TORAL RANK CONJECTURE

VICENTE MUNOZ

Abstract. The Toral Rank Conjecture (TRC) is an interestingproblem in topology which was proposed some 20 years ago andwhich is still unsolved. It has different versions and implications inmultiple areas of Mathematics such as differential geometry, alge-braic topology or commutative algebra. In this short note we shallstate the TRC, recall the definitions that appear in the statementof the conjecture and survey some of the methods used to studysome particular cases. Finally, we shall mention variations of theTRC in different areas of Mathematics.

Introduction

In this text we shall focus on a “small” open problem in topology whichhas been around for over 20 years. It was proposed by S. Halperin in1987 (it appears in [Ha]) and says the following:

Toral Rank Conjecture (TRC). Let T r → E → B be a fibrebundle, where T r = S1 × · · · × S1 is the r-dimensional torus and Bis a compact and simply connected space. Then the dimension of thecohomology of the total space E satisfies:

dimH∗(E) ≥ 2r.

Its statement is extremely simple. The terms that appear are familiarto every undergrad student with some basic knowledge of differentialgeometry and algebraic topology (it is enough to know the notions offibre bundle and cohomology). We shall use a few pages to explain indetail each of the terms that appear in the statement of the TRC.

Perhaps, what makes the TRC attractive is the amount of interactionsit has with a variety of areas in Mathematics, among which we can list

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algebraic topology, differential geometry, commutative algebra or thetheory of Lie groups and algebras. There exist different formulationsof the TRC as well as variations on the statement, which become moreor less general. This makes it possible, for each algebraist, geometer ortopologist, to have his/her favorite version of the TRC at hand. Not toforget the numerous implications of its possible proof! We will explainsome variations of the conjecture and how these interplay with oneanother.

Oddly enough, the TRC is a problem that has not drained much atten-tion. Perhaps this is due to some ancestral creed that if a problem canbe stated in a simple way and does not have a simple solution, thenit must be extremely difficult. It is possible that new techniques mustbe used, or maybe the joint use of machineries from different fields, orperhaps just a new, beautiful idea. We shall mention here some of thetechniques (we can almost call them “classic”) which have been usedso far in relation with the TRC.

We encourage every PhD student in algebra, geometry or topology tofind a path in order to attack the TRC. Each mathematician whoseresearch interests touch one of these areas should be familiar with atleast one version of this problem. Just in case.

Manifolds and CW-complexes

We shall begin by analyzing the different geometric and topological in-gredients which appear in the statement of the Toral Rank Conjecture,as we have stated it in the introduction.

Let us start with speaking about the spaces which one deals within geometry and topology, namely differentiable manifolds and CW-complexes, respectively. Such spaces have just the right properties oneneeds in order to work with them.

Smooth manifolds. (Differential) Geometry deals with smooth man-ifolds: these are spaces that locally (at a reduced scale) look like Rn (sothat one can derive, integrate and perform all the typical manipulationsof multivariable calculus) but which, globally, are allowed to “twist” in

TORAL RANK CONJECTURE 3

some interesting ways. Smooth spaces (those without “corners”) thatwe can think of are of this kind: curves, surfaces, our universe, . . .

The rigorous definition follows from this idea (see [Wa]): A smoothmanifold M , of dimension n, is a (topological) space with a collectionof open sets Uα ⊂M , such that:

(1) The open sets cover M , i. e., M =⋃α

Uα.

(2) There exist homeomorphisms Uαφα−→ φα(Uα) ⊂ Rn.

Figure 1. A local chart

φα is called local chart. It identifies the open set Uα of M withan open set in Rn. Therefore, the local charts allow us to lookat M at a small scale.

(3) If U = Uα ∩ Uβ 6= ∅, then

φα(U)φ−1α−→ U

φβ−→ φβ(U)

is a diffeomorphism (a differentiable map with differentiableinverse). This is a compatibility property of the local charts(using different local charts φα, φβ, we have the same geometricproperties in U). One can think of φ as a map that glues twolocal charts, so that M is built by gluing the open sets Uα.

In general, smooth manifolds model the kind of geometric spaces wewant to study. On the one hand, it is interesting understand, for eachdimension n, what types of manifolds exist. This leads to one of thefundamental problems of geometry: the classification of manifolds (howmany there are, how to identify them, how to know when two of themare equivalent). On the other hand, given a specific manifold, we alsowant to understand what kind of (geometric) properties it has. Thisleads us to the study of different geometric structures (what kind of

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local information must we add to the manifold in order to performgeometric measurements that provide more refined informations on themanifold?)

CW-complexes. Topology undertakes the study of topological spacesand of the properties stemming from the use of continuous functions.Therefore, we can study more general situations: spaces with corners,spaces that do not look locally like Rn, . . . but we will obtain less re-fined information, as we lose the possibility to compute derivatives andintegrals, in the space in which we work.

In order to do this, we can work with topological manifolds: theseshare the same definition as smooth manifolds, except that in point (3)we ask φβ φ−1α to be only a homeomorphism (a continuous map withcontinuous inverse). If we want to study a larger number of spaces,we can deal with all topological spaces, but in full generality they canhave very strange properties which would not allow us to accomplish adeep study of their geometric properties.

For this reason, it is convenient to remain in some intermediate area,with spaces that are more general than manifolds (topological andsmooth) but less general than topological spaces. CW-complexes, firstintroduced by J.H.C. Whitehead (see [Ma]), are spaces that can be cutin pieces similar to Rn; however, the gluing conditions are more generalthan the ones we allow for manifolds.

To be precise, a CW-complex X (of dimension n) is built by gluingcells of different dimensions. A k-cell is a closed k-dimensional ballDk ⊂ Rk. Then

X = Xn ⊃ Xn−1 ⊃ · · · ⊃ X1 ⊃ X0,

where X0 is a (discrete) set of points and

Xk = Xk−1 ∪Dk1 ∪ · · · ∪Dk

rk,

for k = 1, . . . , n, where Dkj is a k-cell glued to Xk−1 along its boundary,

that is, satisfying

∂Dki ⊂ Xk−1 and int(Dk

i ) ∩Xk−1 = ∅.

Xk is called the k-skeleton of X (the k-dimensional part of X). TheCW-complex can be splitted as X0 together with the sets int(Dk

i ),

TORAL RANK CONJECTURE 5

k = 1, . . . , n, i = 1, . . . , rk, all of them open sets in Rk. The maps weuse to glue are not homeomorphisms and this allows much more generalspaces than topological manifolds. As an example, the following CW-complex has dimension n = 2, it has three cells in dimension 0, fourin dimension 1 and one in dimension 2 (we indicated the 1-skeleton).Clearly, it is not a manifold.

Figure 2. A CW-complex

It is interesting to point out here that if M is a smooth manifolds, itcan be triangulated. A triangulation is a simple case of a CW-complex.

Figure 3. Triangulation

The CW-complex structure allows us to perform a combinatorial anal-ysis of our space. If the CW-complex is compact, then it has a finitenumber of cells. If we properly codify the gluing of such cells (bythe mean of incidence relations), we can extract some properties of aCW-complex using elementary algebra.

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Fibre bundles

A fibre bundle F → Eπ−→ B consists of three spaces (manifolds

or CW-complexes, according to the case we want to analyze) and twomaps (we call F the fibre, E the total space and B the base), satisfying:

(1) Fb := π−1(b) ∼= F , ∀ b ∈ B. Fb is the fibre over b. Clearly,E =

⊔b∈B Fb, that is, E is the disjoint union of all fibres.

(2) Every b ∈ B belongs to a small neighborhood U such thatπ−1(U) ∼= U × F (in a compatible way with the identificationof point (1)).

Figure 4. A fibre bundle

Examples:

1. The cylinder. In this case E is a cylinder (E = S1 × [0, 1]), π is theprojection from S1 × [0, 1] to S1, so that F = [0, 1].

2. The Mobius strip. We have again B = S1 y F = [0, 1], but now thetotal space E is the Mobius strip. This bundle is not equivalent to thecylinder, since E is not the product B × F in this case.

3. The torus. E = S1 × S1, π is the projection on the second factorB = S1 and F = S1.

4. The Klein bottle. B and F are as in the previous example, but E isthe Klein bottle.

5. The trivial bundle. The total space of the trivial bundle with baseB and fibre F is E = B × F and π : E → B is the projection fromthe product B×F onto B. The cylinder and the torus are trivial fibrebundles.

TORAL RANK CONJECTURE 7

Figure 5. Cylinder and Mobius band

Figure 6. Torus and Klein bottle

6. Quotients of Lie groups. If K ⊂ G are two compact Lie groups andG/K = [x]K : x ∈ G, [x]K = x ·K, then

K → Gπ−→ G/K,

with π(x) = [x]K , is a fibre bundle with fibre K.

The kind of fibre bundles that appear in the Toral Rank Conjectureare quite simple: the fibre is the r-dimensional torus, which is definedas

T r = S1 × S1 × · · · × S1 ,

with r factors. The base B will be a manifold or a compact CW-complex.

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Figure 7. Quotients of Lie grous

Homotopy and Homology

The final ingredients we need in order to make sense of the statementof Toral Rank Conjecture are the notions of homotopy and homology(see [Ma] and [BT]). Homotopy, homology and cohomology groupsprovide a way to count the “holes” of a reasonable topological space(here “reasonable” includes a large class of spaces, among which wefind CW-complexes). The holes of a space M are a global propertythat give us a great deal of information about M .

Homotopy groups. We shall denote by p0 a distinguished point ofM and call it basepoint.

The k-th homotopy group of M , πk(M), is defined as

πk(M) = γ : Sk →M | γ(N) = p0/ ∼,

whereN is a point in Sk (the North Pole, for instance, N = (0, . . . , 0, 1))and where ∼ is the equivalence relation given by γ0 ∼ γ1 if γ0 and γ1are homotopic maps (that is, if ∃ γt, t ∈ [0, 1], interpolating γ0 and γ1).

Clearly, πk(M) is a topological invariant of M (it does not change ifwe change M to a space homeomorphic to it). What is not so clear,although it is true, is that πk(M) is a group (further, an abelian groupfor k ≥ 2).

The group π1(M) (which is in general not abelian) plays a special role.It is called the fundamental group of M . If π1(M) = 1 (which isequivalent to saying that every loop γ : S1 → M can be contracted tothe basepoint) we say that M is simply connected.

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In the following picture we see two (non-trivial) elements of the homo-topy groups. The first one is a loop [γ] ∈ π1(M), and the second oneis an element A ∈ π2(M).

Homology. Homology groups were invented by H. Poincare. Theyare always abelian groups, and for this reason it is easier to work withthem than it is with homotopy groups. They contain a bit less topo-logical information about the space; however, thanks to the easinesswith which one computes them in practical examples, as well as totheir many algebraic properties, they are among the strongest tools intopology.

Their concrete definition is as follows:

Ck(M) =∑

niTi | Ti : [0, 1]k →M, ni ∈ Z

is the (abelian) group of k-chains of M (these are formal finite sums ofk-cubes in M). There is a boundary operator ∂ : Ck(M) → Ck−1(M)which associates to every k-chain T its boundary ∂T , which is a (k−1)-chain. We introduce:

• Zk(M) = ker ∂ = space of k-cycles. Zk(M) consists of the k-chains T which have no boundary, ∂T = 0. These k-chainscould potentially enclose a hole (see figure).• Bk(M) = im ∂ = space of k-boundaries. If a k-cycle T does

not enclose a hole, then it bounds a (k+ 1)-chain in M , so thatT = ∂S for some S.• Hk(M) = Zk(M)/Bk(M) is the k-th homology group of M . We

consider k-cycles disregarding those that bound a hole. There-fore, the quotient space Hk(M) account for the holes in M .

If we use real coefficients, that is, if we set ni ∈ R in the definitionof Ck(M), we obtain the real homology groups Hk(M,R). It is easy

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Figure 8. Chains in M

to see that Hk(M,R) = Hk(M) ⊗ R, so that we only lose the torsioninformation in homology.

Example: The n-dimensional sphere Sn only has one “hole” in dimen-sion n. More concretely,

Hk(Sn,R) =

R, k = 0, n,0, k 6= 0, n.

In general Hk(M) is easier to compute than πk(M). For example,Hk(M) = 0 for k > n = dimM , while one often has πk(M) 6= 0 fork > n. In the next picture, γ 6= 0 in π1(M), but γ = 0 in H1(M) (theloop can not be contracted, but it does bound a 2-chain).

Figure 9. Homotopy vs. Homology

de Rham cohomology. Let us go back to smooth manifolds. Theirglobal structure can be analized, in particular, by the mean of thetopological invariants that we have just described (homotopy and ho-mology). A surprising result of G. de Rham [BT] tells us that we can

TORAL RANK CONJECTURE 11

compute the real homology by using the differential-geometric infor-mation of the manifold.

Let M be a smooth, n-dimensional manifold. Let

Ωk(M) =

α =

∑i1<···<ik

fi1,...,ik(x) dxi1 ∧ · · · ∧ dxik

be the space of k-forms. The exterior derivative d provides a complex

Ω0(M)d−→ Ω1(M)

d−→ . . .d−→ Ωn(M)

called de Rham complex. Its “cohomology” is given by:

• Zk(M) = ker(d : Ωk(M)→ Ωk+1(M)), space of closed k-forms;• Bk(M) = im (d : Ωk−1(M)→ Ωk(M)), space of exact k-forms;• Hk(M) = Zk(M)/Bk(M), called de Rham cohomology of M .

Theorem: Hk(M) ∼= Hk(M,R)∗.

Properties:

• The numbers bk(M) = dimHk(M) are called Betti numbers ofM . If M is a smooth, oriented and compact manifold, thenbn−k(M) = bk(M). This result is known as Poincare duality.• The Euler-Poincare characteristic of M is defined as the alter-

nate sum of the number of cells: χ(M) =∑

k(−1)krk. It co-incides with the alternate sum of the Betti numbers: χ(M) =∑

k(−1)kbk(M).

Homology of the torus. Set T r = S1 × (r). . .× S1. We start from thefact that, for the circle S1, we have

H0(S1) ∼= R = 〈1〉,H1(S1) ∼= R = 〈dθ〉, θ angular coordinate;

we obtain that, for T r,

Hk(T r) ∼= 〈dθi1 ∧ · · · ∧ dθik | 1 ≤ i1 < · · · < ik ≤ r〉 .Shortening the notation, H∗(T r) '

∧(e1, . . . , er), ej = dθj. There-

fore,

dimHk(T r) =

(r

k

)

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and

dimH∗(T r) =r∑

k=0

(r

k

)= 2r .

Figure 10. The 2-torus

Toral Rank Conjecture

Let us recall the statement of the TRC. If T r → E → B is a fibrebundle such that the base B is compact and simply connected, then

dimH∗(E) ≥ 2r .

Remarks:

(1) In the original statement, we suppose that E and B are man-ifolds. We can also formulate the TRC assuming that E andB are CW-complexes. In this case, we can take by definitionHk(E) = Hk(E,R)∗, or use singular cohomology (see [Ma]).

(2) Since dimH∗(T r) = 2r, the TRC predicts that

dimH∗(E) ≥ dimH∗(F ) ,

where F = T r is the fibre.(3) The conjecture is true for the trivial bundle E = B × T r. In

this case,

dimH∗(E) = dimH∗(T r) · dimH∗(B) = 2r · dimH∗(B).

We see that there is an inclusion H∗(Tr,R) → H∗(E,R), in-

duced by the inclusion T r → E, which justifies the inequalityof the TRC, dimH∗(E) ≥ dimH∗(T r).

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(4) In general, the homology of E need not come from T r, i.e., theinclusion H∗(T

r,R) → H∗(E,R) may not hold. The statementof the TRC concerns the dimension of these spaces. An exampleof this phenomenon is given by the bundle

S1 × S1 → S3 × S3 → S2 × S2 ,

consisting of the product of two Hopf bundles:S1 = U(1)→ S3 = SU(2)→ S2 = SU(2)/U(1).

(5) The TRC can be thought of as a statement on the maximaldimension of a torus T r which can act on a manifold or on aCW-complex E. If N = dimH∗(E), one must have 2r ≤ N ,that is,

r ≤ log2N .

The TRC from different points of view

The Toral Rank Conjecture can be approached from different areasof Mathematics. Indeed, one can state conjectures that are, in oneway or another, related to the TRC. To solve any of those would be agreat success. The techniques vary with the version of the conjecture,according to the area in which one works. One can choose his/herfavorite version of the TRC, according to his/her background and taste.

A. Group actions on manifolds. A formulation of TRC from thepoint of view of differential geometry is the following:

TRC-A: If a simply connected, compact manifold E admits a freeaction of a torus T r, then

dimH∗(E) ≥ 2r .

Examples.

(1) If G is a simply connected, compact Lie group and T r ⊂ G isa maximal torus, then r is usually called the rank of G. In thiscase, it is not hard to prove that H∗(G) is an exterior algebraon r generators. Therefore, dimH∗(G) = 2r and the TRC holdstrue.

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Figure 11. Torus action

(2) Suppose K ⊂ G is a compact subgroup of a compact Lie group.We have a diagram

T s ⊂ T r → T r/T s = T r−s

∩ ∩ ∩K ⊂ G → G/K

where T s and T r are maximal tori in the corresponding Liegroups. Therefore T r−s acts on the manifold M = G/K (thiskind of manifolds are called homogeneous spaces). The inequal-ity

dimH∗(G/K) · dimH∗(K) ≥ dimH∗(G) ,

gives 2s · dimH∗(M) ≥ 2r, so that dimH∗(M) ≥ 2r−s. Again,the TRC holds also for homogeneous spaces.

TRC in its A-incarnation can be attacked with tools from differentialgeometry. For example, one can study Morse functions on the totalspace E which have a generic behavior along the fibres. For a genericfibre Fb, the set of critical points Critb of the restriction of such afunction h to Fb has at least 2r elements. In fact,

Crit :=⋃b∈B

Critb

gives a (branched) cover of degree at least 2r of B. This manifoldCrit ⊂ E could be used in order to show that h has at least 2r criticalpoints on E and then try to get from this dimH∗(E) ≥ 2r.

TORAL RANK CONJECTURE 15

Other ways of thinking can lead to endow the bundle with more struc-ture (for example, considering symplectic bundles), proving the conjec-ture in some particular, albeit important, cases.

B. Spectral sequences. Perhaps, one of the most direct ways toattack the TRC is by the mean of spectral sequences. Every fibrebundle F → E → B (we shall assume B to be simply connected)produces a spectral sequence, called Leray-Serre spectral sequence (see[Mc] o [BT]). Basically, such spectral sequence consists of:

• Pages: E∗,∗k = bi-graded vector spaces, k ≥ 0.• Initial information: the second page is Ep,q

2 = Hp(B)⊗Hq(F ).• Differentials in each page: d2 : Ep,q

2 → Ep+2,q−12 ,

d3 : Ep,q3 → Ep+3,q−2

3 , . . .

• Each page is obtained as the cohomology of the previous one:Ek = H∗(Ek−1, dk−1).• There exists a limit: Ep,q

∞ = Ep,qk , k 0, which satisfies⊕

k=p+q

Ep,q∞ = Hk(E).

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Example: Let us see how to study the TRC for a fibre bundle T 2 →E → S2. We have H∗(T 2) =

∧(e1, e2) and H∗(S2) = 〈1, ω〉. Assume,

for example, that we have the following differential on E2 (the differentcases can be analyzed one by one, we choose a concrete one),

d2(e1) = 0d2(e2) = ω

⇒ d2(e1 e2) = d2(e1)e2 − e1d2(e2) = −ω e1 .

Then, on E3 = H(E2, d2), we have:

Therefore, E∞ = E3 ⇒ dimH∗(E) = 4. The TRC is true in this case.

Remark: For T r → E → B one always has

dimH∗(T r) · dimH∗(B) ≥ dimH∗(E),

so that what we are looking for is the inequality

dimH∗(E) ≥ 2r.

In the previous example we had d3 = d4 = . . . = 0, hence E∞ = E3. Infact, this is a very interesting case, which we state as an aside conjecture(weaker than the original TRC). Set E2 = H∗(B)⊗

∧(e1, . . . , er). Since

ej ∈ E0,12 , there exist elements

yj ∈ H2(B) = E2,02

such that d2(ej) = yj. This clearly determines d2, together with thefact that d2|H∗(B) = 0.

TRC-B: If H∗(B) is a finite-dimensional algebra, then

dimH∗(H∗(B)⊗∧

(e1, . . . , er), d2) ≥ 2r .

TORAL RANK CONJECTURE 17

C. Minimal models. The theory of minimal models, created and de-veloped by D. Sullivan (see [GM]), codifies the homotopic informationof a simply connected CW-complex X (ignoring torsion) in a differ-ential graded algebra (A, d), called minimal model of X. Its basicproperties are the following:

• A = A(VX) = algebra generated by the vector spaces V kX =

〈xk,1, . . . , xk,sk〉, with the property of being “graded commuta-tive”:

x · y = (−1)pq y · x, x ∈ V pX , y ∈ V q

X .

• V kX∼= (πk(X)⊗ R)∗.

• d : An → An+1, a differential with

dxk,α =∑i,j<k

aijkαβγ xi,βxj,γ .

• H∗(A, d) = H∗(X).

Therefore, the minimal model contains all the information of homotopyexcept for the torsion, πk(X) ⊗ R, and of homology except for thetorsion, Hk(X) = Hk(X,R)∗.

Examples.

(1) X = S2n+1

A = A(x2n+1) = R⊕ Rx2n+1 = H∗(S2n+1), d = 0.H∗(A, d) = R⊕ Rx2n+1 = H∗(S2n+1).We deduce that πk(S

2n+1) is torsion for every k 6= 2n + 1 andthat π2n+1(S

2n+1)⊗ R has dimension 1.

(2) X = S2n

A = A(x2n, y4n−1) = 〈1, x2n, x22n, . . . , y4n−1, y4n−1x2n, y4n−1x22n, . . .〉,dy4n−1 = x22n ⇒ d(y4n−1x

i2n) = xi+2

2n .H∗(A, d) = R⊕ Rx2n = H∗(S2n).We deduce that πk(S

2n) is torsion for every k 6= 2n, 4n− 1 andthat π2n(S2n)⊗ R and π4n−1(S

2n)⊗ R have dimension 1.

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(3) X = T r = S1 × · · · × S1

A = A(e1, . . . , er), ei ∈ V 1x , d = 0.

H∗(A, d) =∧

(e1, . . . , er) = H∗(T r).

Minimal models and fibre bundles. If T r → E → B is a fibre bundlewith B simply connected, then there exists a “minimal model” for thisfibre bundle. This consists of a diagram

(A(VB), d)→ (A(VB ⊕ VT r), D)→ (A(VT r), 0) = (∧

(e1, . . . , er), 0)

where D is a differential satisfying

• Dxi,α = dxi,α, xi,α ∈ V iB,

• Dej = yj ∈ V 2B,

• H∗(A(VB ⊕ VT r), D) = H∗(E).

Notice that A(VB⊕VT r) = A(VB)⊗∧

(e1, . . . , er). The version of TRCin the language of minimal models can be stated as follows:

TRC-C: If dimH(A(VB), d) <∞, then

dimH∗(A(VB)⊗∧

(e1, . . . , er), D) ≥ 2r .

It is easy to notice similitaries between TRC-B and TRC-C, whereH∗(B) becomes (A(VB), d) (see [Mu]).

D. Homological algebra. It is quite natural to convert TRC (atleast in its B-incarnation) to a problem in homological algebra. Let usconsider the ring of polynomials in r variables R = R[Y1, . . . , Yr]. Themap

R → H∗(B),Yj 7→ yj

turns M = H∗(B) in an R-module with multiplication defined by Yj ·α = yj · α.

On the other hand, M ′ = R is another R-module: it is enough to definethe multiplication as Yj · λ = 0, ∀λ ∈ R. More algebraically, this iswritten as M ′ = R/(Y1, . . . , Yr).

In homological algebra, one can construct functors, called TorR(−,−)functors, as “derived” functors of the tensor product − ⊗R − functor.

TORAL RANK CONJECTURE 19

To every pair of R-modules M1 and M2, they associate a family of R-modules TorkR(M1,M2). This family has the following fundamentalproperty: if

0→M1 →M2 →M3 → 0

is a short exact sequence of R-modules, then, for every R-module M4,there exists a long exact sequence

. . . → Tor2R(M1,M4) → Tor2R(M2,M4) → Tor2R(M3,M4) →→ Tor1R(M1,M4) → Tor1R(M2,M4) → Tor1R(M3,M4) →→ M1 ⊗RM4 → M2 ⊗RM4 → M3 ⊗RM4 → 0 .

The computation of Tor∗R(M,R) is performed by taking a free resolu-tion of R as an R-module:∧r

⊗R→ · · · →∧2⊗R→

∧1⊗R→ R→ R ,

where we have shortened∧k =

∧k(e1, . . . , er), and the maps consist ofv 7→

∑j Yj · iejv (iej denotes “contraction with ej”). Afterwards, we

tensor the resolution with M and we compute cohomology:

Tor∗R(M,R) = H∗(M ⊗R (∧k

(e1, . . . , er)⊗R))

= H∗(M ⊗∧k

(e1, . . . , er), D) ,

where the differential D is given by Dej = yj.

Therefore, the TRC (in its TRC-B version) becomes

TRC-D: If M is an R-module of finite dimension, then

dim Tor∗R(M,R) ≥ 2r .

E. Commutative algebra. Let R = R[Y1, . . . , Yr] denote again thepolynomial ring in r variables and let us consider an R-module M offinite dimension. The TRC-D version says that one must have

dim Tor∗R(M,R) ≥ 2r.

This Tor groups can be computed by the mean of a free resolution ofM of the form

0→ Rnr fr−→ . . .f2−→ Rn1

f1−→ Rn0f0−→M .

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By the D. Hilbert theorem of syzygies (see [Ei]), the number of freemodules in the resolution is exactly r+1. If M is a graded module (as itis the case, in general, with the TRC), the maps fk have homogeneouspolynomials as entries. The resolution is minimal when no fk has aconstant term. In this case, the numbers nk are called (in commutativealgebra) Betti numbers of the R-module M (the use of this terminologyis quite suggestive, as the nk coincide with the Betti numbers of E inthe original TRC).

In order to compute Tor∗R(M,R), we tensor the resolution of M with− ⊗R R and then calculate the cohomology. Since fk has no constantterm, we are left with

TorkR(M,R) = Rnk .

TRC-E: If M is a finite-dimensional R-module, and nk are the Bettinumbers (of a minimal resolution of M), then

r∑k=0

nk ≥ 2r .

This is a famous conjecture in commutative algebra, proposed in 1977by D. A. Buchsbaum and D. Eisenbud (see [BE]).

F. Lie algebras. There is yet another version of the TRC, which isspecific for Lie algebras. It comes from the C-version of the TRC,applied to minimal models of nilmanifolds. Nilmanifolds are quotientsof simply connected nilpotent Lie groups G by discrete subgroups Γ ⊂G such that M = G/Γ is a compact manifold. A group is nilpotent if,computing repeatedly commutators, we end up obtaining the identityelement. Every simply connected nilpotent Lie group is diffeomorphicto Rn. Therefore, a nilmanifold is never simply connected. Indeed, thetheory of covering spaces [Ma] tells us that

π1(M) = Γ ,

and that πk(M) = 0, ∀k ≥ 2. However, the theory of minimal mod-els also works for spaces with nilpotent fundamental group, such asnilmanifolds.

TORAL RANK CONJECTURE 21

Let M = G/Γ be a nilmanifold and let g denote the Lie algebra of G.Such a Lie algebra is nilpotent (if we compute brackets repeatedly, wealways end up obtaining 0). Thanks to this, we can choose a basis

g = 〈e1, . . . , en〉R ,

ordered in such a way that the brackets obey

[ei, ej] =∑k>i,j

aijk ek .

The minimal model of M has generatorsV 1 = (π1(M)⊗ R)∗ = g∗ = 〈e∗1, . . . , e∗n〉R ,V k = 0, ∀k ≥ 2 .

Hence

AM =∧

(g∗) ,

with differential given by de∗k =∑

i,j<k aijk e∗i e∗j . The cohomology

H∗(M) = H∗(∧

(g∗), d) is known, in Lie algebra theory, as Lie alge-bra cohomology of g (and is usually denoted by H∗(g)).

In the TRC-A version applied to M , we need a torus acting on M . Inour case, we need a subgroup H = Rr ⊂ G such that

T r = Rr/Zr = H/(Γ ∩H) → G/Γ .

For this, H must be contained in the center of the group G. If wechoose H maximal, H = Z(G) = x ∈ G |x y = y x,∀y ∈ G.At the Lie algebra level, the Lie algebra of H must be the center of g,i.e.

h = Z(g) = x ∈ g | [x, y] = 0,∀y ∈ g .

In this way, we obtain our last version of the TRC:

TRC-F: Let g be a nilpotent Lie algebra, with center z of dimensionr = dim z. Then

dimH∗(g) ≥ 2r .

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References

[BE] Buchsbaum D., Eisenbud, D. Algebra structures for finite free resolutions andsome structure theorems for ideals of codimension 3, American J. of Math.99 (1977), 447–485.

[BT] Bott, R., Tu, L. W. Differential forms in algebraic topology. Graduate Textsin Mathematics, 82, Springer Verlag, 1982.

[Ei] Eisenbud, D. Commutative algebra. With a view toward algebraic geometry.Graduate Texts in Mathematics, 150, Springer-Verlag, 1995.

[GM] Griffiths, P. A., Morgan, J. W. Rational homotopy theory and differentialforms. Progress in Mathematics, 16, Birkhauser, 1981.

[Ha] Halperin S. Le complexe de Koszul en algebre et topologie, Ann. Inst. Fourier37 (1987), 77–97.

[Ma] Massey, W. S. A basic course in algebraic topology.raduate Texts in Mathe-matics, 127, Springer-Verlag, 1991.

[Mc] McCleary, J. A User’s Guide to Spectral Sequences. Cambridge Studies inAdvanced Mathematics, 58, 2nd Edition, Cambridge University, 2001.

[Mu] Munoz, V. Torus rational fibrations, J. Pure and Applied Algebra 140 (1999),251–259.

[Wa] Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups.Graduate Texts in Mathematics, 94, Springer-Verlag , 1983

Departamento de Matematicas, CSIC, Serrano 113 bis, 28006 Madrid,Spain

E-mail address: vicente.munoz@imaff.cfmac.csic.es