>m}Z\cdot\overline{P})}$

We use $pr$ to denote the natural projection in this proof. Let $\Lambda
V\otimes_{\Lambda Z}-$ acting on the modules and morphisms of the above
diagram. Observe that $P=\Lambda V\otimes_{\Lambda Z}\overline{P}$, and
$\Lambda V\otimes_{\Lambda
Z}(\overline{P}/(\Lambda^{>m}Z\cdot\overline{P}))=P/(\Lambda^{>m}Z\cdot P)$.
Set $P_{m+1}=\Lambda V\otimes_{\Lambda Z}Q$, which is a $\Lambda V$-semifree
module. Then we have the following commutative diagram of $\Lambda V$-modules.

---
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\scriptstyle{id}$$\textstyle{P}$$\textstyle{P_{m+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P/(\Lambda^{>m}Z\cdot
P)}$

Let $P^{\geq p,\geq q}$ denote the subspace
$(\Lambda^{p}Z\otimes\Lambda^{q}W)\cdot P=\Lambda^{\geq
p}Z\otimes\Lambda^{\geq q}W\otimes U$ of $P$. Note that $P^{\geq p,\geq q}$ is
not closed under differential unless $q=0$. So it is not a DG submodule in
general.

Denote the quotient space $P^{\geq p,\geq q}/P^{\geq p+1,\geq q}$ by
$P^{p,\geq q}$. It is a DG module over $(\Lambda W,\bar{d})$, and is
isomorphic to the module $(\Lambda^{p}Z,0)\otimes(\Lambda^{\geq q}W\otimes
U,\bar{d})$, where $\bar{d}U(j)\subset U(j-1)$. Thus, $P^{p,\geq q}$ is
$\Lambda W$-semifree. Moreover, it is $\Lambda W$-free when $P$ is a minimal
$\Lambda V$-semifree module such that $dU(j)\subset\Lambda^{+}Z\otimes
U(j-1)$.

We use $P^{p,[q,r]}$ denote the quotient $\Lambda W$-module $P^{p,\geq
q}/P^{p,\geq r+1}$. It is isomorphic to $\bigoplus_{q\leq j\leq
r}\Lambda^{p}Z\otimes\Lambda^{j}W\otimes U$ as graded modules.

Then let $I_{m+1}=P^{\geq m+1,\geq 0}$ and define $I_{k}=I_{k+1}+P^{\geq
k,\geq(m+1-k)(n+2)-1}$ inductively for $0\leq k\leq m$. As $dP^{\geq p,\geq
q}\subset P^{\geq p+1,\geq q-1}+P^{\geq p,\geq q}$, $I_{m+1}=P^{\geq m+1,\geq
0}$ is closed under the differential. Then it is a DG submodule.

Now suppose we have constructed $\Lambda V$-semifree modules
$P_{k+1},P_{k+2},\ldots,P_{m+1}$ for some $k<m+1$, such that for $k<i\leq
m+1$, there are surjective quasi-isomorphisms
$P_{i}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/I_{i}$, and
morphisms $P_{i+1}\to P_{i}$, $P_{i}\to P_{i+1}$ make the following diagram
commutative.

---
$\textstyle{P_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\simeq}$$\textstyle{P_{i+1}}$$\textstyle{P_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P/I_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{P/I_{i}}$

Here we put $P_{m+2}=P$ and $I_{m+2}=0$. We will construct $P_{k}$ and the
corresponding morphisms.

First observe that any surjective quasi-isomorphism $\widetilde{P}_{k+1}\to
P_{k+1}$ gives a lift $P_{k+2}\to\widetilde{P}_{k+1}$ which makes the
following diagram commutative.

---
$\textstyle{P_{k+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\simeq}$$\textstyle{P_{k+2}}$$\textstyle{\widetilde{P}_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P/I_{k+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{P/I_{k+1}}$

So we can replace $P_{k+1}$ by $\widetilde{P}_{k+1}$.

Next consider the subspace $(P^{\geq k,\geq(m-k)(n+2)}+I_{k+1})/I_{k+1}$. As

$dP^{\geq k,\geq(m-k)(n+2)}\subset P^{\geq k,\geq(m-k)(n+2)}+P^{\geq
k+1,\geq(m-k)(n+2)-1}\subset P^{\geq k,\geq(m-k)(n+2)}+I_{k+1}$

and $Z\cdot P^{\geq k,\geq(m-k)(n+2)}\subset I_{k+1}$, this subspace is a DG
submodule over $\Lambda V$ where the $\Lambda^{+}Z$-action is trivial. So it
is also a DG module over $(\Lambda W,\bar{d})$, and is isomorphic to
$(\Lambda^{k}Z,0)\otimes(\Lambda^{\geq(m-k)(n+2)}W\otimes U,\bar{d})\cong
P^{k,\geq(m-k)(n+2)}$.

By $mcat_{\Lambda W}(P^{k,\geq(m-k)(n+2)})\leq cat(\Lambda W)=n$, there exists
a $\Lambda W$-semifree module $Q_{k}$ quasi-isomorphic to
$P^{k,\geq(m-k)(n+2)}/(\Lambda^{>n}W\cdot P^{k,\geq(m-k)(n+2)})$ such that the
following diagram commutes.

---
$\textstyle{P^{k,\geq(m-k)(n+2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{pr}$$\textstyle{P^{k,\geq(m-k)(n+2)}}$$\textstyle{Q_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P^{k,\geq(m-k)(n+2)}/(\Lambda^{>n}W\cdot
P^{k,\geq(m-k)(n+2)})}$

$Q_{k}$ can also be recognized as a $\Lambda V$-module where the $Z$-action is
trivial. So we can construct surjective $\Lambda V$-semifree resolutions
$F(P^{k,\geq(m-k)(n+2)})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P^{k,\geq(m-k)(n+2)}$
and
$F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}Q_{k}$.
The latter notation follows from $\Lambda^{>n}W\cdot
P^{k,\geq(m-k)(n+2)}=P^{k,\geq(m-k)(n+2)+n+1}$, which implies that
$F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})$ is also a $\Lambda V$-semifree resolution
of $P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}$. Then we have lifts $f$ and $g$ which
make the following diagram commutative.

---
$\textstyle{F(P^{k,\geq(m-k)(n+2)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{\simeq}$$\textstyle{F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\simeq}$$\textstyle{F(P^{k,\geq(m-k)(n+2)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\simeq}$$\textstyle{P^{k,\geq(m-k)(n+2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{Q_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P^{k,\geq(m-k)(n+2)}}$$\textstyle{P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}}$

Then $(g\circ f)\circ p=p\circ id=p$. As $p$ is a quasi-isomorphism, we have
$g\circ f\sim id$ by Proposition 2.5 (i). Applying Proposition 2.14 (ii), we
can replace $F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})$ by a larger module, and make
the replacements $F,G$ of $f,g$ satisfy $G\circ F=id$. $F$ and $G$ also make
the above commutative. So without loss of generality we can assume that
$g\circ f=id$.

On the other hand, $P^{k,\geq(m-k)(n+2)+n+1}=P^{k,\geq(m+1-k)(n+2)-1}\cong
I_{k}/I_{k+1}$. So $P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}\cong(P^{\geq
k,\geq(m-k)(n+2)}+I_{k+1})/I_{k}$ can be recognized as a subspace of
$P/I_{k}$. It is actually a $\Lambda V$-submodule since differential and
$\Lambda V$-action preserves this subspace. So we have the following
commutative diagram of $\Lambda V$-modules.

|  |
---|---|---
$\textstyle{F(P^{k,\geq(m-k)(n+2)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\simeq}$$\scriptstyle{f}$$\textstyle{P_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P^{k,\geq(m-k)(n+2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{pr}$$\textstyle{P/I_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{P/I_{k}}$

Here $\iota$ is the inclusion and $h$ is a lift obtained by Proposition
2.5(ii). According to Proposition 2.14(i) $h$ can be factored as a injective
morphism $H:F(P^{k,\geq(m-k)(n+2)})\to\widetilde{P}_{k+1}$ and a surjective
quasi-isomorphism
$\widetilde{P}_{k+1}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P_{k+1}$.
Then $P_{k+1}$ and $h$ can be replaced by $\widetilde{P}_{k+1}$ and $H$
respectively. So without loss of generality we assume that $h$ is injective.

Let $(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times P_{k+1})/\sim$ denote the
pushout of $f$ and $h$, with $(f(x),0)\sim(0,h(x))$, i.e. $(f(x),-h(x))\sim 0$
for $x\in F(P^{k,\geq(m-k)(n+2)})$. Use $\phi$ and $\psi$ denote the following
maps.

$\displaystyle\phi:F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}P/I_{k}$
$\displaystyle\psi:P_{k+1}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/I_{k+1}\stackrel{{\scriptstyle
pr}}{{\longrightarrow}}P/I_{k}$

The morphism $F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times P_{k+1}\to P/I_{k}$,
$(a,b)\mapsto\phi(a)+\psi(b)$ vanishes on $(f(x),-h(x))$. So it induces a
morphism $(\phi,\psi)$ from $(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1})/\sim$ to $P/I_{k}$. We claim that $(\phi,\psi)$ is a quasi-
isomorphism.

The proof of this claim is a dual version of Lemma 13.3 in [1]. As $h$ and
$\iota:P^{k,\geq(m-k)(n+2)}\to P_{k+1}$ are injective, we have the following
commutative diagram of short exact sequences.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(P^{k,\geq(m-k)(n+2)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\simeq}$$\textstyle{P_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\text{coker\,}h\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{k,\geq(m-k)(n+2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{P/I_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{coker\,}\iota\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

By the five Lemma the induced morphism $\text{coker\,}h\to\text{coker\,}\iota$
is quasi-isomorphic.

Observe that the morphism
$\chi:F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\to(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1})/\sim$, $a\mapsto(a,0)$ is injective with cokernel $\text{coker\,}h$.
Indeed, if $(a_{1},0)\sim(a_{2},0)$, there exists some $x\in
F(P^{k,\geq(m-k)(n+2)})$ such that $f(x)=a_{1}-a_{2}$ and $-h(x)=0$. As $h$ is
injective, we have $x=0$ and $a_{1}=a_{2}$. So $\chi$ is injective. Its image
is identified with
$(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times\text{im\,}h)/\sim$, then the
cokernel is identified with

$(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1})/(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times\text{im\,}h)\cong\text{coker\,}h.$

On the other hand, $\text{coker\,}\iota$ is also the cokernel of the other
$\iota$ from $P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}$ to $P/I_{k}$. This follows from

$\displaystyle\text{coker\,}\iota$
$\displaystyle=(P/I_{k+1})/P^{k,\geq(m-k)(n+2)}$
$\displaystyle\cong(P/I_{k+1})/((P^{\geq k,\geq(m-k)(n+2)}+I_{k+1})/I_{k+1})$
$\displaystyle\cong(P/I_{k})/((P^{\geq k,\geq(m-k)(n+2)}+I_{k+1})/I_{k})$
$\displaystyle\cong(P/I_{k})/P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}.$

The composition $(\phi,\psi)\circ\chi$ is exactly $\phi$. So we have the
following commutative diagram of short exact sequences.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\scriptstyle{\simeq}$$\textstyle{(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1})/\sim\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi,\psi)}$$\textstyle{\text{coker\,}h\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{P/I_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{coker\,}\iota\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Apply the five lemma again, we have that $(\phi,\psi)$ is a quasi-isomorphism.
So the claim is proved.

Now we construct a morphism from $F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1}\to P_{k+1}$ by $(a,b)\mapsto h\circ g(a)+b$. Then the image of
$(f(x),-h(x))$ where $x\in F(P^{k,\geq(m-k)(n+2)})$ is $h\circ g\circ
f(x)-h(x)=h(x)-h(x)=0$. So it induces a morphism from
$(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times P_{k+1})/\sim$ to $P_{k+1}$. By
definition, this morphism sends $(0,b)$ to $b$, so it is surjective. Construct
a surjective $\Lambda V$-semifree resolution
$P_{k}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1})/\sim$. Then we obtain a lift $P_{k+1}\to P_{k}$ which makes the
following diagram commutative.

---
$\textstyle{P_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{\simeq}$$\textstyle{P_{k+1}}$$\textstyle{P_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(F(P^{k,[(m-k)(n+2),(m-k)(n+2)+n]})\times
P_{k+1})/\sim\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{(\phi,\psi)}$$\textstyle{P/I_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{P/I_{k}}$

Therefore, we can construct $P_{i}\simeq P/I_{i}$ inductively and factor the
identify map as $P_{i+1}\to P_{i}\to P_{i+1}$ for $i\leq m+1$. It leads to the
following commutative diagram.

---
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{pr}$$\scriptstyle{id}$$\textstyle{P}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\simeq}$$\textstyle{P/I_{0}}$

Finally, since $\Lambda^{\geq(m+1)(n+2)-1}V\cdot P\subset I_{0}$, the
projection $P\to P/I_{0}$ factors through $P/(\Lambda^{>(m+1)(n+2)-2}V\cdot
P)$. Choose a surjective $\Lambda V$-semifree resolution
$Q\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}P/(\Lambda^{>(m+1)(n+2)-2}V\cdot
P)$. We have the following commutative diagrams with lifts $\zeta$ and
$\lambda$.

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\scriptstyle{pr}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\scriptstyle{\simeq}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\simeq}$$\textstyle{P/(\Lambda^{>(m+1)(n+2)-2}V\cdot
P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{P/I_{0}}$

So we have $\eta\circ(\lambda\circ\zeta)=\eta\circ\xi$. Since $\eta$ is a
quasi-isomorphism, we have $\lambda\circ\zeta\sim\xi$ by Proposition 2.5 (i).
Apply Proposition 2.14 (ii) again to replace $Q$ by a larger module, we can
make $\lambda\circ\zeta=\xi$. Then we obtain the following commutative
diagram, which implies $mcat_{\Lambda V}(P)\leq(m+1)(n+2)-2$.

---
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\scriptstyle{pr}$$\scriptstyle{id}$$\textstyle{P}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\scriptstyle{\simeq}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P/(\Lambda^{>(m+1)(n+2)-2}V\cdot
P)}$

∎

When $\Lambda V$ itself is a minimal Sullivan algebra, we have $dP^{\geq
p,\leq q}\subset P^{\geq p+2,\leq q-1}$. In this case we can make $I_{k}$ a
little larger. Precisely, set $I_{k}=I_{k+1}+P^{\leq k,\leq(m+1-k)(n+1)}$.
Then the subspace $P^{\geq k,\geq(m-k)(n+1)}+I_{k+1}$ is closed under the
differential. It follows that $(P^{\geq p+2,\leq q-1}+I_{k+1})/I_{k+1}$ is a
DG module over $\Lambda W$, and its submodule $\Lambda^{n+1}W\cdot[(P^{\geq
p+2,\leq q-1}+I_{k+1})/I_{k+1}]$ is isomorphic to $I_{k}/I_{k+1}$. Following
the construction in Theorem 3.1, we can factor $id_{P}$ as

$P\to F(P/(\Lambda^{\geq(m+1)(n+1)}V\cdot P))\to F(P/I_{0})\to P.$

So we have the following estimation.

######  Theorem 3.2.

With the hypothesis of Theorem 3.1, suppose in addition that $\Lambda V$
itself is a minimal Sullivan algebra. If $mcat_{\Lambda Z}(\overline{P})=m$
and $cat(\Lambda W)=n$, then $mcat_{\Lambda V}(P)\leq(m+1)(n+1)-1$.

A minimal Sullivan algebra $\Lambda V$ can be interpreted as a relative
Sullivan algebra $\Lambda V^{1}\otimes\Lambda V^{\geq 2}$. So $cat(\Lambda V)$
is finite if both $cat(\Lambda V^{1})$ and $cat(\Lambda V^{\geq 2})$ are. The
converse statement follows from Proposition 9.6 and Theorem 9.3 of [2]. So we
have obtained Corollary 1.2.

######  Corollary 3.3.

Let $\Lambda V$ be a minimal Sullivan algebra. Then $cat(\Lambda V)<\infty$ if
and only if both $cat(\Lambda V^{1})$ and $cat(\Lambda V^{\geq 2})$ are
finite.

The following two simple examples show that this corollary can not be
generalized to arbitrary relative Sullivan algebras.

######  Example 3.4.

Let $Z$ spanned by $z$ of degree 3, and $W$ spanned by $w$ of degree 2. Set
$dz=0,dw=z$. Then $cat(\Lambda Z\otimes\Lambda W)=0$ but $cat(\Lambda
W)=\infty$.

Although $cat(\Lambda W)\leq cat(\Lambda Z\otimes\Lambda W)$ if $\Lambda
Z\otimes\Lambda W$ is a minimal Sullivan algebra [2, Theorem 9.3], it is still
possible that $cat(\Lambda Z)=\infty$ but $cat(\Lambda Z\otimes\Lambda W)$ is
finite.

######  Example 3.5.

Let $Z$ spanned by $z$ of degree 2, and $W$ spanned by $w$ of degree 3. Set
$dz=0,dw=z^{2}$. Then $\Lambda Z\otimes\Lambda W$ is minimal and its LS
category is 1, but $cat(\Lambda Z)=\infty$.

For any fixed $m$ and $n$, there are examples such that $cat(\Lambda Z)=m$,
$cat(\Lambda W)=n$ and $cat(\Lambda Z\otimes\Lambda W)=(m+1)(n+1)-1$.

######  Example 3.6 (Example 1 of Section 30(b), [1]).

Let $Z$ spanned by $x,y$, and $W$ spanned by $u,v$, where $|x|=2(n+1)$,
$|y|=2(m+1)(n+1)-1$, $u=2$, $v=2(n+1)-1$. Set $dx=du=0$, $dy=x^{m+1}$, and
$dv=u^{n+1}-x$. Then $cat(\Lambda Z)=m$, and $cat(\Lambda W)=n$. $\Lambda
Z\otimes\Lambda W$ is quasi-isomorphic to $\Lambda u/(u^{(m+1)(n+1)})$, whose
LS category is $(m+1)(n+1)-1$.

In this example $\Lambda Z\otimes\Lambda W$ is not a minimal Sullivan algebra.
So it remains open whether there are better estimations for Theorem 3.1 and
3.2.

###  3.2 Estimate the Toomer invariant

The idea of Theorem 3.1 can also be used to estimate the Toomer invarian of
relative Sullivan algebras, although the hypothesis is a little different.
This proof is simpler, as we can apply the four lemma.

######  Lemma 3.7 (Four lemma).

In the following commutative diagram of abelian groups, if $\eta,\lambda$ are
injective and $\xi$ is surjective, then $\zeta$ is injective.

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{\prime}}$

######  Theorem 3.8.

Let $\Lambda V=\Lambda Z\otimes\Lambda W$ be a relative Sullivan algebra, and
$P$ be a $\Lambda V$-semifree module. Then $P$ has a $\Lambda Z$-module
structure and assume that $e_{\Lambda Z}(P)=m$. Additionally, $P^{p,\geq q}$
is a $\Lambda W$-module. If $e_{\Lambda W}(P^{p,\geq q})\leq n$ for any $p$
and $q$, then $e_{\Lambda V}(P)\leq(m+1)(n+2)-2$. Moreover, if $\Lambda V$
itself is a minimal Sullivan algebra, then $e_{\Lambda V}(P)\leq(m+1)(n+1)-1$.

###### Proof.

First observe that $P$ is $\Lambda Z$-semifree, and all $P^{p,\geq q}$ are
$\Lambda W$-semifree. So by the hypothesis that $e_{\Lambda Z}(P)=m$, the
projection $P\to P/(\Lambda^{m}Z\cdot P)$ is injective on cohomology. Then
similar to the Proof of Theorem 3.1, let $I_{m+1}=P^{\geq m+1,\geq 0}$ and
$I_{k}=I_{k+1}+P^{\geq k,\geq(m+1-k)(n+2)-1}$ inductively for $0\leq k\leq m$,
or $I_{k}=I_{k+1}+P^{\geq k,\geq(m-k)(n+2)}$ if $\Lambda V$ is minimal.

By hypothesis all $e_{\Lambda W}(P^{p,\geq q})\leq n$. Thus, the projection

$P^{k,\geq(m-k)(n+2)}\to P^{k,\geq(m-k)(n+2)}/(\Lambda^{>n}W\cdot
P^{k,\geq(m-k)(n+2)})=P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}$

(respectively $P^{k,\geq(m-k-1)(n+1)}\to P^{k,[(m-k-1)(n+1),(m-k)(n+1)-1]}$
when $\Lambda V$ is minimal) is injective on cohomology.

As discussed in the proof of Theorem 3.1,
$P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}\cong(P^{\geq
k,\geq(m-k)(n+2)}+I_{k+1})/I_{k}$ can be recognized as a submodule of
$P/I_{k}$. Also $P^{k,\geq(m-k)(n+2)}=(P^{\geq
k,\geq(m-k)(n+2)}+I_{k+1})/I_{k+1}$. So we have the following commutative
diagram of short exact sequences.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{k,\geq(m-k)(n+2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{P/I_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr}$$\textstyle{P/(P^{\geq
k,\geq(m-k)(n+2)}+I_{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{k,[(m-k)(n+2),(m-k)(n+2)+n]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P/I_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P/(P^{\geq
k,\geq(m-k)(n+2)}+I_{k+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

The left vertical morphism is injective on cohomology, and the right one is
isomorphic. By the four lemma, the middle one is also injective on cohomology.
Therefore, we have obtained a sequence of projections

$P\to P/I_{m+1}\to P/I_{m}\to\ldots\to P/I_{1}\to P/I_{0}$

which are all injective on cohomologies. The composition of these projections
can be factored as $P\to P/(\Lambda^{\geq(m+1-k)(n+2)-1}V\cdot P)\to P/I_{0}$,
which implies that the projection $P\to P/(\Lambda^{\geq(m+1)(n+2)-1}V\cdot
P)$ is injective on cohomology. Therefore, $e_{\Lambda V}(P)\leq(m+1)(n+2)-2$.
A similar discussion shows that $e_{\Lambda V}(P)\leq(m+1)(n+1)-1$ if $\Lambda
V$ is minimal. ∎

In the case that $P=\Lambda V$, $P^{p,\geq
q}\cong(\Lambda^{p}Z,0)\otimes(\Lambda^{\geq q}W,\bar{d})$. So $e_{\Lambda
W}(P^{p,\geq q})\leq n$ if any only if $e_{\Lambda W}(\Lambda^{\geq q}W)\leq
n$. Then Theorem 1.3 follows.

## References

  * [1] Yves Félix, Stephen Halperin, and Jean-Claude Thomas.  Rational homotopy theory, volume 205.  Springer Science & Business Media, 2012.
  * [2] Yves Félix, Steve Halperin, and Jean-Claude Thomas.  Rational homotopy theory II.  World Scientific, 2015.
  * [3] S Halperin and J-M Lemaire.  Notions of category in differential algebra.  Algebraic Topology Rational Homotopy, page 138, 1988.
  * [4] Stephen Halperin.  Lectures on minimal models, volume 9.  Gauthier-Villars, 1983.
  * [5] Kathryn P Hess.  A proof of ganea’s conjecture for rational spaces.  Topology, 30(2):205–214, 1991.
  * [6] L Ljusternik and Lev Schnirelmann.  Méthodes topologiques dans les problèmes variationnels: Espaces à un nombre fini de dimensions. 1ère partie, volume 188.  Hermann, 1934.
  * [7] Dennis Sullivan.  Infinitesimal computations in topology.  Publications Mathématiques de l’IHÉS, 47:269–331, 1977\.

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