E extracta mathematicae Vol. 31, Núm. 2, 235 – 250 (2016)

On LS-Category of a Family of Rational Elliptic Spaces II

Khalid Boutahir, Youssef Rami

Département de Mathématiques et Informatique, Faculté des Sciences,
Université My Ismail, B. P. 11 201 Zitoune, Meknès, Morocco

khalid.boutahir@edu.umi.ac.ma y.rami@fs-umi.ac.ma

Presented by Antonio M. Cegarra Received June 24, 2016

Abstract : Let X be a finite type simply connected rationally elliptic CW-complex with Sul-
livan minimal model (ΛV, d) and let k ≥ 2 the biggest integer such that d =

∑
i≥k di

with di(V ) ⊆ ΛiV . If (ΛV, dk) is moreover elliptic then cat(ΛV, d) = cat(ΛV, dk) =
dim(V even)(k − 2) + dim(V odd). Our work aims to give an almost explicit formula of LS-
category of such spaces in the case when k ≥ 3 and when (ΛV, dk) is not necessarily elliptic.
Key words: Elliptic spaces, Lusternik-Schnirelman category, Toomer invariant.

AMS Subject Class. (2010): 55P62, 55M30.

1. Introduction

The Lusternik-Schirelmann category (c.f. [7]), cat(X), of a topological
space X is the least integer n such that X can be covered by n + 1 open
subsets of X, each contractible in X (or infinity if no such n exists). It is
an homotopy invariant (c.f. [3]). For X a simply connected CW complex,
the rational L-S category, cat0(X), introduced by Félix and Halperin in [2] is
given by cat0(X) = cat(XQ) ≤ cat(X).

In this paper, we assume that X is a simply connected topological space
whose rational homology is finite dimensional in each degree. Such space has a
Sullivan minimal model (ΛV,d), i.e. a commutative differential graded algebra
coding both its rational homology and homotopy (cf. §2).

By [1, Definition 5.22] the rational Toomer invariant of X, or equivalently
of its Sullivan minimal model, denoted by e0(ΛV,d), is the largest integer s
for which there is a non trivial cohomology class in H∗(ΛV,d) represented by
a cocycle in Λ≥sV , this coincides in fact with the Toomer invariant of the
fundamental class of (ΛV,d). As usual, ΛsV denotes the elements in ΛV of
“wordlength” s. For more details [1], [3] and [14] are standard references.

In [4] Y. Felix, S. Halperin and J. M. Lemaire showed that for Poincaré
duality spaces, the rational L-S category coincides with the rational Toomer

235



236 k. boutahir, y. rami

invariant e0(X), and in [9] A. Murillo gave an expression of the fundamental
class of (ΛV,d) in the case where (ΛV,d) is a pure model (cf. §2).

Let then (ΛV,d) be a Sullivan minimal model. The differential d is decom-
posable, that is, d =

∑
i≥k di, with di(V ) ⊆ Λ

iV and k ≥ 2.

Recall first that in [8] the authors gave the explicit formula cat(ΛV,d) =
dim V odd + (k − 2) dim V even in the case when (ΛV,dk) is also elliptic.

The aim of this paper is to consider another class of elliptic spaces whose
Sullivan minimal model (ΛV,d) is such that (ΛV,dk) is not necessarily elliptic.
To do this we filter this model by

Fp = Λ≥(k−1)pV =
∞⊕

i=(k−1)p

ΛiV. (1)

This gives us the main tool in this work, that is the following convergent
spectral sequence (cf. §3):

Hp,q(ΛV,δ) ⇒ Hp+q(ΛV,d). (2)

Notice first that, if dim(V ) < ∞ and (ΛV,δ) has finite dimensional coho-
mology, then (ΛV,d) is elliptic. This gives a new family of rationally elliptic
spaces.

In the first step, we shall treat the case under the hypothesis assuming that
HN (ΛV,δ) is one dimensional, being N the formal dimension of (ΛV,d) (cf.
[5]). For this, we will combine the method used in [8] and a spectral sequence
argument using (2). We then focus on the case where dim HN (ΛV,δ) ≥ 2.
Our first result reads:

Theorem 1. If (ΛV,d) is elliptic, (ΛV,dk) is not elliptic and H
N (ΛV,δ) =

Q.α is one dimensional, then

cat0(X) = cat(ΛV,d) = sup{s ≥ 0, α = [ω0] with ω0 ∈ Λ≥sV}.

Let us explain in what follow, the algorithm that gives the first inequality,

cat(ΛV,d) ≥ sup{s ≥ 0, α = [ω0] with ω0 ∈ Λ≥sV} := r.

i) Initially we fix a representative ω0 ∈ Λ≥rV of the fundamental class α
with r being the largest s such that ω0 ∈ Λ≥sV.



on ls-category of a family of rational elliptic spaces ii 237

ii) A straightforward calculation gives successively:

ω0 = ω
0
0 + ω

1
0 + · · · + ω

l
0

with

ωi0 = (ω
i,0
0 ,ω

i,1
0 , . . . ,ω

i,k−2
0 )∈Λ

(k−1)(p+i)V ⊕ Λ(k−1)(p+i)+1V
⊕ ··· ⊕ Λ(k−1)(p+i)+k−2V.

Using δ(ω0) = 0 we obtain dω0 = a
0
2 + a

0
3 + · · · + a

0
t+l with

a0i = (a
0,0
i ,a

0,1
i , . . . ,a

0,k−2
i )∈Λ

(k−1)(p+i)V ⊕ Λ(k−1)(p+i)+1V
⊕ ··· ⊕ Λ(k−1)(p+i)+k−2V.

iii) We take t the largest integer satisfying the inequality:

t ≤
1

2(k − 1)
(
N − 2(k − 1)(p + l) − 2k + 5

)
.

Since d2 = 0, it follows that a02 = δ(b2) for some

b2 ∈
k−2⊕
j=0

Λ(k−1)(p+2)−(k−1)+jV.

iv) We continue with ω1 = ω0 − b2.

v) By the imposition iii), the algorithm leads to a representative ωt+l−1 ∈
Λ≥rV of the fundamental class of (ΛV,d) and then e0(ΛV,d) ≥ r.

Now, dim(V ) < ∞ imply dim HN (ΛV,δ) < ∞. Notice also that the
filtration (1) induces on cohomology a graduation such that HN (ΛV,δ) =
⊕p+q=NHp,q(ΛV,δ). There is then a basis {α1, ...,αm} of HN (ΛV,δ) with
αj ∈ Hpj,qj (ΛV,δ), (1 ≤ j ≤ m). Denote by ω0j ∈ Λ≥rjV a representative of
the generating class αj with rj being the largest sj such that ω0j ∈ Λ≥sjV.
Here pj and qj are filtration degrees and rj ∈{pj(k−1), . . . ,pj(k−1)+(k−2)}.

The second step in our program is given as follow:

Theorem 2. If (ΛV,d) is elliptic and dim HN (ΛV,δ) = m with basis
{α1, . . . ,αm}, then, there exists a unique pj, such that

cat0(X) = sup{s ≥ 0, αj = [ω0j] with ω0j ∈ Λ≥sV} := rj.



238 k. boutahir, y. rami

Remark 1. The previous theorem gives us also an algorithm to determine
LS-category of any elliptic Sullivan minimal model (ΛV,d). Knowing the
largest integer k ≥ 2 such that d =

∑
i≥k di with di(V ) ⊆ Λ

iV and the formal
dimension N (this one is given in terms of degrees of any basis elements of
V ), one has to check for a basis {α1, . . . ,αm} of HN (ΛV,δ) (which is finite
dimensional since dim(V ) < ∞). The NP-hard character of the problem into
question, as it is proven by L. Lechuga and A. Murillo (cf [12]), sits in the
determination of the unique j ∈{1, . . . ,m} for which a represent cocycle ω0j
of αj survives to reach the E∞ term in the spectral sequence (2).

2. Basic facts

We recall here some basic facts and notation we shall need.

A simply connected space X is called rationally elliptic if dim H∗(X,Q) <
∞ and dim(X) ⊗Q < ∞.

A commutative graded algebra H is said to have formal dimension N if
Hp = 0 for all p > N , and HN 6= 0. An element 0 6= ω ∈ HN is called a
fundamental class.

A Sullivan algebra ([3]) is a free commutative differential graded algebra
(cdga for short) (ΛV , d) (where ΛV = Exterior(V odd) ⊗ Symmetric(V even))
generated by the graded K-vector space V =

⊕i=∞
i=0 V

i which has a well
ordered basis {xα} such that dxα ∈ ΛV<α. Such algebra is said minimal if
deg(xα) < deg(xβ) implies α < β. If V

0 = V 1 = 0 this is equivalent to saying

that d(V ) ⊆
⊕i=∞

i=2 Λ
iV .

A Sullivan model ([3]) for a commutative differential graded algebra (A,d)
is a quasi-isomorphism (morphism inducing isomorphism in cohomology)
(ΛV,d) −→ (A,d) with source, a Sullivan algebra. If H0(A) = K, H1(A) = 0
and dim(Hi(A,d)) < ∞ for all i ≥ 0, then, [6, Th.7.1], this minimal model
exists. If X is a topological space any minimal model of the polynomial dif-
ferential forms on X, APL(X), is said a Sullivan minimal model of X.

(ΛV,d) (or X) is said elliptic, if both V and H∗(ΛV,d) are finite dimen-
sional graded vector spaces (see for example [3]).

A Sullivan minimal model (ΛV,d) is said to be pure if d(V even) = 0 and
d(V odd) ⊂ ΛV even. For such one, A. Murillo [9] gave an expression of a cocycle
representing the fundamental class of H(ΛV,d) in the case where (ΛV,d) is
elliptic. We recall this expression here:

Assume dim V < ∞, choose homogeneous basis {x1, . . . ,xn}, {y1, . . . ,ym}



on ls-category of a family of rational elliptic spaces ii 239

of V even and V odd respectively, and write

dyj = a
1
jx1 + a

2
jx2 + · · · + a

n−1
j xn−1 + a

n
j xn, j = 1, 2, . . . ,m,

where each aij is a polynomial in the variables xi,xi+1, . . . ,xn, and consider
the matrix,

A =



a11 a

2
1

an1

a12 a
2
2

an2

a1m a
2
m a

n
m


 .

For any 1 ≤ j1 < · · · < jn ≤ m, denote by Pj1...jn the determinant of the
matrix of order n formed by the columns i1, i2, . . . , in of A:


a1j1

anj1

a1jn a
n
jn


 .

Then (see [9]) if dim H∗(ΛV,d) < ∞, the element ω ∈ ΛV ,

ω =
∑

1≤j1<···<jn≤m
(−1)j1+···+jnPj1...jny1 . . . ŷj1 . . . ŷjn . . .ym, (3)

is a cocycle representing the fundamental class of the cohomology algebra.

3. Our spectral sequence

Let (ΛV,d) be a Sullivan minimal model, where d =
∑

i≥k di with di(V ) ⊆
ΛiV and k ≥ 2. We first recall the filtration given in the introduction:

Fp = Λ≥(k−1)pV =
∞⊕

i=(k−1)p

ΛiV. (4)

Fp is preserved by the differential d and satisfies Fp(ΛV )⊗Fq(ΛV ) ⊆ Fp+q(ΛV ),
∀p,q ≥ 0, so it is a filtration of differential graded algebras. Also, since



240 k. boutahir, y. rami

F0 = ΛV and Fp+1 ⊆ Fp this filtration is decreasing and bounded, so it
induces a convergent spectral sequence. Its 0th-term is

E
p,q
0 =

(
Fp

Fp+1

)p+q
=

(
Λ≥(k−1)pV

Λ≥(k−1)(p+1)V

)p+q
.

Hence, we have the identification:

E
p,q
0 =

(
Λp(k−1)V ⊕ Λp(k−1)+1V ⊕···⊕ Λp(k−1)+k−2V

)p+q
, (5)

with the product given by:

(u0,u1, . . . ,uk−2) ⊗ (u
′
0,u

′
1, . . . ,u

′
k−2) = (v0,v1, . . . ,vk−2)

for all (u0,u1, . . . ,uk−2), (u
′
0,u

′
1, . . . ,u

′
k−2) ∈ E

p,q
0 with vm =

∑
i+j=m uiu

′
j

and m = 0, . . . ,k − 2.
The differential on E0 is zero, hence E

p,q
1 = E

p,q
0 and so the identification

above gives the following diagram:

E
p,q
1

(
Λ(k−1)pV ⊕ Λ(k−1)p+1V ⊕···⊕ Λ(k−1)p+k−2V

)p+q

E
p+1,q
1

(
Λ(k−1)(p+1)V ⊕ Λ(k−1)(p+1)+1V ⊕···⊕ Λ(k−1)(p+1)+k−2V

)p+q+1
δ

∼=

∼=
δ

dk dk+1 dk
d2(k−1)

d2(k−1)−1 dk

with δ defined as follows,

δ(u0,u1, . . . ,uk−2) = (wk,wk+1, . . . ,w2k−2) with wk+j =
∑
i+i′=j

i′=0,...,k−2

dk+iui′.

Let E
p
1 = E

p,∗
1 =

⊕
q≥0 E

p,q
1 and E

∗
1 =

⊕
p≥0 E

p,∗
1 = ΛV as a graded vector

space. In this general situation, the 1st-term is the graded algebra ΛV provided
with a differential δ, which is not necessarily a derivation on the set V of
generators. That is, (ΛV,δ) is a commutative differential graded algebra, but
it is not a Sullivan algebra. This gives, consequently, our spectral sequence:

E
p,q
2 = H

p,q(ΛV,δ) ⇒ Hp+q(ΛV,d). (6)

Once more, using this spectral sequence, the algorithm completed by
proves of claims that will appear, will give the appropriate generating class of
HN (ΛV,δ) that survives to the ∞ term. Accordingly, the explicit formula of
LS category for this general case, is expressed in terms of the greater filtering
degree of a represent of this class.



on ls-category of a family of rational elliptic spaces ii 241

4. Proof of the main results

4.1. Proof of Theorem 1. Recall that (ΛV,d) is assumed elliptic, so
that, cat(ΛV,d) = e0(ΛV,d) [4]. Notice also that the subsequent notations
imposed us sometimes to replace a sum by some tuple and vice-versa.

4.1.1. The first inequality. In what follows, we put:

r = sup{s ≥ 0, α = [ω0] with ω0 ∈ Λ≥sV}.

Denote by p the least integer such that p(k − 1) ≤ r < (p + 1)(k − 1) and let
then ω0 ∈ Λ≥rV . We have

ω0 ∈ (Λ(k−1)pV ⊕···⊕ Λ(k−1)p+k−2V )
⊕ (Λ(k−1)p+k−1V ⊕···⊕ Λ(k−1)p+2k−3V )
⊕···

Since |ω0| = N and dim V < ∞, there is an integer l such that

ω0 = ω
0
0 + ω

1
0 + · · · + ω

l
0

with ω00 6= 0 and ∀i = 0, . . . , l,

ωi0 = (ω
i,0
0 ,ω

i,1
0 , . . . ,ω

i,k−2
0 ) ∈ Λ

(k−1)(p+i)V ⊕···⊕ Λ(k−1)(p+i)+k−2V.

We have successively:

δ(ωi0) = δ
(
ω
i,0
0 ,ω

i,1
0 , . . . ,ω

i,k−2
0

)
=

(
dkω

i,0
0 ,

∑
i′+i′′=1

dk+i′ω
i,i′′

0 ,
∑

i′+i′′=2

dk+i′ω
i,i′′

0 , . . . ,
∑

i′+i′′=k−2
dk+i′ω

i,i′′

0

)
,

δ(ω0) =
l∑

i=0

δ
(
ω
i,0
0 ,ω

i,1
0 , . . . ,ω

i,k−2
0

)
=

l∑
i=0

(
dkω

i,0
0 ,

∑
i′+i′′=1

dk+i′ω
i,i′′

0 ,
∑

i′+i′′=2

dk+i′ω
i,i′′

0 , . . . ,

∑
i′+i′′=k−2

dk+i′ω
i,i′′

0

)



242 k. boutahir, y. rami

Also, we have dω0 = dω
0
0 + dω

1
0 + · · · + dω

l
0, with:

dω00 = d
(
ω
0,0
0 ,ω

0,1
0 , . . . ,ω

0,k−2
0

)
=

(
dkω

0,0
0 ,

∑
i′+i′′=1

dk+i′ω
0,i′′

0 , . . . ,
∑

i′+i′′=k−2
dk+i′ω

0,i′′

0

)
+ · · ·

∈

(
2k−3⊕
k′=k−1

Λ(k−1)p+k
′
V

)
⊕···

dω10 = d
(
ω
1,0
0 ,ω

1,1
0 , . . . ,ω

1,k−2
0

)
=

(
dkω

1,0
0 ,

∑
i′+i′′=1

dk+i′ω
1,i′′

0 , . . . ,
∑

i′+i′′=k−2
dk+i′ω

1,i′′

0

)
+ · · ·

∈

(
3k−4⊕

k′=2k−2
Λ(k−1)p+k

′
V

)
⊕···

· · ·
dωi0 = d

(
ω
i,0
0 ,ω

i,1
0 , . . . ,ω

i,k−2
0

)
=

(
dkω

i,0
0 ,

∑
i′+i′′=1

dk+i′ω
i,i′′

0 , . . . ,
∑

i′+i′′=k−2
dk+i′ω

i,i′′

0

)
+ · · ·

∈

(
(k−1)p+(i+2)k−(i+3)⊕

k′=(k−1)p+(i+1)k−(i+1)

Λ(k−1)p+k
′
V

)
⊕···

Therefore

dω0 =

l∑
i=0

(
dkω

i,0
0 ,

∑
i′+i′′=1

dk+i′ω
i,i′′

0 , . . . ,
∑

i′+i′′=k−2
dk+i′ω

i,i′′

0

)

+

l∑
i=0

(
d2k−2ω

i,1
0 + (d2k−2 + d2k−3)ω

i,2
0 + · · · + (d2k−2 + d2k−3 + · · ·

+ dk+1)ω
i,k−2
0

)
+

∑
k′>2k−2

dk′ω0



on ls-category of a family of rational elliptic spaces ii 243

that is:

dω0 = δ(ω0) +

l∑
i=0

(
d2k−2ω

i,1
0 + (d2k−2 + d2k−3)ω

i,2
0 + · · · + (d2k−2 + . . .

+ dk+1)ω
i,k−2
0

)
+

∑
k′>2k−2

dk′ω0.

As δ(ω0) = 0, we can rewrite:

dω0 = a
0
2 +a

0
3 +· · ·+a

0
t+l with a

0
i = (a

0,0
i , . . . ,a

0,k−2
i ) ∈

k−2⊕
j=0

Λ(k−1)(p+i)+jV.

Note also that t is a fixed integer. Indeed, the degree of a0t+l is greater
than or equal to 2

(
(k − 1)(p + t + l) + k − 2

)
and it coincides with N + 1, N

being the formal dimension of (ΛV,d).
Then

N + 1 ≥ 2
(
(k − 1)(p + t + l) + k − 2

)
.

Hence

t ≤
1

2(k − 1)
(
N − 2(k − 1)(p + l) + 5 − 2k

)
.

In what follows, we take t the largest integer satisfying this inequality.
Now, we have:

d2ω0 = da
0
2 + da

0
3 + · · · + da

0
t+l

= d(a
0,0
2 ,a

0,1
2 , . . . ,a

0,k−2
2 ) + d(a

0,0
3 ,a

0,1
3 , . . . ,a

0,k−2
3 ) + · · ·

+ d(a
0,0
t+l,a

0,1
t+l, . . . ,a

0,k−2
t+l ),

with

d(a
0,0
2 ,a

0,1
2 , . . . ,a

0,k−2
2 ) = dk(a

0,0
2 ,a

0,1
2 , . . . ,a

0,k−2
2 )

+ dk+1(a
0,0
2 ,a

0,1
2 , . . . ,a

0,k−2
2 ) + · · ·

=

(
dka

0,0
2 ,

∑
i′+i′′=1

dk+i′a
0,i′′

2 , . . . ,
∑

i′+i′′=k−2
dk+i′a

0,i′′

2

)

+
(
d2k−1a

0,0
2 + d2k−2a

0,1
2 + · · · , . . .

)
+ · · ·



244 k. boutahir, y. rami

d(a
0,0
3 ,a

0,1
3 , . . . ,a

0,k−2
3 ) = dk(a

0,0
3 ,a

0,1
3 , . . . ,a

0,k−2
3 )

+ dk+1(a
0,0
3 ,a

0,1
3 , . . . ,a

0,k−2
3 ) + · · ·

=

(
dka

0,0
3 ,

∑
i′+i′′=1

dk+i′a
0,i′′

3 , . . . ,
∑

i′+i′′=k−2
dk+i′a

0,i′′

3

)

+
(
d2k−1a

0,0
3 + d2k−2a

0,1
3 + · · · , . . .

)
+ · · ·

· · ·

It follows that:

d2ω0 =

(
dka

0,0
2 ,

∑
i′+i′′=1

dk+i′a
0,i′′

2 , . . . ,
∑

i′+i′′=k−2
dk+i′a

0,i′′

2

)

+
(
d2k−1a

0,0
2 + d2k−2a

0,1
2 + · · · , . . .

)
+ · · ·

+
(
d2k−1a

0,0
3 + d2k−2a

0,1
3 + · · · , . . .

)
+ · · ·

Since d2ω0 = 0, we have(
dka

0,0
2 ,

∑
i′+i′′=1

dk+i′a
0,i′′

2 , . . . ,
∑

i′+i′′=k−2
dk+i′a

0,i′′

2

)
= δ(a02) = 0

with a02 = (a
0,0
2 , . . . ,a

0,k−2
2 ) ∈

⊕k−2
j=0 Λ

(k−1)(p+2)+jV . Consequently a02 is a
δ-cocycle.

Claim 1. a02 is an δ-coboundary.

Proof. Recall first that the general rth-term of the spectral sequence (6) is
given by the formula:

Ep,qr = Z
p,q
r /Z

p+1,q−1
r−1 + B

p,q
r−1,

where
Zp,qr =

{
x ∈ [Fp(ΛV )]p+q | dx ∈ [Fp+r(ΛV )]p+q+1

}
and

Bp,qr = d
(
[Fp−r(ΛV )]p+q−1

)
∩Fp(ΛV ) = d

(
Z
p−r+1,q+r−2
r−1

)
.



on ls-category of a family of rational elliptic spaces ii 245

Recall also that the differential dr : E
p,q
r → E

p+r,q−r+1
r in E

∗,∗
r is induced

from the differential d of (ΛV,d) by the formula dr([v]r) = [dv]r, v being any
representative in Z

p,q
r of the class [v]r in E

p,q
r .

We still assume that dim HN (ΛV,δ) = 1 and adopt notations of §4.1.1.
Notice then ω0 ∈ Z

p,q
2 and it represents a non-zero class [ω0]2 in E

p,q
2 .

Otherwise ω0 = ω
′
0 + d(ω

′′
0 ), where ω

′
0 ∈ Z

p+1,q−1
1 and ω

′′
0 ∈ B

p,q
1 , so that

α = [ω0] = [ω
′
0 − (d− δ)(ω

′′
0 )]. But ω

′
0 − (d− δ)(ω

′′
0 ) ∈ Λ

≥r+1 is a contradic-
tion to the definition of ω0. Now, using the isomorphism E

∗,∗
2
∼= H∗,∗(ΛV,δ),

we deduce that, [ω0]2 ∈ E
p,q
2 (being the only generating element) must sur-

vive to E
p,q
3 , otherwise, the spectral sequence fails to converge. Whence

d2([ω0]2) = [a
0
2]2 = 0 in E

p+2,q−1
2 , i.e., a

0
2 ∈ Z

p+3,q−2
1 + B

p+2,q−1
1 . How-

ever a02 ∈
⊕k−2

j=0 Λ
(k−1)(p+2)+jV , so a02 ∈ B

p+2,q−1
1 , that is a

0
2 = d(x), x ∈⊕k−2

j=0 Λ
(k−1)(p+1)+jV . By wordlength argument, we have necessary a02 = δ(x),

which finishes the proof of Claim 1.

Notice that this is the first obstruction to [ω0] to represent a non zero class
in the term E

∗,∗
3 of (6). The others will appear progressively as one advances

in the algorithm.

Let then b2 ∈
⊕k−2

j=0 Λ
(k−1)(p+2)−(k−1)+jV such that a02 = δ(b2) and put

ω1 = ω0 − b2. Reconsider the previous calculation for it:

dω1 = dω0 −db2
= (a02 + a

0
3 + · · · + a

0
t+l) − (dkb2 + d4b2 + · · ·),

with

dkb2 = dk(b
0
2,b

1
2, . . . ,b

k−2
2 ) = (dkb

0
2,dkb

1
2, . . . ,dkb

k−2
2 ) ∈

k−2⊕
j=0

Λ(k−1)(p+2)+jV,

dk+1b2 = dk+1(b
0
2,b

1
2, . . . ,b

k−2
2 )

= (dk+1b
0
2,dk+1b

1
2, . . . ,dk+1b

k−2
2 ) ∈

k−2⊕
j=0

Λ(k−1)(p+2)+j+1V,

· · ·



246 k. boutahir, y. rami

This implies that

dω1 = a
0
2 + a

0
3 + · · · + a

0
t+l −

(
dkb

0
2,
∑

i′+i′′=1

dk+i′b
i′′
2 , . . . ,

∑
i′+i′′=k−2

dk+i′b
i′′
2

)

−
(
d2k−1b

0
2 + · · · , . . .

)
= a02 −δ(b2) + a

0
3 −

(
d2k−1b

0
2 + · · · , . . .

)
+ · · ·

= a03 −
(
d2k−1b

0
2 + · · · , . . .

)
+ · · · ,

and then:

dω1 = a
1
3 + a

1
4 + · · · + a

1
t+l, with a

1
i ∈

k−2⊕
j=0

Λ(k−1)(p+i)+jV.

So,

d2ω1 = da
1
3 + da

1
4 + · · · + da

1
t+l

=

(
dka

1,0
3 ,

∑
i′+i′′=1

dk+i′a
1,i′′

3 , . . . ,
∑

i′+i′′=k−2
dk+i′a

1,i′′

3

)

+
(
d2k−1a

1,0
3 + · · · , . . .

)
+ · · ·

Since d2ω1 = 0, by wordlength reasons,(
dka

1,0
3 ,

∑
i′+i′′=1

dk+i′a
1,i′′

3 , . . . ,
∑

i′+i′′=k−2
dk+i′a

1,i′′

3

)
= δ(a13) = 0.

We claim that a13 = δ(b3) and consider ω2 = ω1 − b3.
We continue this process defining inductively ωj = ωj−1−bj+1, j ≤ t+l−2

such that:

dωj = a
j
j+2 + a

j
j+3 + · · · + a

j
t+l, with a

j
i ∈

k−2⊕
h=0

Λ(k−1)(p+i)+hV

and a
j
j+2 a δ-cocycle.



on ls-category of a family of rational elliptic spaces ii 247

Claim 2. a
j
j+2 is a δ-coboundary, i.e., there is

bj+2 ∈
k−2⊕
j=0

Λ(k−1)(p+j+2)−(k−1)+jV

such that δ(bj+2) = a
j
j+2; 1 ≤ j ≤ t + l− 2.

Proof. We proceed in the same manner as for the first claim. Indeed, we
have clearly for any 1 ≤ j ≤ t+l−2, ωj = ωj−1−bj+1 = ω0−b2−b3−···−bj+1 ∈
Z
p,q
j+2 and it represents a non zero class [ωj]j+2 in E

p,q
j+2 which is also one

dimensional. Whence as in Claim 1, we conclude that, a
j
j+2 is a δ-coboundary

for all 1 ≤ j ≤ t + l− 2.

Consider ωt+l−1 = ωt+l−2 − bt+l, where δ(bt+l) = at+l−2t+l . Notice that
|dωt+l−1| = |dωt+l−2| = N +1, but by the hypothesis on t, we have d(ωt+l−2) =
at+l−2t+l and then

|d(ωt+l−2 −bt+l)| = |at+l−2t+l −δ(bt+l) − (d−δ)bt+l| = |− (d−δ)bt+l| > N + 1.

It follows that dωt+l−1 = 0, that is ωt+l−1 is a d-cocycle. But it can’t be a
d-coboundary. Indeed suppose that ωt+l−1 = (ω

0
0 + ω

1
0 + · · · + ω

l
0) − (b2 +

b3 + · · · + bt+l), were a d-coboundary, by wordlength reasons, ω00 would be a
δ-coboundary, i.e., there is x ∈

⊕k−2
j=0 Λ

(k−1)p−(k−1)+jV such that δ(x) = ω00.
Then

ω0 = δ(x) + ω
1
0 + · · · + ω

l
0.

Since δ(ω0) = 0, we would have δ(ω
1
0 +· · ·+ω

l
0) = 0 and then [ω0] = [ω

1
0 +· · ·+

ωl0]. But ω
1
0 + · · ·+ ω

l
0 ∈ Λ

>rV , contradicts the property of ω0. Consequently
ωt+l−1 represents the fundamental class of (ΛV,d).

Finally, since ωt+l−1 ∈ Λ≥rV we have

e0(ΛV,d) ≥ r.

4.1.2. For the second inequality. Denote s = e0(ΛV,d) and let
ω ∈ Λ≥sV be a cocycle representing the generating class α of HN (ΛV,d).

Write ω = ω0 + ω1 + · · · + ωt, ωi ∈ Λs+iV . We deduce that:

dω =

(
dkω0 +

∑
i+i′=1

dk+iωi′ + · · · +
∑

i+i′=k−2
dk+iωi′

)
+ dkωk−1 + d2k−1ω0 + · · ·

= δ(ω0,ω1, . . . ,ωk−2) + · · ·



248 k. boutahir, y. rami

Since dω = 0, by wordlength reasons, it follows that δ(ω0,ω1, . . . ,ωk−2) =
0. If (ω0,ω1, . . . ,ωk−2), were a δ-boundary, i.e., (ω0,ω1, . . . ,ωk−2) = δ(x),
then

ω −dx = (ω0,ω1, . . . ,ωk−2) −δ(x) + (ωk−1 + · · · + ωt) − (d−δ)(x)
= (ωk−1 + · · · + ωt) − (d−δ)(x),

so, ω − dx ∈ Λ≥s+k−1V , which contradicts the fact s = e0(ΛV,d). Hence
(ω0,ω1, . . . ,ωk−2) represents the generating class of H

N (ΛV,δ). But
(ω0,ω1, . . . ,ωk−2) ∈ Λ≥sV implies that s ≤ r. Consequently, e0(ΛV,d) ≤ r.

Thus, we conclude that
e0(ΛV,d) = r.

4.2. Proof of Theorem 2. It suffices to remark that since (ΛV,d) is
elliptic, it has Poincaré duality property and then dim HN (ΛV,d) = 1. The
convergence of (6) implies that dim E

∗,∗
∞ = 1. Hence there is a unique (p,q)

such that p+q = N and E
∗,∗
∞ = E

p,q
∞ . Consequently only one of the generating

classes α1, . . . ,αm had to survive to E∞. Let αj this representative class and
(pj,qj) its pair of degrees.

Example 1. Let d =
∑

i≥3 di and (ΛV,d) be the model defined by V
even =

< x2,x
,
2 >, V

odd =< y5,y7,y
,
7 > , dx2 = dx

,
2 = 0, dy5 = x

3
2, dy7 = x

,4
2 and

dy
,
7 = x

2
2x
,2
2 , in which subscripts denote degrees.

For k ≥ 3, l ≥ 0, we have

xk2x
,l
2 = x

k−3
2 x

3
2x
,l
2 = d

(
y5x

k−3
2 x

,l
2

)
.

For k ≥ 4, l ≥ 0,

x
,k
2 x

l
2 = x

l
2x
,k−4
2 x

,4
2 = d

(
xl2x

,k−4
2 y7

)
.

Clearly we have

dim H(ΛV,d) < ∞ and dim H(ΛV,d3) = ∞.

Using A. Murillo’s algorithm (cf. §2) the matrix determining the funda-
mental class is:

A =


 x

2
2 0

0 x
,3
2

x2x
,2
2 0


 ,



on ls-category of a family of rational elliptic spaces ii 249

so, ω = −x22x
,3
2 y

,
7 + x2x

,5
2 y5 ∈ Λ

≥6V is a generator of this fundamental coho-
mology class.

It follows that e0(ΛV,d) = 6 6= m + n(k − 2).

Example 2. Let d =
∑

i≥3 di and (ΛV,d) be the model defined by V
even =

< x2,x
,
2 >, V

odd =< y5,y9,y
,
9 > , dx2 = dx

,
2 = 0, dy5 = x

3
2, dy9 = x

,5
2 and

dy
,
9 = x

3
2x
,2
2 .

Clearly we have

dim H(ΛV,d) < ∞ and dim H(ΛV,d3) = ∞.

Using A. Murillo’s algorithm (cf. §2) the matrix determining the funda-
mental class is:

A =


 x

2
2 0

0 x
,4
2

x22x
,2
2 0


 ,

so, ω = −x22x
,4
2 y

,
9 + x

2
2x
,6
2 y5 ∈ Λ

≥7V is a generator of this fundamental coho-
mology class.

It follows that e0(ΛV,d) = 7 6= m + n(k − 2).

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