arXiv:2207.00839v1 [math.AT] 2 Jul 2022
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL
ELLIPTIC SPACES
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
Abstract. We establish some upper and lower bounds of the rational topological
complexity for certain classes of elliptic spaces. Our techniques permit us in
particular to show that the rational topological complexity coincides with the
dimension of the rational homotopy for some special families of coformal elliptic
spaces.
1. Introduction
Let S be a topological space. We recall that the topological complexity TC(S )
of S , introduced by Farber [9], is defined as the least integer m such that there
exists a family of open subsets U0 , · · · , Um covering S × S and of local continuous
sections si : Ui → S [0,1] of the map ev0,1 : S [0,1] → S × S , λ → (λ(0), λ(1)).
The integer TC is a homotopy invariant which satisfies cat(S ) ≤ TC(S ) ≤ 2cat (S )
where cat (S ) is the Lusternik–Schnirelmann category of S . We refer to [10] and
[8] for more information on these invariants.
If S is a simply-connected CW-complex of finite type and if S0 is its rationalization, then lower bounds of cat(S ) and TC(S ) are given respectively by the rational
LS-category cat 0 (S ) := cat(S0 ) and rational topological complexity TC 0 (S ) :=
TC(S0 ). As is known, these invariants can be characterized in terms of a Sullivan
model (ΛV, d) of S , see Sections 2 and 3 below for more details. We here recall
that, when S is formal (that is, (ΛV, d) → (H ∗ (X; Q), 0) is a quasi-isomorphism),
we have cat 0 (S ) = cl Q (S ) and TC 0 (S ) = zcl Q (S ) where cl Q (S ) and zcl Q (S ) are
respectively the cuplength and zero-divisor cuplength of H ∗ (S ; Q). However, in
general cat 0 and TC 0 can be larger than these cohomological lower bounds.
In this article we study the rational topological complexity of elliptic spaces. Recall that S is elliptic if π∗ (S ) ⊗ Q and H ∗ (S ; Q) are both finite dimensional, see [12,
Ch 6] as a general reference. Explicit expressions for the rational LS-category of
certain classes of elliptic spaces have been established in [1], [18], [16] and [20].
In the continuity of these works, our general goal is to study the rational topological complexity of an elliptic space in terms of its LS-category. Many elliptic
spaces, for instance the homogeneous spaces G/H, admit a pure minimal Sullivan
model (ΛV, d), where pure means dV even = 0 and dV odd ⊂ ΛV even . We will call
such a space a pure elliptic space and our work focus on the study of the rational
topological complexity of these spaces. Considering the formal case, we first note:
2010 Mathematics Subject Classification. 55M30, 55P62.
Key words and phrases. Rational topological complexity, Elliptic spaces.
1
2
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
Theorem 1.1. Let S be an elliptic pure formal space. We have
TC 0 (S ) = 2cat 0 (S ) + χπ (S )
where χπ (S ) = dim πeven (S ) ⊗ Q − dim πodd (S ) ⊗ Q.
Note that the ellipticity of S implies that χπ (S ) ≤ 0. If χπ (S ) = 0 then S is
formal and is called an F0 -space. In order to study the non-formal case we next
establish the following theorem:
Theorem 1.2. Let S be a pure elliptic space and let (ΛV, d) be its minimal Sullivan
model. If there exists an extension (ΛZ, d) ֒→ (ΛV, d) where Z even = V even and
(ΛZ, d) is the model of an F0 -space R, then
TC 0 (S ) ≤ 2cat 0 (R) − χπ (S ).
If moreover there exists k such that dV ⊂ Λk V, then
TC 0 (S ) ≤ 2cat 0 (S ) + χπ (S ).
When S is elliptic and coformal (that is dV ⊂ Λ2 V) we have [11]
cat 0 (S ) = dim πodd (S ) ⊗ Q
and, under the hypothesis of the Theorem 1.2, we obtain
TC 0 (S ) ≤ dim π∗ (S ) ⊗ Q.
The second part of the article is dedicated to the study of the rational topological
complexity of (pure) elliptic coformal spaces. In particular, the following theorem
follows from Theorem 5.1 in Section 5.2. The invariant L0 which appears in the
statement is a certain cuplength defined in the same section.
Theorem 1.3. Let S be a pure elliptic coformal space. Then
cat 0 (S ) + L0 (S ) ≤ TC 0 (S ).
S U(6)
This permits us, for instance, to see that the homogeneous space S = S U(3)×S
U(3)
satisfies TC 0 (S ) = dim π∗ (S ) ⊗ Q = 2cat 0 (S ) + χπ (S ) = 5.
Finally we establish the equality TC 0 (S ) = dim π∗ (S ) ⊗ Q (= 2cat 0 (S ) + χπ (S ))
for some special families of coformal spaces which include spaces for which Theorem 1.3 might not be sufficient to reach the equality.
2. Preliminaries
Along this paper space means a simply-connected CW-complex of finite type
and we work over Q. We refer to [12] and [13] as general references on Rational
Homotopy Theory. A cdga model of a space S is a commutative (cochain) differ≃
≃
ential graded algebra (A, d) with a chain of quasi-isomorphisms (A, d) ←
− ··· −
→
APL (S ) where APL is the Sullivan functor of polynomial forms. Quasi-isomorphism
here means a cdga morphism which induces an isomorphism in cohomology. In
particular H ∗ (A, d) = H ∗ (S ; Q). If (H ∗ (S ; Q), 0) is a cdga model of S then S is said
formal.
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
3
A Sullivan model of S is a cdga model of the form (ΛV, d) where ΛV
Lis free as
a commutative graded algebra and there exists a decomposition V =
k≥0 V(k)
L
≥2
such that dV(0) = 0 and d(V(i)) ⊂
k≤i−1 V(k). If moreover dV ⊂ Λ V, the
model is said minimal. In this case the graded vector space V is isomorphic to the
dual of the rational homotopy vector space π∗ (S ) ⊗ Q of S .
A KS-extension of (ΛV, d) is a cdga
(ΛV ⊗ ΛZ, d) together with a cdga inclusion
L
(ΛV, d) ֒→ (ΛV ⊗ ΛZ, d) where Z =
k≥0 Z(k) and d(Z(k)) ⊂ ΛV ⊗ (Z(0) ⊕ · · · ⊕
Z(k − 1)). Such a KS-extension models a fibration p : T → S over S such that the
quotient algebra
!
ΛV ⊗ ΛZ
, d̄ (ΛZ, d̄)
Λ+ V ⊗ ΛZ
is a model for the fiber of p.
Any cdga morphism (ΛV, d) → (A, d) admits a relative Sullivan model, that is a
decomposition of the form
// (A, d)
88
♣
♣
ξ ♣♣
♣
♣♣♣≃
♣♣♣
(ΛV, d)◆t
◆◆
◆◆◆
◆◆
i ◆◆◆''
(ΛV ⊗ ΛZ, D)
where i is a KS-extension and ξ is a quasi-isomorphism.
Finally we recall the following lifting lemma:
Lemma 2.1 (Lifting lemma). Let consider the solid commutative diagram
ΛZ
_
∃rs
i
s
ΛZ ⊗ ΛX
s
s
// ΛV
s99
ψ ≃
// ΛW
in which i is a KS-extension and ψ is a surjective quasi-isomorphism. Then there is
a (cdga) morphism r : ΛZ⊗ΛX → ΛV which makes the two triangles commutative.
3. Rational topological complexity of certain extensions
Let S be a simply-connected CW-complex of finite type. Recall that the rational
LS-category and topological complexity of S are defined by
cat 0 (S ) := cat (S0 ) and TC 0 (S ) = TC(S0 ),
where S0 is the rationalization of S , and satisfy
cat 0 (S ) ≤ cat (S ) and TC 0 (S ) ≤ TC(S ).
Suppose that (ΛV, d) is a Sullivan model of S . Through the well-known FélixHalperin characterization of cat 0 from [11] we have cat 0 (S ) = cat(ΛV, d) where
cat(ΛV, d) (or simply cat (ΛV)) is the least integer m such that the projection
ΛV
(ΛV, d) → ( >m , d̄)
Λ V
admits a homotopy retraction. In [5], solving a conjecture posed by Jessup, Murillo
and Parent [15], Carrasquel established a characterization of TC 0 which is in the
4
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
same spirit as Félix-Halperin’s characterization of cat 0 . More precisely, let ker µΛV
be the kernel of the multiplication µΛV : ΛV ⊗ ΛV → ΛV of the algebra (ΛV, d)
and denote by ρm the canonical projection
!
ΛV ⊗ ΛV
(ΛV ⊗ ΛV, d) →
,
d̄
.
(ker µΛV )m+1
By denoting by TC(ΛV, d), or simply TC(ΛV), the least integer m for which ρm
admits a homotopy retraction we have:
Theorem 3.1. [5, Theorem 8] TC(ΛV) = TC 0 (S ).
We recall that ρm admits a homotopy retraction if there exists a cdga morphism
i
r : ΛV ⊗ ΛV ⊗ ΛZ → ΛV ⊗ ΛV such that r ◦ i = idΛV⊗ΛV , where ΛV ⊗ ΛV ֒→
ΛV ⊗ΛV ⊗ΛZ is a relative Sullivan model of ρm . This is expressed by the following
diagram
r
✐
❜
ΛV ⊗99 ΛV ⊗ ΛZ
r
rrr
r
r
rr
✠
≃ ξ
rrr
r
r
✌
r i
r
r
✑
rr
✓ + rrr
// ΛV⊗ΛVm+1 .
ΛV ⊗ ΛV
ρ
(ker µ )
r
⑦
m
ΛV
It follows also from Carrasquel’s results that the invariants MTC(S ) and HTC(S )
defined in [14] and [7] can be respectively identified to:
• MTC(ΛV) which is defined as the least integer m for which ρm admit a
homotopy retraction as a morphism of ΛV ⊗ ΛV-module,
• HTC(ΛV) which is defined as the least integer m for which ρm is injective
in cohomology.
These invariants together with the rational zero-divisor cuplength zcl Q (H ∗ (ΛV)) :=
zcl Q (H ∗ (S ; Q)), defined as the largest integer m such that there exist α1 , · · · , αm ∈
ker(H ∗ (µΛV )) satisfying α1 · · · αm , 0, are ordered as follows:
zcl Q (H ∗ (ΛV)) ≤ HTC(ΛV) ≤ MTC(ΛV) ≤ TC(ΛV).
When S is formal, it follows from [17] that all the invariants coincide but as shown
in [14] and [7] the invariants HTC and MTC can in general be larger than the
zero-divisor cuplength. We refer to the survey [3] for more details on the rational
approximations of the topological complexity.
We now state and prove our main theorem in this section.
Theorem 3.2. Let (ΛV, d) be a Sullivan model and let (ΛV ⊗Λu, d) be an extension
of (ΛV, d) with u is an element of odd degree. Then
(i) TC(ΛV ⊗ Λu) ≤ TC(ΛV) + 1.
(ii) MTC(ΛV ⊗ Λu) ≤ MTC(ΛV) + 1.
(iii) HTC(ΛV ⊗ Λu) ≤ HTC(ΛV) + 1.
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
5
Proof. Suppose that TC(ΛV) = m. Then m is the least integer for which the proΛV⊗ΛV
admits a homotopy retraction r as represented
jection ρm : ΛV ⊗ ΛV → (ker
µΛV )m+1
in the following diagram
r
r
✐
❜
ΛV ⊗88 ΛV ⊗ ΛW
rr
rr
rrr
r
r
r
✠
≃ ξ
rrri
✌
r
r
r
r
✏
r
✓ + rrr
ΛV⊗ΛV
/
/
ΛV ⊗ ΛV
ρ
(ker µ )m+1
⑥
m
ΛV
where ξ is a quasi-isomorphism and i is a relative model of ρm . Through the
extension ΛV ⊗ ΛV ֒→ ΛV ⊗ Λu ⊗ ΛV ⊗ Λu ΛV ⊗ ΛV ⊗ Λu ⊗ Λu we
obtain the morphisms ρm ⊗ idΛu ⊗ idΛu , i ⊗ idΛu ⊗ idΛu , ξ ⊗ idΛu ⊗ idΛu and
r ⊗ idΛu ⊗ idΛu which satisfy (ξ ⊗ idΛu ⊗ idΛu )(i ⊗ idΛu ⊗ idΛu ) = ρm ⊗ idΛu ⊗ idΛu
and (r ⊗ idΛu ⊗ idΛu )(i ⊗ idΛu ⊗ idΛu ) = id. Note also that ξ ⊗ idΛu ⊗ idΛu is a
quasi-isomorphism. On the other hand, let consider
≃
(ΛV ⊗Λu)⊗(ΛV ⊗Λu) ֒→ (ΛV ⊗Λu)⊗(ΛV ⊗Λu)⊗ΛZ −
→
(ΛV ⊗ Λu) ⊗ (ΛV ⊗ Λu)
(ker µΛV⊗Λu )m+2
a relative Sullivan model of ρm+1 : (ΛV ⊗ Λu) ⊗ (ΛV ⊗ Λu) → (ΛV⊗Λu)⊗(ΛV⊗Λu)
.
(ker µΛV⊗Λu )m+2
2
Since |u| is odd, we have (ker µΛu ) = 0. This fact together with
ker µΛV⊗Λu ker µΛV ⊗ Λu ⊗ Λu + ΛV ⊗ ΛV ⊗ ker µΛu ,
implies that the composition
ρm ⊗idΛu ⊗idΛu
(ker µΛV⊗Λu )m+2 ֒→ (ΛV ⊗ ΛV) ⊗ (Λu ⊗ Λu) −−−−−−−−−−→
ΛV ⊗ ΛV
⊗ Λu ⊗ Λu
(ker µΛV )m+1
is trivial. Therefore ρm ⊗ idΛu ⊗ idΛu can be factorized as
(ΛV ⊗ ΛV) ⊗ (Λu❙ ⊗ Λu)
ρm ⊗idΛu ⊗idΛu
❙❙❙
❙❙❙
❙❙❙
ρm+1 ❙❙❙❙❙
❙))
//
ΛV⊗ΛV
(ker µΛV )m+1
⊗ (Λu ⊗ Λu)
❧55
❧❧❧
❧
❧
❧
❧❧❧
❧❧❧
(ΛV⊗Λu)⊗(ΛV⊗Λu)
.
(ker µΛV⊗Λu )m+2
Using the relative model of ρm+1 we obtain the diagram
ΛV ⊗ ΛV ⊗ _ Λu ⊗ Λu
❨❨❨
❨❨❨❨❨❨
❨❨❨❨❨ρ❨m ⊗idΛu ⊗idΛu
❨❨❨❨❨❨
❨❨❨❨❨❨
❨❨❨❨❨,,
(ΛV⊗Λu)⊗(ΛV⊗Λu)
//
/
/
(ΛV ⊗ Λu) ⊗ (ΛV ⊗ Λu) ⊗ ΛZ
(ker µ
)m+2
ΛV⊗Λu
ΛV⊗ΛV
(ker µΛV )m+1
⊗ Λu ⊗ Λu.
6
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
We can then form the following commutative solid diagram
i⊗idΛu ⊗idΛu
// (ΛV ⊗ ΛV ⊗ ΛW) ⊗ (Λu ⊗ Λu)
❡ ❡ ❡22
❡
τ ❡ ❡ ❡
❡
≃ ξ⊗idΛu ⊗idΛu
❡
❡ ❡ ❡
❡
❡
❡
ΛV⊗ΛV
/
/
(ΛV ⊗ Λu) ⊗ (ΛV ⊗ Λu) ⊗ ΛZ // (ΛV⊗Λu)⊗(ΛV⊗Λu)
⊗
(Λu ⊗ Λu)
(ker µ
)m+2
(ker µ )m+1
ΛV ⊗ ΛV ⊗ _ Λu ⊗ Λu
ΛV⊗Λu
ΛV
and, by the lifting lemma, there is a morphism τ which makes the complete diagram
commutative. As a result, the composition (r ⊗ IdΛu ⊗ IdΛu ) ◦ τ is the desired
homotopy retraction of ρm+1 : (ΛV ⊗ Λu) ⊗ (ΛV ⊗ Λu) → (ΛV⊗Λu)⊗(ΛV⊗Λu)
, and we
(ker µΛV⊗Λu )m+2
conclude that TC(ΛV ⊗ Λu) ≤ m + 1.
Assertions (ii) and (iii) are obtained in the same way as (i) by taking respectively a
ΛV ⊗ΛV-retraction and a linear differential retraction instead of a (cdga) homotopy
retraction.
By induction we can generalize this result to get the following corollary, which
can be seen as a version for TC of [12, Prop 30.7].
Corollary 3.1. If (ΛV, d) ֒→ (ΛV ⊗ ΛU, d) is an extension of (ΛV, d), where U is a
graded vector space concentrated in odd degrees with d(U) ⊂ ΛV and dim U = n,
then
(i) TC(ΛV ⊗ ΛU) ≤ TC(ΛV) + n.
(ii) MTC(ΛV ⊗ ΛU) ≤ MTC(ΛV) + n.
(iii) HTC(ΛV ⊗ ΛU) ≤ HTC(ΛV) + n.
4. An upper bound of TC 0 for certain pure elliptic spaces
We recall that a space S is said elliptic if π∗ (S ) ⊗ Q and H ∗ (S ; Q) are finite
dimensional and pure if it admits a Sullivan model (ΛV, d) such that dV even = 0
and dV odd ⊂ ΛV even . If (ΛV, d) is minimal then V π∗ (S ) ⊗ Q and the homotopy
characteristic of S given by χπ (S ) = dim πeven (S ) ⊗ Q − dim πodd (S ) ⊗ Q coincides
with χπ (ΛV) = dim V even − dim V odd . Abusing language we use the terminology
"pure", "formal", "elliptic" for both space and its minimal Sullivan model. Recall
that by ellipticity we always have χπ (ΛV) ≤ 0 and that an elliptic minimal model
(ΛV, d) is called an F0 -model if χπ (ΛV) = 0. Theorem 1.1 follows from
Theorem 4.1. If (ΛV, d) is a pure elliptic minimal Sullivan model which is formal
then
TC(ΛV) = 2cat (ΛV) + χπ (ΛV).
Proof. Since (ΛV, d) is formal, it can be decomposed as
(ΛV, d) = (ΛV ′ , d) ⊗ (Λ(z1 , · · · , zl ), 0)
where (ΛV ′ , d) is an F0 -model and zi are all generators of odd degree (see for
instance, [2, Lemma 1.5]). As (ΛV, d) is formal, we have by [17]
TC(ΛV) = MTC(ΛV) = zcl Q (H ∗ (ΛV)).
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
7
As is well-known we also have cat (ΛV) = cl Q (H ∗ (ΛV)) [12]. By [15] (see also
[4, th.12]) we have that MTC(ΛV) = MTC(ΛV ′ )+l. Moreover since (ΛV ′ , d) is an
F0 -model, we have H ∗ (ΛV ′ ) = H even (ΛV ′ ). Therefore we have zcl Q (H ∗ (ΛV ′ )) =
2cl Q (H ∗ (ΛV ′ )) [6, Corollary 30]. Since ΛV ′ is formal, it follows that MTC(ΛV ′ ) =
2 · cat (ΛV ′ ) and therefore
TC(ΛV) = 2cat (ΛV ′ ) + l.
On the other hand, by the additivity of cat 0 with respect to the product (see for
instance, [12, Th 30.2]), we have
2cat (ΛV) + χπ (ΛV) = 2(cat (ΛV ′ ) + l) − l
= 2cat (ΛV ′ ) + l.
Finally we get TC(ΛV) = 2cat (ΛV) + χπ (ΛV).
Our next result, which corresponds to Theorem 1.2 from the introduction, will
permits us to obtain, under some conditions, an upper bound for TC(ΛV) in the
non-formal case.
Theorem 4.2. Let (ΛV, d) be a pure elliptic minimal Sullivan model. If there exists
an extension (ΛZ, d) ֒→ (ΛV, d) where Z even = V even and (ΛZ, d) is an F0 -model,
then
TC(ΛV) ≤ 2cat (ΛZ) − χπ (ΛV).
If moreover there exists k such that dV ⊂ Λk V, then
TC(ΛV) ≤ 2cat (ΛV) + χπ (ΛV).
Remark 1. According to [16, Lemma 3.3] (see the remark after the proof of
Lemma 3.3), if V even is concentrated in a single degree then it is always possible to
construct an F0 -model (ΛZ, d) as required in hypothesis of Theorem 4.2
Proof of Theorem 4.2. In the conditions of the theorem we can suppose that V =
Z ⊕ U where U is a subspace of V concentrated in odd degrees. By Corollary 3.1,
we have
TC(ΛV) = TC(Λ(Z ⊕ U), d)
≤ TC(ΛZ, d) + dim U.
Since (ΛZ, d) is an F0 -model, we have, by Theorem 4.1, TC(ΛZ) = 2 · cat(ΛZ)
and we obtain
TC(ΛV) ≤ 2 · cat (ΛZ) + dim U. (∗)
On the other hand dim V even = dim Z even = dim Z odd implies that
dim U =
=
=
=
dim V − dim Z
dim V even + dim V odd − dim Z even − dim Z odd
dim V odd − dim V even
−χπ (ΛV).
We then obtain TC(ΛV) ≤ 2cat (ΛZ) − χπ (ΛV).
8
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
We now suppose that the differential d is homogeneous of rank k (d = dk ).
By Lechuga-Murillo formula for the rational LS-category of such an elliptic space
[18], if dim V even = n and dim V odd = m, thus
cat(ΛZ) = n(k − 2) + n = n(k − 1)
and
cat (ΛV) = n(k − 2) + m.
As we have
2 · cat (ΛV) + χπ (ΛV) =
=
=
=
2[n(k − 2) + m] + n − m
2n(k − 1) − 2n + 2m + n − m
2n(k − 1) + m − n
2cat (ΛZ) + dim U,
inequality (∗) finally implies TC(ΛV) ≤ 2cat (ΛV) + χπ (ΛV).
5. The coformal case
Along this section, we consider a pure coformal model (ΛV, d) where dim V is
finite. Recall that coformal means that dV ⊂ Λ2 V. We denote by X = V even and
Y = V odd . As the model is pure we have dX = 0 and dY ⊂ ΛX. We note that
(ΛV, d) is not required to be elliptic. However, given a basis B = {x1 , · · · , xn } of X,
we associate with (ΛV, d) the elliptic extension
ΛWB = Λ(X ⊕ Y ⊕ U),
where U is the graded vector space generated by u1 , · · · , un with dui = x2i for all
i = 1, · · · , n. Let point out that the elliptic extension (ΛWB , d) associated to (ΛV, d)
satisfies the inequality in the proposition below:
Proposition 5.1. The elliptic extension (ΛWB , d) of the model (ΛV, d) satisfies
TC(ΛWB ) ≤ dim WB = 2 dim V even + dim V odd .
Proof. It suffices to consider the extension (Λ(X ⊕ U), d) ֒→ (ΛWB , d) and then the
result follows by Theorem 4.2.
We will show later that this construction permits us to obtain a lower bound
of the rational topological complexity of (ΛV, d). In our calculations, it will be
useful to reduce the algebra ΛWB to a smaller algebra AB through the surjective
quasi-isomorphism




Λ(xi )
ϕ : (ΛWB , d) ։ AB = 2 ⊗ ΛY, d̄
(xi )
defined by ϕ(x) = x, ϕ(y) = y, ϕ(u) = 0, for x ∈ X, y ∈ Y and u ∈ U. In the sequel
we will simply write ΛW (resp. A) instead of ΛWB (resp. AB ) if no confusion
arises.
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
9
5.1. The cocycle Ω. Recall that an elliptic space is a Poincaré duality space. In
[19], Lechuga and Murillo gave an efficient method to obtain an explicit expression
of a cocycle representing the fundamental class of any pure elliptic model. We first
use this process to construct a representing cocycle ω for the fundamental class of
the elliptic extension ΛW of ΛV described above. Let X̄ = sX be the suspension of
X with d extended by d x̄ = x, ∀x ∈ X. Using the same notations as above we have
W even = X and W odd = Y ⊕ U.
We work in ΛW ⊗ ΛX̄. Let {y1 , · · · , ym } be a basis of Y. Since d is a quadratic
differential, for each j = 1, · · · , m, we may express d(y j ) as
X
X
j
β p,q
x p · xq
d(y j ) =
αkj x2k +
p<q
k
= d{
X
j
αk xk
· x̄k +
where
j
β p,q x p · x̄q }
p<q
k
j
j
αk , β p,q
X
∈ Q. In addition
d(ui ) = x2i
= d(xi · x̄i ).
Q
According to [19], ω can be obtained as the cofficient of ni=1 x̄i after developing
the product
m
n
Y
X
X
Y
j
j
(y j −
αk xk · x̄k −
β p,q x p · x̄q ) ·
(ui − xi · x̄i ).
j=1
p<q
k
i=1
It is clear to see that
ϕ(ω) = (−1)n x1 · · · xn · y1 · · · ym .
By setting x[n] = x1 · · · xn , y[m] = y1 · · · ym and ωA = (−1)n · ϕ(ω) = x[n] · y[m] ,
we remark that ωA denote the unique (up to a scalar) representing element for the
fundamental class of the algebra A.
By adapting the technique above, we shall construct an important cocycle Ω ∈
(ker µΛW )m+n , which will be a key ingredient to obtain a lower bound of the TC(ΛV).
More precisely, we consider a second copy ΛW ′ of ΛW and we consider ΛW ⊗
ΛW ′ Λ(W ⊕ W ′ ) writing x′i , u′i and y′j for the elements corresponding to xi , ui and
y j . Working in ΛW ⊗ X̄ ⊗ ΛW ′ ⊗ X̄ ′ , the odd generators of ker µΛW , where µΛW is
the multiplication over ΛW, satisfy
d(ui − u′i ) = d[(xi − x′i )( x̄i + x̄′i )], for all i = 1, · · · , n,
and
d(y j − y′j ) =
X
k
j
αk (x2k − x′ 2k ) +
X
j
β p,q (x p xq − x′p x′q )
p<q
X
1X j
j
= d[
αk (xk − x′ k )( x̄k + x̄′k ) +
β p,q (x p − x′ p )( x̄q + x̄′q )
2
p<q
k
1X j
β (xq − x′ q )( x̄ p + x̄′p )], for all j = 1, · · · , m.
+
2 p<q p,q
10
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
By setting
φj =
X
αkj (xk −x′ k )( x̄k + x̄′k )+
k
1X j
1X j
β p,q (x p −x′ p )( x̄q + x̄′q )+
β (xq −x′ q )( x̄ p + x̄′p )
2 p<q
2 p<q p,q
we obtain that, the elements
• y j − y′j − φ j , with j = 1, · · · , m
• ui − u′i − (xi − x′i )( x̄i + x̄′i ), with i = 1, · · · , n
are all cocycles in ΛW ⊗ ΛX̄ ⊗ ΛW ′ ⊗ ΛX̄ ′ .
Q
We define Ω to be the coefficient of ni=1 ( x̄i + x̄′i ) gotten after developing
m
n
Y
Y
[y j − y′j − φ j ] ·
[ui − u′i − (xi − x′i )( x̄i + x̄′i )].
j=1
i=1
Remark that
ΩA := (−1)n · (ϕ ⊗ ϕ)(Ω) =
n
Y
(xi − x′i ) ·
i=1
m
Y
(y j − y′j ).
j=1
We will see through the following lemma, that Ω is a cocycle with non-zero cohomology class.
Lemma 5.1. Ω is a cocycle in (ker µΛW )m+n with non-zero cohomolgy class.
Proof. By construction it is clear that Ω ∈ (ker µΛW )n+m . Now, considering the
decomposition
ΛW ⊗ ΛW ′ ⊗ Λ≤n (X̄ ⊕ X̄ ′ ) = ΛW ⊗ ΛW ′ ⊗ Λn (X̄ ⊕ X̄ ′ ) ⊕ ΛW ⊗ ΛW ′ ⊗ Λ<n (X̄ ⊕ X̄ ′ ),
we have
m
n
n
i
Y
Y
Y
X
Y
′
′
′
′
′
[y j −y j −φ j ]· [ui −ui −(xi − xi )( x̄i + x̄i )] = Ω· ( x̄i + x̄i )+
Θi · ( x̄ik + x̄′ik ),
j=1
i=1
i=1
0≤i<n
k=1
where Θi ∈ ΛW ⊗ ΛW ′ , for i = 1, · · · , n − 1.
Applying the differential d to the equation above we get, after identification, dΩ =
0.
To see that [Ω] , 0, we proceed by contradiction and suppose that [Ω] = 0. Then
[ΩA ] = 0. Let A′ be a second copy of the algebra A. Recall that ω′A = x′[n] · y′[m]
with x′[n] = x′1 · · · x′n and y′[m] = y′1 · · · y′m is the unique element (up to a scalar)
representing the fundamental class of A′ . Since [ΩA ] = 0, we have
[ΩA ] · [ω′A ] = 0.
(1)
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
11
On the other hand
n
m
Y
Y
[ΩA ] · [ω′A ] = [ (xi − x′i ) ·
(y j − y′j )] · [x′[n] · y′[m] ]
i=1
n
Y
j=1
(xi − x′i ) · x′[n] ·
= [
i=1
m
Y
(y j − y′j ) · y′[m] ]
j=1
= [x[n] x′[n] y[m] y′[m] ]
= [ωA ][ω′A ]
, 0,
which contradicts (1).
5.2. Lower bound of TC(ΛV). As previously we consider a pure coformal model
(ΛV, d) with dim V < ∞ and the elliptic extension
ΛWB = Λ(X ⊕ Y ⊕ U),
associated to the basis B = {x1 , · · · , xn } of X = V even . We recall that U is the
graded vector space generated by u1 , · · · , un , with dui = x2i , and we also fix a basis
{y1 , · · · , ym } of Y = V odd . Since d is a quadratic differential, it splits into as the sum
of the following operators
d p,q : Λ p X ⊗ Λq (Y ⊕ U) → Λ p+2 X ⊗ Λq−1 (Y ⊕ U).
We then obtain a (lower) bigradation on the cohomology of ΛW given by
ker(d p,q )
.
H p,q (ΛW) =
Im(d p−2,q+1 )
Recall the quasi-isomorphism
ϕ : ΛWB = Λ(xi , y j , ui ) → A =
Λ(xi )
⊗ Λ(y j ).
(x2i )
Denoting by M p (X) the vector space ϕ(Λ p X), the differential of A splits into the
sum of
d p,q : M p (X) ⊗ Λq Y −→ M p+2 (X) ⊗ Λq−1 Y
and we obtain a lower bigradation of H(A) given by
ker(d p,q )
H p,q (A) =
.
Im(d p−2,q+1 )
This yields an isomorphism compatible with the bigradation structure
H∗,∗ (ϕ) : H∗,∗ (ΛWB ) −
→ H∗,∗ (A).
We then define the following cuplength which takes into account only the cohomology classes whose wordlength in X = V even is odd.
L(ΛV, B) := max{r : ∃α1 , · · · , αr ∈ Hodd,∗ (A) such that α1 · · · αr , 0 in H(A)}.
This gives a lower bound to the rational topological complexity through the following theorem.
12
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
Theorem 5.1. Let (ΛV, d) be a pure coformal model with dim V < ∞. For every
basis B of V even , we have
dim(V odd ) + L(ΛV, B) ≤ TC(ΛV).
If (ΛV, d) is the minimal Sullivan model of S then the invariant L0 (S ) mentioned
in Theorem 1.3 is defined as
L0 (S ) = sup{L(ΛV, B) : B is a basis for V even }
and Theorem 1.3 of the introduction follows directly from Theorem 5.1 together
with the fact that, for an elliptic coformal space cat 0 (S ) = dim πodd (S ) ⊗ Q.
Before to begin proving our result, we need to set some lemmas, which will be
the keys of the demonstration.
As before we consider ΛW ′ a second copy of ΛW as well as a second copy A′ of
A.
Lemma 5.2. Let 1 ≤ q ≤ m and 1 ≤ j1 < · · · < jq ≤ m. In both ΛW ⊗ ΛW ′ and
A ⊗ A′ we have
m
m
Y
Y
′
(yl − yl ) · y j1 · · · y jq =
(yl − y′l ) · y′j1 · · · y′jq .
j=1
l=1
Proof. First, we have
m
Y
(yl − y′l ) · y j1
=
l=1
j1 −1
m
Y
Y
(yl − y′l ) · (y j1 − y′j1 ) ·
(yl − y′l ) · y j1
l=1
=
l= j1 +1
(−1)ε ·
j1 −1
Y
(yl − y′l ) · (y j1 − y′j1 ) · y j1 ·
l=1
Qm
m
Y
(yl − y′l ).
l= j1 +1
y′l )||y j1 |.
where ε = | l= j1 +1 (yl −
As |y j1 | is odd, (y j1 −
· y j1 = (y j1 − y′j1 ) · y′j1
and it follows that
j1 −1
m
m
Y
Y
Y
(yl − y′l ) · y j1 = (−1)ε ·
(yl − y′l ) · (y j1 − y′j1 ) · y′j1 ·
(yl − y′l )
l=1
y′j1 )
l=1
=
j1 −1
Y
=
m
Y
l= j1 +1
(yl − y′l ) · (y j1 − y′j1 ) ·
l=1
m
Y
(yl − y′l ) · y′j1
l= j1 +1
(yl − y′l )y′j1 .
l=1
Hence, the result is true for y j1 .
Next, by induction suppose that the equality is satisfied up to the rank q − 1, which
means
m
m
Y
Y
(yl − y′l ) · y j1 · · · y jq−1 =
(yl − y′l ) · y′j1 · · · y′jq−1 .
l=1
l=1
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
13
Thus, we get
m
Y
(yl − y′l ) · y j1 · · · y jq
l=1
m
Y
= ( (yl − y′l ) · y j1 · · · y jq−1 ) · y jq
l=1
m
Y
= ( (yl − y′l ) · y′j1 · · · y′jq−1 ) · y jq
l=1
|
= (−1)
Qm
′
′
′
l=1 (yl −yl )||y j1 ···y jq−1 |
m
Y
· y′j1 · · · y′jq−1 · ( (yl − y′l ) · y jq ).
l=1
By the same calculation as in the first step, we have
m
m
Y
Y
(yl − y′l ) · y jq =
(yl − y′l ) · y′jq .
l=1
l=1
Then
m
Y
(yl − y′l ) · y j1 · · · y jq
|
= (−1)
Qm
′
′
′
l=1 (yl −yl )||y j1 ···y jq−1 |
m
Y
· y′j1 · · · y′jq−1 · ( (yl − y′l ) · y′jq )
l=1
l=1
m
Y
=
(yl − y′l ) · y′j1 · · · y′jq .
l=1
Lemma 5.3. Let z1 , · · · , zr ∈ A such that for any 1 ≤ i ≤ n, zi ∈ M2si +1 (X) ⊗ ΛY
with si ∈ N. We have
ΩA · (z1 − z′1 ) · · · (zr − z′r ) = 2r · ΩA · z1 · · · zr
where ΩA = (−1)n · (ϕ ⊗ ϕ)(Ω).
Proof. An element z ∈ M2s+1 (X) ⊗ ΛY can be written as
X
αI,J xI · yJ
I,J
where αI,J ∈ Q, I = {1 ≤ i1 < i2 < · · · < i2p+1 ≤ n}, J = {1 ≤ j1 < j2 · · · < jq ≤ m},
xI := xi1 · · · xi2p+1 and yJ := y j1 · · · y jq . By setting Iˆ = {1, · · · , n} \ I, and using the
oddness of card(I) we have
ΩA · z =
n
m
X
Y
Y
αI,J
(xk − x′k ) ·
(yl − y′l ) · xI · yJ
I,J
k=1
l=1
m
Y
X
Y
(xk − x′k ) ·
(yl − y′l ) · yJ .
= − αI,J · xI · x′I
I,J
k∈ Iˆ
l=1
14
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
Analogously, and using Lemma 5.2, we get
n
m
X
Y
Y
ΩA · z′ =
αI,J
(xk − x′k ) ·
(yl − y′l ) · x′I · y′J
I,J
=
=
k=1
l=1
αI,J
I,J
m
Y
Y
′
′
(xk − xk ) ·
(yl − y′l ) · y′J
· xI · xI
X
αI,J
m
Y
Y
(xk − x′k ) ·
· xI · x′I
(yl − y′l ) · yJ .
X
k∈ Iˆ
l=1
k∈ Iˆ
I,J
l=1
Therefore
ΩA · (z − z′ ) = −2
X
αI,J · xI · x′I
m
Y
Y
(xk − x′k ) ·
(yl − y′l ) · yJ
k∈ Iˆ
I,J
l=1
= 2ΩA · z.
(2)
By induction, we suppose that
ΩA ·
r−1
r−1
Y
Y
(zk − z′k ) = 2r−1 ΩA ·
zk
k=1
k=1
and we have
ΩA ·
r
r−1
Y
Y
(zk − z′k ) = ΩA ·
(zk − z′k ) · (zr − z′r )
k=1
k=1
= 2r−1 ΩA ·
r−1
Y
zk · (zr − z′r )
k=1
= 2r−1 (−1)|ΩA ||
Qr−1
k=1 zk |
·
r−1
Y
zk · ΩA · (zr − z′r ).
k=1
By (2), ΩA · (zr − z′r ) = 2 · ΩA · zr , which implies
ΩA ·
r
r−1
Y
Y
Qr−1
(zk − z′k ) = 2r−1 (−1)|ΩA || k=1 zk | ·
zk · (2)ΩA · zr
k=1
k=1
= 2r · ΩA ·
r
Y
zk .
k=1
Proof of Theorem 5.1. Suppose L(ΛV, B) = r. Using the notations introduced before as well as the cocycle Ω defined in Section 5.1, we first construct a cocycle in
(ker µΛWB )m+n+r whose cohomology class is non-zero. This will permit us to see
that HTC(ΛWB ) ≥ m+n+r. Since L(ΛV, B) = r there exists r cohomology classes,
[z1 ], · · · , [zr ] ∈ Hodd,∗ (A) such that [z1 ] · · · [zr ] , 0. Recall that the bigradation of
H(A) comes from the splitting of d as the sum of the operators
d p,q : M p (A) ⊗ Λq Y → M p+2 (A) ⊗ Λq−1 Y.
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
15
We can then assume that each cocycle zi ∈ M2ki +1,hi (X) ⊗ Λhi Y, ∀i = 1, · · · , r.
It follows that the elements
Q z1 , · · · , zr satisfy the conditions of Lemma 5.3. In
addition, the element z = rk=1 zk is also a cocyle in A, and x[n] · y[m] is the unique
cocycle (up to a scalar) in A representing the fundamental class of H ∗ (A). Then, by
Poincaré duality in H ∗ (A), there is another cocycle ẑ ∈ A satisfying z · ẑ = x[n] · y[m] .
Now, using the Lemma 5.3 we obtain that
r
r
Y
Y
ΩA ·
(zk − z′k ) · ẑ = 2r · ΩA ·
zk · ẑ
k=1
k=1
= 2r ΩA · x[n] · y[m]
= 2r · (−1)n+m x′[n] · y′[m] · x[n] · y[m]
, 0.
Since ϕ is a quasi-isomorphism, there exists for each i, [αi ] ∈ H ∗ (ΛWB , d) such that
[ϕ(αi )] = [zi ], as well as a cocycle α̂ ∈ ΛW, such that [ϕ(α̂)] = [ẑ]. We therefore
obtain
r
r
Y
Y
′
(ϕ ⊗ ϕ)(Ω ·
(αk − αk ) · α̂) = ΩA ·
(zk − z′k ) · ẑ
k=1
k=1
, 0.
Q
Now, we can see clearly that Ω · rk=1 (αk − α′k ) · α̂ is a cocycle in (ker µΛWB )m+n+r ,
satisfying
r
Y
[Ω ·
(αk − α′k ) · α̂] , 0.
k=1
We then have
m + n + L(ΛV, B) ≤ HTC(ΛWB , d) ≤ TC(ΛWB , d).
On the other hand it follows from Corollary 3.1, that
TC(ΛWB ) ≤ TC(ΛV) + n.
We finally obtain
dim(V odd ) + L(ΛV, B) ≤ TC(ΛV).
Example 1. We consider a Sullivan model of the form
(ΛV, d) = (Λ(x1 , x2 , y1 , y2 , y3 ), d)
where |x1 | and |x2 | are even and the differential is given by dx1 = dx2 = 0, dy1 = x21 ,
S U(6)
dy1 = x21 and dy3 = x1 x2 . For instance the homogeneous space S U(3)×S
U(3) admits
such a model with |x1 | = 4 and |x2 | = 6.
Since we have an extension (ΛZ, d) ֒→ (ΛV, d) where (ΛZ, d) = (Λ(x1 , x2 , y1 , y2 ), d)
is an F0 -model and χπ (ΛV) = −1 we have by Theorem 4.2 TC(ΛV) ≤ 5. We
have cat (ΛV) = dim V odd = 3 and we can check that zcl Q (H(ΛV)) = 3. We
now compute L(ΛV, B) where B = {x1 , x2 }. The extension ΛWB is given by
16
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
ΛWB = (ΛV ⊗ ΛU, d) where U =< u1 , u2 > with du1 = x21 and du2 = x22 . We
consider the quasi-isomorphism


 Λ(xi )


ϕ : (ΛWB , d) = (ΛV ⊗ ΛU, d) → (A, d̄) =  2 ⊗ Λ(y1 , y2 , y3 ), d̄
(xi )
and note that d̄y1 = d̄y2 = 0 and d̄y3 = x1 x2 . The elements of Hodd,∗ (A) corresponds to the cocycles of the form
x1 γ,
x2 γ,
x1 y3 γ,
x2 y3 γ where γ ∈ Λ(y1 , y2 )
The product of z1 = x1 and z2 = x2 y3 gives a non-trivial class of H(A) so that
L(ΛV, B) ≥ 2. Since cat (ΛV) = dim V odd = 3 we can then conclude by Theorem 5.1 that TC(ΛV) ≥ 5 and therefore TC(ΛV) = dim V = 5. In particular, as
S U(6)
indicated in the introduction TC 0 ( S U(3)×S
U(3) ) = 5.
Remark 2. In the example above we clearly have
A = Â ⊗ (Λ(y1 , y2 ), 0))



 Λ(xi )
where  =  2 ⊗ Λy3 , d̄
(xi )
and Hodd,∗ (A) Hodd,∗ (Â) ⊗ Λ(y1 , y2 ). Consequently the length L(ΛV, B) can be
calculated by considering only the element of Hodd,∗ (Â).
Example 2. We consider (ΛV, d) given by
ΛV = Λ(x1 , x2 , x3 , x4 , y1 , y2 , y3 , y4 , y5 )
with dyi = x2i for i = 1, · · · , 4 and dy5 = x1 x2 − x3 x4 .
As in Example 1 we can write ΛV as an extension ΛZ ⊗ Λy5 where ΛZ is an F0 model and we have TC(ΛV) ≤ dim V = 9.
Considering the extension ΛWB = ΛV ⊗ΛU associated to B = {x1 , · · · , x4 } and the
observation
made in
Remark 2 we calculate L(ΛV, B) by considering the algebra
i)
⊗ Λy5 , d̄ where d̄y5 = x1 x2 − x3 x4 . Note that the fundamental class of
 = Λ(x
(x2i )
the algebra is given by the cocycle x1 x2 x3 x4 y5 . We can see that the generators of
Hodd,∗ (Â) correspond to the cocycles
x1 , · · · , x4 ,
x1 x2 x3 y5 ,
x1 x2 x4 y5 ,
x1 x3 x4 y5 ,
x2 x3 x4 y5 .
The maximal non-trivial products of the corresponding cohomology classes are obtained from either products of the form xi x j with i , j or products of two cocycles
resulting in the fundamental cocycle x1 x2 x3 x4 y5 . We then obtain that L(ΛV, B) = 2
and the inequality of Theorem 5.1 gives us TC(ΛV) ≥ 7, while as mentioned above
TC(ΛV) ≤ 9.
It then appears that the approach through Theorem 5.1 is not sufficient to completely determine the topological complexity of this example. However, the more
specific calculations we do in the next section will permit us to see that in this
particular case the actual value is 9 (see Theorem 5.2).
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
17
5.3. Special families. In this section, we are going to determine the rational topological complexity of some families of spaces each of which can be seen as the
elliptic extension of a (non-necessarily elliptic) pure coformal space. More exactly we consider elliptic spaces of the form ΛW = (Λ(xi , ui , y j ), d), dui = x2i and
dy j ∈ Λ(xi ) where i = 1, · · · , n and j = 1, · · · , m. Such a space can be seen as
the elliptic extension of the (non-necessarly elliptic) model Λ(xi , y j ). Since the differential is quadratic P
and dui = x2i , through a change of variables we can always
αi, j xi x j , αi, j ∈ Q. We will use the construction of the
suppose that dy j =
i< j
cocycle Ω as well as the quasi-isomorphism



 Λ(xi )

ϕ : (Λ(xi , ui , y j ), d) ։ (A, d) =  2 ⊗ Λ(y j ), d̄ ,
(xi )
given by ϕ(xi ) = xi , ϕ(y j ) = y j and ϕ(ui ) = 0.
Recall that ΛW ′ (resp. A′ ) denotes a second copy of ΛW (resp. A).
5.3.1. The family ΛW = Λ(xi , ui , y) (case m = 1). As described in Section 5.1,
let consider the cocycle Ω associated to the particular case ΛW = Λ(xi , ui , y). In
order to find a lower bound for TC(ΛW), we are going to construct a cocycle
β ∈ (ker µΛW )n such that Ω·β represents exactly the fundamental class of ΛW ⊗ΛW.
Through the following lemma we assert the existence of such a cocyle β.
Lemma 5.4. Suppose that ΛW = Λ(xi , ui , y) where 1 ≤ i ≤ n. There exists γ ∈
n
Q
(ker µΛW )n such that β := (xi − x′i ) · y′ − γ ∈ (ker µΛW )n is a cocycle and
i=1
n
(ϕ ⊗ ϕ)(γ) = −
n
1 XY
(xi − x′i )(y − y′ ).
2 l=1 i=1
i,l
Proof. In what follows we use the following notations
n
Q
• πh0i = (xk − x′k ) .
k=1
• πhli =
n
Q
k,l
n
Q
(xk − x′k ), for all i, j = 1, · · · , n.
k,i,
Qj
=
(xk − x′k ), for all i, j, l = 1, · · · , n.
• πhi, ji =
• πhi, j,li
(xk − x′k ), for all l = 1, · · · , n.
k,i, j,l
Let consider the element θ ∈ (ker µΛW )n given by
θ = πh0i · y′ +
n
X
1
l=1
Recall that
dy′
2
πhli · (y − y′ ).
is given by
d(y′ ) =
X
i< j
αi, j x′i x′j ,
αi, j ∈ Q.
18
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
Then, we have
dθ = πh0i ·
X
αi, j x′i x′j +
i< j
=
X
n
X
1
l=1
αi, j πhi, ji (xi −
x′i )(x j
X
2
x′l · πhli
−
x′j )x′i x′j
αi, j (xi x j − x′i x′j )
i< j
i< j
+
X
i< j
αi, j
n
X
1
l=1
2
x′l · πhli (xi x j − x′i x′j ).
Using the following identity in the first sum
(xi − x′i )(x j − x′j )x′i x′j =
+
+
1 ′
1
x (x j − x′j )(x′i x′j − xi x j ) + x′j (xi − x′i )(x′i x′j − xi x j )
2 i
2
1 ′ 2
1
x (x − x′j 2 )(xi − x′i ) + x′j (x2i − x′i 2 )(x j − x′j )
2 i j
2
1
1 ′2
x (x j − x′j )2 + x′j 2 (xi − x′i )2 ,
2 i
2
we have
dθ =
X
i< j
1
1
αi, j πhi, ji { x′i (x j − x′j )(x′i x′j − xi x j ) + x′j (xi − x′i )(x′i x′j − xi x j )
2
2
1 ′ 2
1
xi (x j − x′j 2 )(xi − x′i ) + x′j (x2i − x′ 2i )(x j − x′j )
2
2
1 ′2
1 ′2
′ 2
+
x (x j − x j ) + x j (xi − x′i )2 }
2 i
2
X
X
1 ′
1
+
αi, j xi πhii (xi x j − x′i x′j ) +
αi, j x′j · πh ji (xi x j − x′i x′j )
2
2
i< j
i< j
X
X1
+
αi, j
x′l · πhli (xi x j − x′i x′j ).
2
i< j
l,i, j
+
After reduction we obtain
X
1
1
d(θ) =
αi, j πhi, ji { x′i (x2j − x′j 2 )(xi − x′i ) + x′j (x2i − x′i 2 )(x j − x′j )
2
2
i< j
1
1 ′2
x (x j − x′j )2 + x′j 2 (xi − x′i )2 }
2 i
2
X
X1
x′l πhli (xi x j − x′i x′j ).
+
αi, j
2
i< j
l,i, j
+
(3)
For l , i, j we can write πhli = πhi, j,li · (xi − x′i )(x j − x′j ). Using the following
decomposition
(xi − x′i )(x j − x′j )(xi x j − x′i x′j ) =
=
1
(xi − x′i )(x j − x′j ){(xi − x′i )(x j + x′j ) + (x j − x′j )(xi + x′i )}
2
1
1
(xi − x′i )2 (x2j − x′j 2 ) + (x j − x′j )2 (x2i − x′i 2 )
2
2
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
19
in the last sum of (3) we obtain
X
1
1
dθ =
αi, j πhi, ji { x′i (x2j − x′j 2 )(xi − x′i ) + x′j (x2i − x′i 2 )(x j − x′j )
2
2
i< j
1 ′2
1
xi (x j − x′j )2 + x′j 2 (xi − x′i )2 }
2
2
X
X1
1
1
′
+
αi, j
xl πhi, j,li { (xi − x′i )2 (x2j − x′j 2 ) + (x j − x′j )2 (x2i − x′i 2 )}.
2
2
2
i< j
l,i, j
+
Since x2i = dui , we have d(θ) = d(θ̂) where
θ̂ =
X
i< j
1
1
αi, j πhi, ji { x′i (u j − u′j )(xi − x′i ) + x′j (ui − u′i )(x j − x′j )
2
2
1 ′
1
ui (x j − x′j )2 + u′j (xi − x′i )2 }
2
2
X
X1
1
1
′
xl · πhi, j,li { (xi − x′i )2 (u j − u′j ) + (x j − x′j )2 (ui − u′i )}
+
αi, j
2
2
2
i< j
l,i, j
+
P
Note that θ̂ ∈ (ker µΛW )n and (ϕ ⊗ ϕ)(θ̂) = 0. By setting γ = θ̂ − nl=1 12 πhli · (y − y′ )
and β = πh0i · y′ − γ we have β = θ − θ̂ ∈ (ker µΛW )n , d(β) = 0 and remark that γ
n
P
satisfies (ϕ ⊗ ϕ)(γ) = − 21 πhli · (y − y′ ).
l=1
As we shall show in the following theorem, the elements γ and β are the main
ingredients needed to determine TC(ΛW).
Theorem 5.2. For any elliptic extension ΛW = Λ(xi , ui , y) we have
TC(ΛW) = dim W
Proof. By Lemma 5.4 there exists γ ∈ (ker µΛW )n satisfying
(ϕ ⊗ ϕ)(γ) = −
n
X
1
l=1
2
πhli · (y − y′ )
such that β = πh0i · y′ − γ ∈ (ker µΛW )n is a cocycle. From the construction of Ω
(see Section 5.1), we have
(ϕ ⊗ ϕ)(Ω) = (−1)n
n
Y
(xk − x′k )(y − y′ ) = (−1)n πh0i (xk − x′k )(y − y′ ).
k=1
It follows that
[(ϕ ⊗ ϕ)(Ω · β)] = [(ϕ ⊗ ϕ)(Ω · (πh0i · y′ − γ))]
n
Y
= (−1)n
(xk − x′k )(y − y′ ) · (πh0i · y′ − (ϕ ⊗ ϕ)(γ)).
k=1
20
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
Since (ϕ ⊗ ϕ)(γ) = −
n
P
l=1
1
2 πhli
· (y − y′ ) we deduce that
n
n
n
Y
Y
X
1
′
′
′ ′
[(ϕ ⊗ ϕ)(Ω · β)] = (−1) [ (xk − xk )(y − y ) ·
(xk − xk )y +
πhli · (y − y′ )]
2
k=1
k=1
l=1
n
n
n
Y
Y
′
′
= (−1) [ (xk − xk )(y − y ) ·
(xk − x′k )y′ ]
n
k=1
k=1
= 2n [x[n] yx′[n] y′ ].
Recall that x[n] y and x′[n] y′ represent respectively the fundamental classes of A and
A′ (see Section 5.1). Moreover, x[n] yx′[n] y′ represents the fundamental class of
A⊗A′ . Then Ω·β is a cocycle in (ker µΛW )2n+1 , representing a non-zero cohomology
class. As a result, we conclude that TC(ΛW) ≥ 2n + 1. The other inequality
(TC(ΛW) ≤ 2n + 1) follows from Proposition 5.1 and finally TC(ΛW) = dim W =
2n + 1.
5.3.2. A special case where m > 1. We now consider a model ΛW = Λ(xi , ui , y j )
with m > 1, and we assume that satisfies the following conditions




dy1 ∈ Λ(x1P, · · · , xn−1 )
j

i


dy j = xn · αi xi , α j ∈ Q for all j ≥ 2.
(4)
i
We will see that TC(ΛW) = dim W.
First we consider the subalgebra ΛW̃ = Λ(x1 , · · · , xn−1 , u1 , · · · , un−1 , y) of ΛW.
We have ker µΛW̃ ⊂ ker µΛW . Applying Lemma 5.4 to ΛW̃, there exists γ1 ∈
n−1
Q
(ker µΛW̃ )n−1 ⊂ (ker µΛW )n−1 such that β1 = (xi − x′i )y′1 − γ1 ∈ (ker µΛW )n−1 is a
i=1
cocycle and (ϕ ⊗ ϕ)(γ1 ) =
− 21
n−1
P n−1
Q
l=1
i=1
(xi −
x′i )(y1
− y′1 ).
i,l
Secondly, it is clear that xn · y2 · · · ym is a cocycle in the algebra A. Since ϕ is a
quasi-isomorphism there exists ε ∈ ker ϕ such that d(xn · y2 · · · ym − ε) = 0.
Finally considering the cocycle Ω ∈ (ker µΛW )n+m for the algebra ΛW, we construct
the cocycle
α := Ω · β1 · (xn · y2 · · · ym − ε − x′n · y′2 · · · y′m + ε′ ) ∈ (ker µΛW )2n+m .
ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
21
A calculation shows that
n−1
m
Y
Y
[(ϕ ⊗ ϕ)(α)] = [ (xi − x′i )2 ·
(y j − y′j ) · (xn − x′n ) · y′1 · (xn y2 · · · ym − x′n y′2 · · · y′m )]
i=1
j=1
n−1
m
Y
Y
= ±(−2)n−1 [
xi x′i · (xn − x′n ) · y1 y′1
(y j − y′j ) · (xn y2 · · · ym − x′n y′2 · · · y′m )]
i=1
j=2
= ±(−2)n−1 [x[n] x′[n] · y1 y′1
m
Y
(y j − y′j ) · (y2 · · · ym + y′2 · · · y′m )]
j=2
n
= ±2 [x[n] y[m] ·
, 0.
x′[n] y′[m] ]
In fact, the last equality is gotten from
m
Y
(y j − y′j )(y2 · · · ym + y′2 · · · y′m ) = y2 · · · ym · y′2 · · · y′m + (−1)m−1 y′2 · · · y′m · y2 · · · ym
j=2
′
′
= (1 + (−1)m−1 · (−1)|y2 ···ym ||y2 ···ym | )y2 · · · ym · y′2 · · · y′m
= (1 + (−1)m−1 · (−1)(m−1)(m−1) )y2 · · · ym · y′2 · · · y′m
= (1 + (−1)(m−1)m )y2 · · · ym · y′2 · · · y′m
= 2 · y2 · · · ym · y′2 · · · y′m .
As a result 2n + m ≤ TC(ΛW). Consequently, Proposition 5.1 implies TC(ΛW) =
2n + m.
Let us finish with the following observation
• For n = 2: We can always suppose (through a change of variables) that
there exists an y1 such that d(y1 ) = 0. Then the conditions (4) are satisfied
and it follows that TC(ΛW) = m + 4.
• For n = 3: If there exists an element y1 such that dy1 = λx1 x2 (with
λ , 0) then we can suppose that dy j ∈ Λ+ (x3 ) ⊗ Λ(x1 , x2 ) for all j ≥ 2.
Consequently, the conditions (4) are satisfied and we have TC(ΛW) =
m + 6.
As a consequence of this observation and of the techniques used before, if (ΛV, d) =
(Λ(x1 , · · · , xn , y1 , · · · , ym ), d) is an elliptic pure coformal model with either
• n = 2 and m > 2
or
• n = 3, m > 3 and dy1 = αx21 + βx1 x2 + γx22 where β , 0
then TC(ΛV) ≥ dim V.
Acknowledgements
This work has been partially supported by Portuguese Funds through FCT –
Fundação para a Ciência e a Tecnologia, within the projects UIDB/00013/2020 and
UIDP/00013/2020. A portion of this work has been discussed during the BIRSCMO workshop Topological Complexity and Motion Planning, Oaxaca (Mexico),
22
SAID HAMOUN, YOUSSEF RAMI, AND LUCILE VANDEMBROUCQ
May 2022, and S.H and L.V thank the Casa Matemática Oaxaca for its support
and hospitality during this workshop. S.H would like to thank the Moroccan center
CNRST –Centre National pour la Recherche Scientifique et Technique for providing him with a research scholarship grant number: 7UMI2020.
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ON THE RATIONAL TOPOLOGICAL COMPLEXITY OF COFORMAL ELLIPTIC SPACES
23
My Ismail University of Meknès, Department of Mathematics, B. P. 11 201 Zitoune, Meknès,
Morocco.
Email address: s.hamoun@edu.umi.ac.ma
Email address: y.rami@umi.ac.ma
Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal.
Email address: lucile@math.uminho.pt