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REALIZATION OF OBJECTS IN A TRIANGULATED CATEGORY

KATSUHIKO KURIBAYASHI

Abstract. The level of a module over a differential graded algebra measures the number of steps to build the module in an appropriate triangulated cat- egory. We define the levels for spaces and investigate the invariant of spaces over aK-formal space. In particular, the level of the total space of a bundle over the 4-dimensional sphere is computed with the aid of Auslander-Reiten theory over spaces due to Jørgensen. We also discuss a realization problem of indecomposable objects in the derived category of thed-dimensional sphere.

The Hopf invariant brings a criterion for the realization.

1. Introduction

Categorical representation theory provides important technical tools and ideas in the study of many areas of mathematics including finite group theory, algebraic geometry and algebraic topology. Triangles and quivers which appear in Auslander- Reiten theory are indeed such tools; see, for example, [15], [16] and [17].

The singular (co)chain complex functor develops algebraic model theory for topo- logical spaces; see [1], [3], [11], [14] and [30]. Thus we anticipate that the singular cochain functor brings, moreover, new insights into the connection between algebra and topology when it works together with the tools mentioned above in the realm of categorical representation theory; see, for related results, [4], [6], [7], [9] and [23].

In this paper, with such expectation in mind, we introduce and investigate a topological invariant, which is called the level of a space. The notion of levels of objects in a triangulated category was originally introduced by Avramov, Buch- weitz, Iyengar and Miller in [2]. Roughly speaking, the level of an object M in a triangulated categoryT counts the number of steps required to build M out of a fixed object via triangles in T. We then define the level of a spaceX to be that of the differential graded moduleC^∗(X;K) obtained by converting the space with the singular cochain functorC^∗( ;K), whereKis a field.

In the rest of this section, we overview briefly our main results. We first give a reduction theorem (Theorem 2.3) for computing the level of the pullback associ- ated withK-formal spaces. An explicit calculation with the theorem tells us that a ‘nice’ space such as the total space E of a bundle over the sphere S^d is of low level; see Propositions 2.4 and 2.5. This means that the objectC^∗(E;K) consists of indecomposable ones, which is called molecules, with low level in an appropri- ate triangulated subcategory T of the derived category D(C^∗(S^d;K)). As for the molecules which constituteC^∗(E;K), these are visualized with black vertices in the

2000 Mathematics Subject Classification: 16E45, 18E30, 55R20, 13D07.

Key words and phrases.Level, Auslander-Reiten quiver, triangulated category, formal space.

Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan e-mail:kuri@math.shinshu-u.ac.jp

1

Auslander-Reiten quiver ofT as drawn below. Observe that the vertices are put on the first and second rows; see Lemma 3.3. For more details, see also Theorem 2.8, which is a remarkable result due to Jørgensen.

· · · ◦

Z₂

""

EE EE

E ◦

""

EE EE

E ◦

""

EE EE

E ◦

Σ^−3(d−1)Z₂

""

EE EE

E · · ·

◦

Z₁

""

EE EE E

<<

yy yy

y ◦

Σ^−(d−1)Z₁

""

EE EE E

<<

yy yy

y ◦

Σ^−2(d−1)Z₁

""

EE EE E

<<

yy yy

y ◦ · · ·

◦

""

EE EE E

<<

yy yy

y ◦

""

EE EE E

<<

yy yy

y •

""

EE EE E

<<

yy yy

y •

Σ^−2(d−1)Z₀

""

EE EE E

<<

yy yy y

· · · ◦

<<

yy yy

y •Z0

<<

yy yy

y •_Σ−(d−1)Z₀

<<

yy yy

y • · · ·

Here the vertexZ0 denotes the differential graded moduleC^∗(S^d;K) and Σ is the shift operator ofT. Thus one has a new algebraic aspect of a topological object.

We further deal with a problem of realizing a vertex (molecule) in an Auslander- Reiten quiver by a topological space via the singular cochain functor. In conse- quence, we see that almost all molecules appeared in the quiver over the sphere are not realized by finite CW complexes. In fact Theorem 2.12 states that, in the Auslander-Reiten quiver mentioned above, the arrow

Z0 • //• ^{Σ−(d−1)Z1}

is only realizable. This gives a new topological aspect of the Auslander-Reiten quiver.

2. Results

In this section, we state our main results more precisely. The results, which directly motivate us to work in a triangulated category over a singular cochain algebra, are also described.

Let A be a simply-connected differential graded algebra over a field of charac- teristic zero and D(A) the derived category of differential graded modules overA.

Recently, Jørgensen [18] has proved that the full subcategory D^c(A) of D(A), which consists of compact objects, has the Auslander-Reiten triangles if the cohomology ofAis a Poincar´e duality algebra. It is also proved in [19] that each component of the Auslander-Reiten quiver is of the formZA_∞.

Very recently, Schmidt [36] has shown that the result on Auslander-Reiten com- ponents holds even if the characteristic of the underlying field is positive; see also [20]. Thus if X is a simply-connected space whose cohomology H^∗(X;K) with coefficients in a filed K is a Poincar´e algebra, then the singular cochain complex functorC^∗( ;K) makes an appropriate space overXinto an object in D^c(C^∗(X;K)) in which Auslander-Reiten theory is applicable; see below for more details.

To describe our result on the level of a space, we begin by recalling from [2] the definition of the thickening of a triangulated categoryT. For a given objectGinT, we first define the 0th thickening by thick⁰_T(G) ={0}and thick¹_T(G) by the smallest strict full subcategory which containsGand is closed under taking finite coproducts, retracts and all shifts. Moreover for n > 1 define inductively the nth thickening thickⁿ_T(G) by the smallest strict full subcategory ofT which is closed under retracts and contains objectsM admitting a distinguished triangleM1→M →M2→ΣM1

in T for which M₁ and M₂ are in thickⁿ_T⁻¹(G) and thick¹_T(G), respectively. As mentioned in [2, 2.2.4], the thickenings provide a filtration of the thick subcategory

Thick_T(G) ofT containing the objectG:

{0}= thick⁰_T(G)⊂ · · · ⊂thickⁿ_T(G)⊂ · · · ⊂ ∪ⁿ≥0thickⁿ_T(G) = Thick_T(G).

For an objectM inT, we define a numerical invariant level^G_T(M), which is called the level ofM with respect toGinT, by

level^G_T(M) := inf{n∈N∪ {0} |M ∈thickⁿ_T(G)}. The dimension dimT of a triangulated categoryT [5] [35] is defined by

dimT = inf{n∈N|there exists an objectG∈ T with thickⁿ⁺¹_T (G) =T }. Thus the notion of levels is closely related to the dimension of T and to the thick subcategory ofT as well.

Let A be a differential graded algebra (abbreviated DGA) over a field K and D(A) the derived category of differential graded rightA-modules (abbreviated DG modules), which is viewed as a triangulated category. Observe that a triangle in D(A) comes from a cofibre sequence of the form M →^f N → Cf → ΣM in the homotopy category of DG modules overA. HereCf denotes the mapping cone and ΣM is the suspension ofM defined by (ΣM)ⁿ=Mⁿ⁺¹. In what follows, we denote by level_D(A)(M) the invariant level^A_D(A)(M) for any objectM in D(A).

We shall say that a graded vector space M is locally finite if Mⁱ is of finite dimension for anyi. Unless otherwise explicitly stated, it is assumed that a space has the homotopy type of a CW complex whose cohomology with coefficients in the underlying field is locally finite. LetX be a simply-connected space andT OP/X the category of connected spaces overX; that is, objects are maps with the target X and morphisms from α : Y → X to β : Z → X are maps f : Y → Z such that βf=α. For an objectα:Y →X inT OP/X, the singular cochain complex C^∗(Y;K) is regarded as a DG module over the DGAC^∗(X;K) with the morphism of DGA’s induced byα. Thus we have a functor

(2.1) C^∗( ;K) :T OP/X→D(C^∗(X;K)).

Let Y → X be an object in T OP/X. We then write level_D(C∗(X;K))(Y) for the invariant level_D(C∗(X;K))(C^∗(Y;K)) and refer to it asthe levelof the spaceY.

Let m_X : T V_X →^' C^∗(X;K) be a minimal TV-model for a simply-connected space in the sense of Halperin and Lemaire [14]; that is, T V_X is a DGA whose underlyingK-algebra is the tensor algebra generated by a graded vector spaceV_X and, for any element v ∈ V_X, the image ofv by the differential is decomposable;

see also Appendix.

Recall that a space X is K-formal if it is simply-connected and there exists a quasi-isomorphism from a minimalT V-model forX to the cohomologyH^∗(X;K).

Thus, in the case, we have a sequence of quasi-isomorphisms H^∗(X;K) T VX

φ_X

oo ' ^mX

' //C^∗(X;K),

where mX :T VX →C^∗(X;K) denotes a minimalT V-model forX. Observe that spheresS^d withd >1 areK-formal for any fieldK[8][34] and a simply-connected space whose cohomology with coefficients inK is a polynomial algebra generated by elements with even degree isK-formal; see [32, Section 7].

Definition 2.1. Letq:E→B andf :X →Bbe maps betweenK-formal spaces.

The pair (q, f) is relativelyK-formalizable if there exists a commutative diagram up to homotopy

H^∗(E;K) T VE φE

oo ' ^mE

' //C^∗(E;K) H^∗(B;K)

H^∗(q)

OO

H^∗(f)

²²

T VB φ_B

oo ' ^mB

' //

e q

OO

e f

²²

C^∗(B;K)

q^∗

OO

f^∗

²²H^∗(X;K) T VX

φ_X

oo ' ^mX

' //C^∗(X;K), in which horizontal arrows are quasi-isomorphisms.

We here comment on a map between K-formal spaces. In general, for a given quasi-isomorphismsφE,mE,φBandmBas in Definition 2.1, there exist DGA maps e

q1andqe2which make the right upper square and left that homotopy commutative, respectively. However, in general, one cannot choose a map qewhich makes upper two squares homotopy commutative simultaneously even if the mapsφE, mE, φB

andm_B are replaced by other quasi-isomorphisms; see Remark 4.5.

The following proposition, which are deduced from the proof of [26, Theorem 1.1], gives examples of relativelyK-formalizable pairs of maps.

Proposition 2.2. A pair (q, f) of maps between K-formal spaces with the same target is relatively K-formalizable if the two maps q and f satisfy either of the following three conditions concerning a mapπ:S→T respectively.

(i) H^∗(S;K) and H^∗(T;K) are polynomial algebras with at most countably many generators in which the operation Sq₁ vanishes when the characteristic of the field Kis 2. HereSq₁x=Sqⁿ⁻¹xforxwith degreen; see[32, 4.9].

(ii)Heⁱ(S;K) = 0 for anyi withdimHeⁱ⁻¹(ΩT;K)−dim(QH^∗(T;K))ⁱ6= 0.

Let q : E → B be a fibration over a space B and f : X → B a map. Let F denote the pullback diagram

E×^BX //

²²

E

q

²²X

f //B.

Our main theorem concerning the level of a space is stated as follows.

Theorem 2.3. Suppose that the spacesX,B andEin the diagramF areK-formal and the pair (q, f)is relatively K-formalizable. Then

level_D(C∗(X;K))(E×^BX) = level_D(H∗(X;K))(H^∗(E;K)⊗^LH^∗(B;K)H^∗(X;K)).

In general, the equality in Theorem 2.3 does not hold even if the spacesX, B andE inF areK-formal; see Example 3.2.

By virtue of Theorem 2.3 and Proposition 2.2, we have

Proposition 2.4. Let G be a simply-connected Lie group and G→ E_f → S⁴ a G-bundle with the classifying map f : S⁴ → BG. Suppose that H^∗(BG;K) is a

polynomial algebra on generators with even degree. Then level_D(C∗(S⁴;K))(Ef) =

½ 2 if H⁴(f;K)6= 0, 1 otherwise.

Proposition 2.5. Let G be a simply-connected Lie group andH a maximal rank subgroup. LetG/H→E_g →S⁴be the pullback of the fibrationG/H→BH→^π BG by a map g:S⁴→BG. Suppose that H^∗(BG;K) andH^∗(BH;K)are polynomial algebras on generators with even degree. Then

level_D(C∗(S⁴;K))(E_g) = 1.

Propositions 2.4 and 2.5 make one expect that a ‘nice’ object in T OP/X is almost of low level. The intriguing feature is investigated further in subsequent work [27]. On the other hand, the following result guarantees existence of an object inT OP/S^d with the level greater than given arbitrary number.

Theorem 2.6. Suppose that the underlying field Kis of characteristic zero. For any integerl≥1, there exists an objectPl→S^d inT OP/S^d such that

level_D(C∗(S^d;K))(Pl)≥l.

In order to describe our result on a realization problem for objects in the category D(C^∗(S^d;K)) by topological ones, we here give an overview of Jørgensen’s result in [18].

LetT be a triangulated category. We say that an object inT is indecomposable if it is not a coproduct of nontrivial objects. Recall that a triangleL→^u M →^v N →^w ΣL inT isan Auslander-Reiten triangle[15] if the following conditions are satisfied:

(i)LandN are indecomposable.

(ii)w6= 0.

(iii) Each morphismN⁰→N which is not a retraction factors throughv.

As mentioned below, those particular triangles and a quiver associated with them visualize objects in the triangulated categoryT.

We say that a morphismf :M →N in T isirreducibleif it is neither a section nor a retraction, but satisfies that in any factorizationf =rs, eithersis a section or ris a retraction. The categoryT is said to have Auslander-Reiten triangles if, for each objectN with local endomorphism ring, there exists an Auslander-Reiten triangle withN as the third term from the left. Recall also that an objectK inT iscompactif the functor Hom_T(K, ) preserves coproducts; see [33, Chapter 4].

Definition 2.7. The Auslander-Reiten quiver ofT has as vertices the isomorphism classes [M] of indecomposable objects. It has one arrow from [M] to [N] when there is an irreducible morphismM →N and no arrow from [M] to [N] otherwise.

Let A be a DGA over a field K. We denote by D^c(A) the full subcategory of the derived category D(A) consisting of the compact objects. For a DG moduleM overA, letDM be the dual Hom_K(M,K) toM.

We assume thatAis locally finite and simply-connected in the sense thatH⁰(A) = K and H¹(A) = 0. Observe that the cochain algebra C^∗(X;K) for a simply- connected spaceX satisfies the condition for the DGA under our assumption for a space.

Put d:= sup{i | HⁱA 6= 0}. One of the main results in [18] asserts that both D^c(A) and D^c(A^op) have Auslander-Reiten triangles if and only if there are iso- morphisms of graded HA-modules HA(DHA) ∼= HA(Σ^dHA) and (DHA)HA ∼= (Σ^dHA)_HA; that is, H^∗(A) is a Poincar´e duality algebra. Moreover we observe that the condition for A is equivalent to the Gorensteinness of A in the sense of F´elix, Halperin and Thomas [9].

The form of the Auslander-Reiten quiver of D^c(A) is clarified in [18] and [19]

for a DGAAwhose cohomology is a Poincar´e duality algebra. The key lemma [18, Lemma 8.4] to proving results in [18, Section 8] is obtained by using the rational formality of the spheres. Since the spheres are also K-formal for any field K, the assumption of the characteristic of the underlying field is removed from all the results in [18, Section 8]. In particular, we have

Theorem 2.8. [18, Theorem 8.13][18, Proposition 8.10]LetS^dbe thed-dimensional sphere with d >1 andK an arbitrary field. Then the Auslander-Reiten quiver of the categoryD^c(C^∗(S^d;K))consists ofd−1 components, each isomorphic toZA_∞. The component containingZ₀∼=C^∗(S^d;K) is of the form

...

Z₃

... ... ...

· · · ◦

ÂÂ@

@@

@@ ◦

Z₂

ÂÂ@

@@

@@ ◦

ÂÂ@

@@

@@ ◦

Σ^−2(d−1)Z₂

ÂÂ@

@@

@@ · · ·

◦

ÂÂ@

@@

@@

??~

~~

~~ ◦

Z₁

ÂÂ@

@@

@@

??~

~~

~~ ◦

Σ^−(d−1)Z₁

ÂÂ@

@@

@@

??~

~~

~~ ◦ · · ·

◦

ÂÂ@

@@

@@

??~

~~

~~ ◦

ÂÂ@

@@

@@

??~

~~

~~ ◦

Σ^d−1@Z@@₀@@ÂÂ

??~

~~

~~ ◦

Σ^−(d−1)Z0

ÂÂ@

@@

@@

??~

~~

~~

· · · ◦

??~

~~

~~ ◦

??~

~~

~~ ◦_Z0

??~

~~

~~ ◦ · · ·

Moreover, the cohomology of the indecomposable objectΣ^−lZmhas the form Hⁱ(Σ^−lZm)∼=

½ K fori=−m(d−1) +l andd+l, 0 otherwise.

Remark2.9. The latter half of Theorem 2.8 implies that indecomposable objects in D^c(C^∗(S^d;K)) are characterized by the cohomology. Moreover, those objects are also classified bythe amplitudeof the cohomology of the objects up to shifts. Here the amplitude of a DG moduleM, denoted ampM, is defined to be

ampM := sup{i∈Z|Mⁱ6= 0} −inf{i∈Z| Mⁱ6= 0}.

We are aware that the cohomology of Σ^−(d−1)Z1 is isomorphic toH^∗(S^2d−1;K) as a graded vector space and that there is an irreducible map which induces H^∗(S^d;K) =H^∗(Z0)→H^∗(Σ^−(d−1)Z1) a morphism of H^∗(S^d;K)-modules. Thus one would expect that realizability of the object by a space is related to the Hopf in- variantH :π2d−1(S^d)→Z; see below for the explicit definition of the realizability.

In fact, we establish

Proposition 2.10. Letφ:S^2d−1→S^dbe a map. The cochain complexC^∗(S^2d−1;K) endowed with the C^∗(S^d;K)-module structure induced by the map φ:S^2d−1→S^d is in D^c(C^∗(S^d;K)) if and only if H(φ)_K is nonzero, where H(−)_K denotes the composite of the Hopf invariant with the reduction Z →Z⊗K. In this case, the induced map φ^∗ : C^∗(S^d;K) → C^∗(S^2d−1;K) coincides with the irreducible map Z0→Σ^−(d−1)Z1 up to scalar multiple.

Definition 2.11. An objectMin the category D^c(C^∗(X;K)) isrealizableby an ob- jectf :Y →X inTOP /X ifM is isomorphic to the cochain complexC^∗(Y;K) en- dowed with theC^∗(X;K)-module structure induced byf^∗:C^∗(X;K)→C^∗(Y;K).

Since the 0th cohomology of a space is nonzero and the negative part of the coho- mology is zero, the only indecomposable objects of the form Σ^−m(d−1)Zm(m≥0), which are on the line connectingZ₀and Σ^−(d−1)Z₁, may be realizable. However, the following proposition states that most of indecomposable objects in D^c(C^∗(X;K)) arenotrealizable by finite CW complexes.

Theorem 2.12. Suppose that the characteristic of the underlying field is greater than 2 or zero. An indecomposable object of the form Σ⁻ⁱZ_l in D^c(C^∗(S^d;K))is realizable by a finite CW complex if and only if i = d−1, l = 1 and d even, or i= 0 andl= 0.

The rest of this paper is organized as follows. Section 3 is devoted to proving Theorems 2.3 and 2.6. In section 4, we prove Proposition 2.10 and Theorem 2.12.

The explicit computations of levels described in Propositions 2.4 and 2.5 are made in Section 5.

We conclude this section with comments on our work. Let X be a simply- connected space whose cohomology with coefficients in a fieldKis a Poincar´e duality algebra. The Auslander-Reiten quiver of D^c(C^∗(X;K)) then visualizes indecompos- able objects, which will be calledmoleculesof the full subcategory, and irreducible morphisms between them. Even though a molecule in D^c(C^∗(X;K)) is not in the image of the singular cochain complex functor from TOP /X, it may be needed to construct C^∗(Y;K) for a space Y over X as a C^∗(X;K)-module. In fact, it follows from the proof of Propositions 2.4 and 2.5 that some molecules are retracts of C^∗(S⁴;K)-modules C^∗(Ef;K) and C^∗(Eg;K) while they are not realizable; see also Example 5.3.

A CW complexZis made of disks, which are called cells, by repeating attachment of them. The result [12, Theorem 4.18] asserts that a CW complexZ has a cellular chain model. In consequence, the cochain complex C^∗(Z;K) is isomorphic to the dual to the cellular chain complex ofZ. ThusC^∗(Z;K) may be also regarded as

‘a set of cells’ and hence it seems a creature in some sense. As mentioned above, indecomposable objects in D^c(C^∗(X;K)) are needed when considering images by the functor C^∗(−;K) from the viewpoint of Auslander-Reiten theory. Therefore one might regard such an object as structural one smaller than a cell. This is the reason why we give indecomposable objects in D^c(C^∗(X;K)) the name ‘molecules’.

3. Proofs of Theorems 2.3 and 2.6

LetN be a left DG module over a DGAA. we first observe that the left derived functor− ⊗^LAN is defined byM⊗^LAN :=F⊗^AN for any right DG moduleM over A, whereF →M →0 is a semi-free resolution in the sense of F´elix, Halperin and Thomas [12,§6]; see Appendix. We also recall the fact that the bar resolution of a DG module is a semi-free resolution. This is extracted from the same argument as in the proof of [11, Lemma 4.3 (ii)]; see also [12, Lemma 6.3].

Let X be a simply-connected formal space and m_X : T V_X →^' C^∗(X;K) be a minimal model. We then have the following equivalences of triangulated categories;

see [23, Proposition 4.2], D(C^∗(X;K))

' m^∗_X

//D(T VX)^−⊗

L

T VXH^∗(X;K)

' //D(H^∗(X;K)),

where m^∗_X is the pullback functor; that is, m^∗_XM for a C^∗(X;K)-module M is regarded as a T V_X-module with the DGA map m_X. We denote byF_X the com- position of the functors. Observe that the functorF_X leaves the cohomology of an object unchanged.

Lemma 3.1. Under the same hypothesis as in Theorem 2.3, the differential graded module FX(C^∗(E×^BX;K))is isomorphic to H^∗(E;K)⊗^LH^∗(B;K)H^∗(X;K)in the categoryD(H^∗(X;K)).

Proof. We use the same notation as in Introduction and writeC^∗(−) forC^∗(−;K).

Let H : T V_B ∧I → C^∗(E) and K : T V_B ∧I → C^∗(X) be homotopies from q^∗◦m_B tom_E◦qeand fromf^∗◦m_B tom_E◦fe, respectively. HereT V_B∧Idenotes the cylinder object due to Baues and Lemaire [10] in the category of DGA’s; see Appendix. The homotopiesH andKmakeC^∗(E) andC^∗(X) into a rightT VB∧I- module and a leftT VB∧I-module, respectively. We have aC^∗(X)-module of the formC^∗(E)⊗^LT V_B∧IC^∗(X). Then there exists a sequence of quasi-isomorphisms of T VX-modules

C^∗(E×^BX) C^∗(E)⊗^LC^∗(B)C^∗(X)

'

oo EM C^∗(E)⊗^LT V_BC^∗(X)

1⊗ε01 '²²

1⊗_mB1

oo '

T VE⊗^LT V_BT VX ' mE⊗1mX

//C^∗(E)⊗^LT V_BC^∗(X) ^1⊗ε11

' //C^∗(E)⊗^LT V_B∧IC^∗(X), where EM denotes the Eilenberg-Moore map; see [37, Theorem 3.2]. Therefore we see thatm^∗_X(C^∗(E×^BX)) is isomorphic toT VE⊗^LT VBT VX in D(T VX). Since T V_E⊗^LT VBT V_Xis a freeT V_X-module, it follows that (T V_E⊗^LT VBT V_X)⊗^LT VXH^∗(X) is isomorphic toT VE⊗^LT V_BH^∗(X) in D(H^∗(X)). Then the same sequence of quasi- isomorphisms as above connectsT VE⊗^LT VBH^∗(X) withH^∗(E)⊗^LH^∗(B)H^∗(X) in D(H^∗(X)). In fact we have quasi-isomorphisms

T V_E⊗^LT V_BH^∗(X)

' φ_E⊗1

//H^∗(E)⊗^LT V_BH^∗(X)

' 1⊗ε01

//H^∗(E)⊗^LT V_B∧IH^∗(X)

H^∗(E)⊗^LH∗(B)H^∗(X) ^' H^∗(E)⊗^LT V_BH^∗(X).

1⊗_φB1

oo

1⊗ε11 'OO

This completes the proof. ¤

Proof of Theorem 2.3. The result [2, Proposition 3.4 (1)] enables us to deduce that level_D(C∗(X;K))(M) = level_D(H∗(X;K))(FXM) for any object M in D(C^∗(X,K)).

Thus the result follows from Lemma 3.1. ¤

Example3.2. Letν :S⁷→S⁴be the Hopf map andE_ν the pullback ofν:S⁷→S⁴ by itself. We have a fibrationS³→E_ν →S⁷. Then it follows that

(2.1) level_D(C∗(S⁷;K))(E_ν)6= level_D(H∗(S⁷;K))(H^∗(S⁷;K)⊗^LH^∗(S⁴;K)H^∗(S⁷;K)).

In fact, we have a Koszul resolution of the form

(Γ[w]⊗ ∧(s⁻¹x₄)⊗H^∗(S⁴;K), δ)→K→0

withδ(s⁻¹x4) =x4 andδ(ω) =s⁻¹x4⊗x4. This gives rise to a semifree resolution H^∗(S⁷;K)⊗Γ[w]⊗ ∧(s⁻¹x4)⊗H^∗(S⁴;K)→H^∗(S⁷;K)→0

ofH^∗(S⁷;K) as anH^∗(S⁴;K)-module. Thus we have

M :=H^∗(S⁷;K)⊗^LH∗(S⁴;K)H^∗(S⁷;K) = (H^∗(S⁷;K)⊗Γ[w]⊗∧(s⁻¹x4)⊗H^∗(S⁷;K),0).

Since dimH(M) =∞, it follows thatM is not in thickⁿ_D(H∗(S⁷;K))(H^∗(S⁷;K)) for anyn≥0. This implies that the right hand side of the inequality (2.1) is infinite.

On the other hand, by Lemma 4.3 below, we see thatC^∗(Eν;K) is in D^c(C^∗(S⁷;K)) and hence levelD(C∗(S⁷;K))(Eν) <∞; see [21, Theorem 5.3] and [18, Lemma 3.2].

We refer the reader to Example 5.2 for the explicit calculation of the level ofE_ν. We recall a result on the level of a molecule in D^c(C^∗(S^d;K)) due to Schmidt before proving Theorem 2.6.

Lemma 3.3. [36, Proposition 6.6] Let Z_i be the molecule inD^c(C^∗(S^d;K)) de- scribed in Theorem 2.8. Then level_D(C∗(S^d;K))(Z_i) =i+ 1.

Proof of Theorem 2.6. In the case wherel= 1, the sphereS^dis the space we require in the theorem. In what follows, we assume thatl≥2. Let∧(xi;i∈J) denote the free graded commutative algebra generated by elements x_i (i ∈J). Letm be an integer sufficiently larger thanld.

Assume thatdis even. We have a minimal modelB= (∧(x, ξ), δ) overQforS^d withδ(ξ) =x², where degx=d. Consider a Koszul-Sullivan extension of the form

B →(∧(x, ξ, ρ, w₀, ..., w_l−1), D) for which the differentialD is defined by

D(ρ) =x, D(w0) = 0 andD(wi) = (ρx−ξ)wi−1

fori≥1, where degwi=i(2d−1) + (2m−1)−i. Letπ:Pl→S^d be the bullback of the fibration|Ml| → |B|=S_Q^d, which is the spacial realization of the extension, by the localizing map S^d → S_Q^d; see [12, Proposition 7.9]. Let Fπ be the fibre of the fibrationπ:P_l→S^d. Then we see that

H^∗(F_π;K)∼=H^∗(F_π;Q)⊗QK∼=H^∗(M_l⊗^BQ)⊗QK

=H^∗(∧(ρ, w0, w1, ..., wl−1), D)⊗QK. Since the generators ofMl⊗^BQare of odd degree, it follows from Lemma 3.3 below that C^∗(Pl;K) is in D^c(C^∗(S^d;K)) ; see also Remark 4.4. Moreover, we see that H^∗(Pl;K)∼=H^∗(Pl;Q)⊗QK∼=H^∗(Ml)⊗QK.

By using the manner in [28, Section 7] for computing the homology of a DGA (or by the direct calculation), we have elements 1,ξ,w0 and (ρx−ξ)wl−1, which form a basis ofH^∗(M_l) with degree less than or equal to l(2d−1) + (2m−1)−(l−1).

LetZ be a decomposable direct summand ofC^∗(P_l;K) in D(C^∗(S^d;K)) containing a cocycle of degree zero. By virtue of Theorem 2.8, we see that Z = Σ^−k(d−1)Zk

for somek ≥0; see Remark 2.9. Suppose thatZ contains a representative ofw0, (ρx−ξ)wl−1 or a cohomology class with degree greater thanl(2d−1) + (2m−1)− (l−1). Theorem 2.8 implies that Hⁱ(Z) = Kif and only if i = (k+ 1)d−k or i= 0. It follows that (k+ 1)d−k≥2m−1 ≥2ld−1 and hencek ≥2l−1 ≥l.

Then Lemma 3.3 allows us to conclude that level_D(C∗(S^d;K))(Pl)≥l+ 1.

Suppose thatZ contains a representative of the elementξ. By Theorem 2.8, we see thatZ= Σ^−(d−1)Z₁. In this case, letZ⁰ be a decomposable direct summand of

C^∗(Pl;K) containing a representative ofw0. Observe thatZ⁰6=Z. IfZ⁰contains a representative of the element (ρx−ξ)w_l−1, thenZ⁰= Σ^−(2m−1)Σ^{−(2l−1)(d−1)}Z_2l−1 since dimH^∗(Z⁰) = 2 and the amplitude ofZ⁰should be 2ld−2l+ 1. IfZ⁰contains a representative of the cohomology class with degree greater thanl(2d−1) + (2m− 1)−(l−1), thenZ⁰ = Σ^−(2m−1)Σ^{−(2l−1)(d−1)}Zk for somek≥2l−1. Lemma 3.3 yields that level_D(C∗(S^d;K))(P_l)≥2l.

Suppose thatdis odd. We have a Koszul-Sullivan extension of the form (∧(x),0)→(∧(x, w₀, w₁, ..., w_l−1), D) =:N_l

for which the differentialD is defined by D(x) =D(w0) = 0 and D(wi) =xwi−1

for i ≥ 1, where degx = d and degw0 = 2m−1. We assume that the integer m is sufficiently larger than ld. Observe that degwi = id+ (2m−1)−i. Let π : Pl → S^d be the bullback of the fibration |Nl| → |(∧(x),0)| = S_Q^d by the localizing mapS^d→S_Q^d. The same argument as above does work well to show that level_D(C∗(S^d;K))(Pl)≥l. This completes the proof. ¤ Remark 3.4. The Whitehead length WL(Y) of a simply-connected space Y is re- garded as one of the homotopy invariants which measures geometrical complexity of a space.

Let LY be the rational homotopy Lie algebra π_∗(ΩY)⊗Q. We define a sub- space [L_Y, L_Y]^(l)ofL_Y by [L_Y,[L_Y,[..,[L_Y, L_Y]...]] (l-times) and [L_Y, L_Y]⁽⁰⁾ =L_Y, where [, ] denotes the Lie bracket ofL_Y. Then the invariant WL(Y) is defined to be the greatest integer nsuch that [LY, LY]⁽ⁿ⁾6= 0. The result [29, Theorem 2.5]

implies that WL(Pl) =l−1 for the spacePl constructed in the proof of Theorem 2.6.

4. Realization of molecules in D^c(C^∗(S^d;K))

We recall briefly the Hopf invariant. Let φ : S^2d−1 → S^d be a map. Choose generators [x_2d−1]∈H^2d−1(S^2d−1;Z) and [x_d]∈H^d(S^d;Z). Let ρbe an element ofC^∗(S^2d−1;Z) such thatφ^∗(x_d) =dρ. Since [x_d]² = 0 inH^∗(S^d;Z), there exists an element ξofC^∗(S^d;Z) such that dξ =x²_d. We then have a cocycle of the form ρφ^∗(xd)−φ^∗(ξ). The Hopf invariantH(φ)∈Zis defined by the equality

[ρφ^∗(xd)−φ^∗ξ] =H(φ)[x2d−1].

Remark4.1. Ifdis odd, thenH(φ) is zero in general.

We relate the Hopf invariant with a differential of the Eilenberg-Moore spectral sequence (EMSS). LetFφ be the homotopy fibre of a mapφ:S^2d−1→S^d. Then Fφ fits into the pullback diagramF⁰ :

ΩS^d

²²

ΩS^d

²²Fφ //

²²

P S^d

²²π

S^2d−1 _φ //S^d.

Here ΩS^d→P S^{d π}→S^ddenotes the path-loop fibration. We consider the EMSS as- sociated withF⁰converging toH^∗(F_φ;K) withE₂^∗,∗∼= Tor^∗_H^,∗_∗(Sd;K)(H^∗(S^2d−1;K),K).

The Koszul resolution ofKas an H^∗(S^d;K)-module allows us to conclude that E₂^∗,∗∼=

½ H^∗(S^2d−1;K)⊗ ∧(s⁻¹xd)⊗Γ[τ] ifdis even, H^∗(S^2d−1;K)⊗Γ[s⁻¹xd] ifdis odd,

where bidegs⁻¹x_d= (−1, d) and bidegτ = (−2,2d); see [38, Lemma 3.1] and also [24, Proposition 1.2].

Observe that the Eilenberg-Moore map induces an isomorphism from the homol- ogy of the bar complex (B(C^∗(S^2d−1;K), C^∗(S^d;K),K), δ1+δ2) toH^∗(Fφ;K). By the definitions of differentialsδ1 andδ2, we see that

δ₁([x_d|x_d]) = (−1)^dφ^∗(x_d)[x_d] + (−1)^d(−1)^d+1[x²_d]

= δ₂((−1)^dρ[x_d] + 1[ξ]), δ₁((−1)^dρ[x_d] + 1[ξ]) = (−1)^d{(−1)^d−1ρφ^∗x_d}+φ^∗ξ

= −(ρφ^∗(xd)−φ^∗ξ).

It follows from [22, Lemma 2.1] thatd2([xd|xd]) =H(φ)_Kx2d−1 in theE2-term of the EMSS. Moreover, by the same argument as in [24, Lemma 1.5], we have

Lemma 4.2. The element[xd|xd]in the torsion productTor_H∗(S^d;K)(H^∗(S^2d−1;K),K), which is computed by the bar complex mentioned above, coincides with the element τ ∈Γ[τ] if dis even and with the element γ2(s⁻¹xd)∈Γ[s⁻¹xd] ifd is odd. Thus one hasd2(τ) =H(φ)_Kx2d−1 if dis even and d2(γ2(s⁻¹xd)) =H(φ)_Kx2d−1 = 0if dis odd.

In order to study Auslander-Reiten triangles, in [19], Jørgensen introduced the functionϕ: D(A)→Z∪ {∞}defined by

ϕ(M) := dimH^∗RHomA(M,K) = dimH^∗(M ⊗^LAK).

This gives a criterion for a given object in D(A) to be compact.

Lemma 4.3. [36, Lemma 3.9]Let Abe a simply-connected DGA withdimH(A)<

∞. An objectM inD(A) is compact if and only ifdimH^∗(M)<∞andϕ(M)<

∞. In particular, for a map φ : Y → X from a space Y to a simply-connected space Xwith dimH^∗(X;K) <∞, if the total dimension of the cohomology of the homotopy fibre of the mapφ is finite, thenC^∗(Y;K)is inD^c(C^∗(X;K)).

Remark 4.4. Let Fφ be the homotopy fibre of a mapφ: Y →X. The latter half of Lemma 4.3 follows from the fact thatH^∗(C^∗(Y;K)⊗^LC∗(X;K)K)∼=H^∗(F_φ;K) as a graded vector space; see [12, Theorem 7.5]. In view of the Leray-Serre spectral sequence, we see that the dimension of H^∗(Y;K) is finite if so are the dimensions ofH^∗(F_φ;K) andH^∗(X;K).

Let (TOP /X)^f,K be the full subcategory ofTOP /X consisting of objects φ : Y → X with dimH^∗(Y;K) < ∞ and dimH^∗(Fφ;K) < ∞. Lemma 4.3 asserts that the singular cochain functor (1.1) restricted to the full subcategory yields the functor

C^∗( ;K) : (TOP /X)^f,K→D^c(C^∗(X;K)).

By using Lemmas 4.2 and 4.3, we prove Proposition 2.10.

Proof of Proposition 2.10. Let{Ee_r^∗,∗,der} be the EMSS converging toH^∗(ΩS^d;K).

Observe that

e E₂^∗,∗∼=

½ ∧(s⁻¹xd)⊗Γ[τ] ifdis even, Γ[s⁻¹xd] ifdis odd,

where bidegs⁻¹xd = (−1, d) and bidegτ = (−2,2d). The result [10, Theorem III] implies that the EMSS for the fibre square F⁰ is a right DG comodule over {Ee^∗_r^,∗,de