FREE TORUS ACTIONS AND TWO-STAGE SPACES BARRY JESSUP AND GREGORY LUPTON Abstract.We prove the toral rank conjecture of Halperin in some new case* *s. Our results apply to certain elliptic spaces that have a two-stage Sulli* *van minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work. 1.Introduction A well-known conjecture due to Halperin concerns torus actions on a space (see [Hal85, Prob.1.4]). If X is a space on which an n-dimensional torus acts, we say the action is almost-free if each isotropy subgroup is a finite group. The lar* *gest integer n 1 for which X admits an almost-free n-torus action is called the to* *ral rank of X, and is denoted rk(X). If X does not admit an almost-free circle acti* *on, then rk(X) = 0. Halperin's conjecture gives an upper bound for the toral rank of X in terms of its cohomology, as follows: Conjecture 1.1 (The toral rank conjecture). If X is simply connected, then dimH(X; Q) 2rk(X). We shall henceforth assume that all our spaces are 1-connected, finite cell c* *om- plexes. There are some technical conditions on the topology of the space X in Halperin's original formulation, but as these are satisfied by finite cell comp* *lexes, we will not mention them explicitly here. The main tool we shall use is the Sullivan minimal model, and a basic referen* *ce is [FHT01 ]. For our purposes, we note that to any 1-connected space X there co* *rre- sponds, in a contravariant way, a commutative differential graded algebra (uniq* *ue up to isomorphism) ( W, d), called the minimal model of X, which algebraically models the rational homotopy type of the space. By W we mean the free graded commutative algebra generated by the graded vector space W . The differential d* * of any element of W is a polynomial in W with no linear term, which in particular means that there is a homogeneous basis {wi}i 1 of W for which dwi 2 W, and we also denote W by (w1, . .,.wm ) in this case, often omitting explicit reference to the differential. We note that models with* * W and H( W, d) both finite-dimensional are called elliptic, and a space X with an elliptic minimal model is called an elliptic space. Topologically, this means * *that both ß(X) Q and H(X; Q) are finite dimensional. Conjecture 1.1 is already known to hold for homogeneous spaces G=H, for G connected, and H closed and connected [Hal85, Prop.1.5]. Such spaces have two- stage minimal models [FHT01 , Prop.15.16], where a minimal model ( W, d) is said to be two-stage if W decomposes as W ~= U V with d U = 0 and d V U. By the remark in the previous paragraph, it is easy to see that there are many other examples of spaces, indeed smooth manifolds, with two-stage models, and it is these spaces that we shall study (see for example Corollary 2.4). We note for later reference that there may be several ways to display a minim* *al model as a two-stage model. In particular, a generator in V that is a cocycle c* *ould just as well be included in U. We will generally be interested in choosing a tw* *o-stage decomposition in which V is as large as possible. By a K-S extension (Koszul-Sullivan), or simply an extension, we mean a se- quence of the form ( W1, d1) ! ( W1 W2, D) ! ( W2, d2) with ( W1, d1) a minimal model, in which D restricts to d1 on W1 1, and for which there is an ordered basis of W2 = .with Dwi 2 W1 (w1, . .,.wi-1) for each i. A K-S extension is the minimal-model analogue of a Serre fibration (cf. [FHT01 , Sec.15(a)]), and for this reason, W1, W1 W2 and W2 are known respectively as the base, the total space and the fibre of the extension. The connection between minimal models and the toral rank is originally due to Allday and Halperin [AH78 ] (cf. [Hal85, Prop.4.2]): If X has minimal model ( W, d) and admits an n-dimensional torus action, then there is an extension (1) An ! An W ! W in which An = with each ai of degree 2. If the action is almost f* *ree, then dimH( An W, D) is finite-dimensional [Hal85, Prop.4.2]. In principle, this result allows an upper bound for rk(X) to be obtained by a direct analysis of t* *he minimal model of X, and this direct approach has been carried out to great effe* *ct in some situations. In general, however, the computational problems involved he* *re appear to be quite substantial. However, since we are really only considering the rational homotopy type of X, we are led to the following variation of the toral rank: the rational toral * *rank rk0(X) of X is defined by rk0(X) = max {rk(Y ) | Y 'Q X}. Clearly, we have rk0(X) rk(X), hence an upper bound on rk0(X) will serve as one on rk(X). See [Hal85, Prop.4.2] for the precise relationship between these two numbers. T* *he characterization of rk0(X) in terms of a minimal model of X is as follows. If FREE TORUS ACTIONS AND TWO-STAGE SPACES 3 ( W, d) is a minimal model of X, then rk0(X) is the largest n (if such exists) * *for which there is a K-S-extension of the form (1), for which dimH( An W, D) is finite-dimensional. We will also denote rk0(X) by rk0( W ). Now we briefly summarize the contents of the paper. In Section 2, Theorem 2.3 gives a lower bound for the dimension of the cohomology of any space with an el* *liptic two-stage minimal model. This result is proved using standard tools familiar fr* *om algebraic topology, namely the Wang sequence and the Serre spectral sequence. Also in Section 2 we give two results that, under certain additional hypotheses, give upper bounds on the rational toral rank_hence on the toral rank_of a space with two-stage minimal model (Theorem 2.9 and Theorem 2.10). Combining these results allows us to establish Conjecture 1.1 in some new cases (Corollary 2.11* *). In addition, our results can be used to provide new and essentially elementary pro* *ofs of the conjecture in some previously-known cases, which we discuss after Theorem 2* *.3. In Section 3, we give a number of examples and suggest several directions for f* *uture investigation. Acknowledgements. It is our pleasure to thank Yves F'elix and Steve Halperin for their input. They have each generously shared with us their expertise in concoc* *ting examples of the sort germane to this paper. In particular, Example 3.3 and the discussion that follows it are entirely due to Steve Halperin. 2. Cohomology and Toral Rank of Two-Stage Minimal Models In this section we give several results about the cohomology and the toral ra* *nk of two-stage minimal models. 2.1. Cohomology of two-stage minimal models. We begin by establishing a lower bound for the dimension of the cohomology of an elliptic space with two-s* *tage minimal model. Our bound is of a form similar to that featured in the toral rank conjecture. If (U, V ) is an elliptic two-stage minimal model, then the Mapping theorem [FHT01 , P.375] implies that V has generators of odd degree only. On the other hand, U may have generators of odd or even degree. For the remainder of the paper, unless otherwise specified, we assume that all minimal models are two-st* *age, elliptic. For details about elliptic spaces and their minimal models, see [FHT* *01 , Sec.32]. First, we establish our lower bound in a special case. The main result is then established by reducing to this special case. Proposition 2.1. Let (U, V ) be a two-stage, elliptic minimal model with odd degree generators only and suppose that d: V ! 2U is an isomorphism. Then dimH (U, V ) 2dimV. Proof.Suppose that dim U = n. Since d: V ! 2U an isomorphism, and since U is an exterior algebra, we have dim V = n2. Thus we must show that n dimH (U, V ) 2(2). We will proceed by induction on n. First, we give an explicit description of the model (U, V ). Write (U, V ) as Mn = ( (Un, Vn), dn), with Un = and Vn = <{vi,j}1 i. The differential is dn(Un) = 0 and dn(vi,j) = uiuj for each 0 i < j n. For n = 1, we have dim U1 = 1 and V1 = 0, and the proposition 4 BARRY JESSUP AND GREGORY LUPTON is trivial. For n = 2, we have M2 = (u1, u2, v1,2). It is easily checked that dimH(M2) = 6 21. This starts the induction. n Now suppose inductively that dimH(Mn) 2(2)for some n 2. We adjust the notation somewhat and write Mn+1 as follows. Write Un+1 = = Un. Also, set V0 = , so that Vn+1 = V0 Vn. The differ* *ential dn+1 of Mn+1 extends that of Mn. Thus, we have dn+1(Un+1) = 0, dn+1(vi,j) = dn(vi,j) = uiuj_for_1 i < j n and dn+1(v0,j) = u0uj for 1 j n. Further, if M n+1 denotes Mn V0, we have the following extension: ___ __ (2) u0 ! (Mn+1, dn+1) ! (M n+1, dn+1). Consider the short exact sequence of differential graded vector spaces ___ j p ___ (3) 0_____//Mn+1_____//Mn+1 ____//_Mn+1____//_0, where p denotes the projection and j the map defined by j(Ø) = u0Ø for Ø 2 W . The ensuing long exact sequence in cohomology_has a connecting_homomorphism_ that may be described as follows: On M n+1, define `* : M n+1 ! M n+1 by u0` = dn+1-d~n+1._Then, one can easily check that ` is actually a derivation of* * the algebra_M_n+1 of degree 1 -_|u0|_which commutes with ~dn+1, and so induces a map `* : H(M n+1) ! H*-|u0|+1(M n+1) which is the connecting homomorphism of (3). We will call the derivation `* the Wang derivation for (2), for reasons indicat* *ed in Remark 2.2 below. This Wang derivation can_be explicitly_computed as follows. Because dn+1(V0) u0 . Un, we see that H(M n+1, dn+1) = H(Mn) V0. Now, for Ø 2 Mn V0, we find that `(Mn) = 0, directly from the definition. Since dn+1(v0,i) = u0ui, * *we have `(v0,i) = ui for each i = 1, . .,.n. On passing to cohomology, therefore, * *we find that `* H(Mn) = 0 and `*(v0,i) = ui2 H(Mn) for i = 1, . .,.n. As usual, we can condense the long exact sequence coming from (3) into 0 ! coker`* ! H(Mn+1) ! ker`* ! 0, so that to complete the induction, it simply remains to show that dim ker`* + dimcoker`* = 2 dimker`* n+1 n is at least 2( 2). Our inductive assumption is that dimH(Mn) 2(2). Therefore n n n+1 dim H(Mn) . V0 2(2)x 2n. Since 2 + n = 2 , our inductive assumption n+1 implies that dim H(Mn) V0 2( 2). We claim that (`*)2 = 0, after which elementary linear algebra completes the proof. The key observation here is the following: Denote by [ui] 2 H(Mn) the class represented by the cocycle ui. Then the ideal of H(Mn) generated by all two-fold products {[ui][uj]}1 i and V = , all generators being of degree 1. If we set dv1 = u1u3 - u1u2 + u2u3, then each basis element of U certainly occurs in a non-triv* *ial differential. However, V is not maximal here. We can see this by re-writing dv1 = (u1 + u2)(u1 + u3), and noting that only 2 linearly independent elements of U actually occur. In other words, we can change the basis in U, and move the `spare' generator of U into V . Specifically, define OE(u1) = u1, OE(u2) = u2 * *+ u1, OE(u3) = u3 + u1 and OE(v1) = v1, then extend to an algebra automorphism of (U, V ). If we set d0= OE-1dOE, then OE becomes an isomorphism of minimal mode* *ls OE: (U, V ; d0) ! (U, V ; d). A simple check reveals that (U, V ; d0) is tw* *o-stage with d0(v1) = u2u3. Thus we can write it as (U0, V 0; d0), U0 = and V 0= . In this latter case, V 0is now of maximal dimension. Definition 2.7. Suppose ( (U, V ), d) is a two-stage minimal model. We say that* * V has maximal dimension, or that the two-stage decomposition displays V with max- imal dimension, if, for any isomorphic two-stage minimal model ( (U0, V 0), d0)* * ~= ( (U, V ), d), we have dimV 0 dimV . Since we assume that the spaces of generators are finite-dimensional, it is c* *lear that every two-stage minimal model has a decomposition that displays V with max- imal dimension. Also, for the two-stage decomposition to display V with maximal dimension, then it is clearly necessary (but not sufficient) that every generat* *or from U appear in some differential. We now give a consequence of the two-stage decomposition being chosen so as to display V of maximal dimension. This result does not cover all two-stage cas* *es, but is sufficient for our purposes. We focus on the case in which the different* *ial is quadratic. For parity of degree reasons, in this case we have d(V ) 2(Uodd) 2(Ueven). Let Uodd = , Ueven= and V = denote bases. From now on, we will adopt this as standard notation, for the case of a two-stage minimal model that may contain even-degree generators. Since we assume the differential is quadratic, for each basis element vk of V* * , we may write X X dvk = aki,juiuj+ bki,jwiwj. 1 i j 0 if i = j 8 BARRY JESSUP AND GREGORY LUPTON 2 3 M1 Now, let M = 64...75be the pr x p block marix formed by the Mk as rows. Mr Lemma 2.8. If V has maximal dimension in (U, V ), then dimM = p. In partic- ular, there is a p x rp matrix N such that NM is the p x p identity matrix. Proof.Let U* denote the dual space of U, and for u* 2 U*, let iu* denote the derivation of (U, V ) of degree -1 extending the linear map iu* : U V ! Q defined by iu*(x + y) = u*(x), for x 2 U, y 2 V . We first show that, under our hypotheses, the Lie derivative L: U* ! Hom (V, U) defined by *| L(u*) = iu*d - (-1)|u diu* = iu*d is injective. (In fact, the injectivity of L is equivalent to the maximality of* * dimV .) Let K = kerL and choose a complement X U* for K so that U* = K X and U = K? X? , where, if W U*, W ?= {u 2 U | 8w 2 W, w(u) = 0}. Then, as U = ( X? X? ) ( X? + K? ) ( + K? + K? ), the definition of K shows that dV + K? + K? . Now define U1 = K? and V1 = V X? , with d0U1= 0, d0|V= d|Vand d0X?= 0. Then U1 V1 = U V , and this induces an isomorphism of two-stage minimal models (U1, V1; d0) ~= (U, V ; d). Since V has maximal dimension, K must be 0. Let {u1, . .,.up} be a basis of Uodd, and {u*1, . .,.u*p} the dual basis. The* *n, as V has maximal dimension, we know that in particular the maps L(u*j) = iu*jd : V ! Uodd for jP= 1, . .,.p arePlinearly independent. In other words, if c = (c1, . * *.,.cp)t2 Qp, then jcjiu*jdvk = i,jMki,juicj = 0 for k = 1, . .,.r, implies that c = * *0. Because the ui are linearly independent, this is equivalent to the statement X Mki,jcj = 0 fori = 1, . .,.p and k = 1, . .,.r =) c = 0. j That is, Mc = 0 =) c = 0. Hence, dimM = p. To phrase our results, we use the following terminology: We say that a graded vector space V is n-co-connected if V i= 0 for i n. Theorem 2.9. Let (U, V ) be an elliptic, two-stage minimal model. Assume that dV 2U, and that the two-stage decomposition displays V with maximal dimen- sion. Finally, assume that Uodd and Uevensatisfy one of the following connectiv* *ity and co-connectivity hypotheses: (A) Uodd is (2r - 1)-connected and Uevenis (2r + 2)-co-connected. (B) Uodd is (2s + 1)-co-connected and Uevenis (4s - 4)-connected. Then rk0 (U, V ) dimV - dimUeven. Proof.If rk0 (U, V ) = n, then we have a K-S-extension ( An (U, V ), D) as in (1) that has finite-dimensional cohomology. We claim that in any such minimal model, we can assume that D(Uodd) is contained in I(Uodd V ), the ideal in An (U, V ) generated by Uodd V . Allowing this claim for the time-being, we appeal to some results of Halperin concerning elliptic minimal models. First of all, to any elliptic minimal model ( W, d), there is an associated pure model, denoted ( W, doe), which is defined* * by adjusting the differential d to doeas follows: We set doe= 0 on each even degree FREE TORUS ACTIONS AND TWO-STAGE SPACES 9 generator of W , and on each odd degree generator w 2 W , we set doe(w) equal to the part of d(w) contained in (W even). One checks that this defines a differe* *ntial doeon W , and thus we obtain a pure (minimal) model ( W, doe). Then, by [Hal77, Prop.1] (cf. also [FHT01 , Prop.32.4]), dim H( W, d) is finite-dimensional if a* *nd only if dimH( W, doe) is finite-dimensional. Applying all this to the minimal m* *odel ( An (U, V ), D), we obtain the following: From the claim, ( An (U, V ), Doe) satisfies Doe(Uodd) = 0, and therefore ( An (U, V ), Doe) ~=( Uodd, Doe= 0) ( An (Ueven, V ), Doe). Since ( An (U, V ), Doe) has finite-dimensional cohomology, so does ( An (Ueven, V ), * *Doe). It follows that ( An (Ueven, V ), Doe) is an elliptic minimal model, and the* *re- fore that its homotopy Euler characteristic is non-positive [Hal77, Th.1] (cf. * *also [FHT01 , Prop.32.10]). This implies that n dimV - dimUeven, as required. It only remains to establish the claim. We do this by a careful analysis of t* *he terms that can occur in the differentials. Our argument is essentially the same* * as that in [AP85b , Th.4.6]. First we write, for each vk in the basis of V , X X (7) Dvk = aki,juiuj + bki,jwiwj + terms in the ideal I(An). 1 ilAn U. Consequently, when we equate terms in lAn U that arise in the equation 0 = D2v = D(dv) + D(Ø), we have 0 = ffi(dv). By the assumption that non-zero ffiui's are linearly independent, it follows from an easy argumen* *t that ffiui= 0 for each ui that occurs in the differential dv. Finally, our assumptio* *n that V is taken as large as possible implies that each generator uidoes occur in at * *least one differential dv. This implies that the shortest length terms ffiuiare zero,* * and an induction completes the proof of our claim, namely that Dui= 0 for each ui2 U. So far, we have argued that in any extension ( An (U, V ), D), our hypotheses imply that D(U) = 0. Now an argument as in the second paragraph of the proof of Theorem 2.9 shows that n dimV . Corollary 2.11. Let X be a simply connected, elliptic space with two-stage mini* *mal model. If the minimal model satisfies either the hypotheses of Theorem 2.9, or * *those of Theorem 2.10, then Conjecture 1.1 holds for X. Proof.Combine Theorem 2.3 with the results mentioned in the statement. It is natural to ask whether rk0(X) dimV - dimUevenfor a general two-stage, elliptic space. This is not the case in general, as we illustrate in Example 3.* *1 below. Indeed, under hypotheses that include all cases in which the two-stage minimal model has odd-degree generators only, we can actually reverse the direction in * *this inequality. Lemma 2.12. Suppose U V is an elliptic two-stage minimal model with Ueven= U2n for some fixed n 2 N. Then, rk0( U V ) dimV - dimUeven. Proof.Consider the extension sequence ( Uodd, 0) ! ( Uodd Ueven V, d) ! ( Ueven V, ~d). Since V = V odd, the right-hand term is now pure (and elliptic) with Ueven= U2n. By [Jes90, Lem.3.3], there is an isomorphism of two-stage models æ: ( Ueven V, ~d) ! ( Ueven (V 0 V 00), ~d0) 12 BARRY JESSUP AND GREGORY LUPTON such that (1) dim H( Ueven V 0, ~d) < 1, (2) dim V 0= dimUeven, (3) æ is the identity on Ueven, and (4) æ(V ) V 0 V 00 ( + Ueven (V 0 V 00)). These conditions imply that æ induces an isomorphism V ~=V 0 V 00, and that æ-1(V 0 V 00) V ( + Ueven V ). We now extend æ to an isomorphism of algebras ~æ: U V ! U (V 0 V 00) by letting it be the identity on U. Define a derivation d0on U (V 0 V 00* *) by setting d0U = 0, and d0v0= dæ-1(v0), for v02 (V 0 V 00). In order to show that ~æ:( U V, d) ! ( U (V 0 V 00), d0) is an isomorphism of models, it suffices to show that (d0)2v0= 0 for v02 V 0 * *V 00, so we compute: (d0)2v0= d0(dæ-1(v0)) 2 d0(d(V ( + Ueven V )) d0( U) = {0}. Thus, we may assume that V = V odd= V 0 V 00, with dimH( U V 0) < 1, and dimV 0= dimUeven. Now suppose V 00= , and define a K-S-extension ( An, 0) ! ( An (U, V ), D) ! ( (U, V ), d), |vi|+1_ in which An = with|ai| = 2, by Dvi = dvi + ai 2 , and d = D otherwise. A standard argument (using the associated pure model, as in the pro* *of of Theorem 2.9), now shows that dim H( An (U, V ), D) is finite-dimensional, so that rk0 (U, V ) n = dimV 00= dimV - dimUeven. Consequently, we obtain values for the rational toral rank in the following * *cases. Corollary 2.13. Let X be a simply connected elliptic space with two-stage mini* *mal model (U, V ). Suppose that the two-stage decomposition displays V with maxim* *al dimension. (2.13.1)If the minimal model has quadratic differential, satisfies one of the (* *co-) connectivity conditions of Theorem 2.9 and also satisfies Ueven= U2n for some fixed n, then rk0(X) = dimV - dimUeven. (2.13.2)If the minimal model has odd-degree generators only, and satisfies the * *(co-) connectivity conditions of Theorem 2.10, then rk0(X) = dimV . Proof.Combine Lemma 2.12 with the results mentioned in the statement. Remarks 2.14. We can specialize (2.13.1) in a number of interesting directions. For example, if Ueven= 0, then we retrieve the identification rk0(X) = dim V of Theorem 2.5. As a second example, we can restrict to the case in which Uodd = * *0. This is the case in which the minimal model is pure with quadratic differentia* *l, and also satisfies Ueven= U2n for some fixed n. Here, we obtain rk0(X) = -Øß(X). FREE TORUS ACTIONS AND TWO-STAGE SPACES 13 3.Examples, Comments and Questions In this section we mention various examples and results. Our focus here is mo* *re on the exact toral rank, rather than the bound of Conjecture 1.1. 3.1. The relation between rk0(X) and dim V - dimUeven. We begin with the simplest example that we can find to illustrate that the inequality rk0(X) di* *mV - dimUevendoes not hold in general for a two-stage, elliptic space with both even and odd generators. Example 3.1. Consider the two-stage minimal model (u1, u2, u3, u4, u5, u6, w, * *v), with |u1| = |u2| = |u3| = 3, |u4| = 7, |u5| = 23, |u6| = 27, |w| = 18 and |v| =* * 35, and the single non-trivial differential given by dv = w2-u1u2u4u5-u1u2u3u6. The associated pure model satisfies ( (U, V ), doe) ~=( (u1, u2, u3, u4, u5, u6), d* *oe= 0) ( (w, v), doe), with doe(w) = 0 and doe(v) = w2. Now H( (w, v), doe) ~= (w)=(w2* *), so ( (U, V ), d) is elliptic. The two-stage decomposition U = and V = displ* *ays V with maximal dimension, so we have dimV - dimUeven= 0. We now show that rk0 (U, V ) ) 1. Let a be a generator of degree 2, and define a differential D on a) (U, V ) as follows: Da = Du1 = Du2 = Du3 = Du4 = Dw = 0, Du5 = a3w, Du6 = a2wu1u2 and Dv = w2 - u1u2u4u5 - u1u2u3u6 + a18- au6u4. A straightforward check shows that this defines a differential. We show that a (U, V ), D has finite-dimensional cohomology. The associated pure model in th* *is case is ( a (U, V ), Doe) ~=( (u1, u2, u3, u4, u6), Doe= 0) ( (a, w, u5, v* *), Doe), with Doea = 0, Doe(w) = 0, Doeu5 = a3w and Doev = w2 + a18. We observe that Doe(a3v) = a3w2 + a21 = Doe(u5w) + a21, hence Doe(a3v - u5w) = a21, so [Hal77, Prop.1] shows that H( a (U, V ), Doe) is finite-dimensional. It fol* *lows that H( a (U, V ), D) is finite dimensional. Thus rk0 (U, V ) ) 1. We assert that more work will show rk0 (U, V ) ) = 1 in Example 3.1. We also remark that whilst Conjecture 1.1 does not follow from Theorem 2.3 in this example, it is nonetheless easily confirmed here. Next, we specialize to the case of odd generators only. Unfortunately, this r* *e- striction alone does not give us the inequality rk0(X) dim V . The following example lies immediately outside the hypotheses of Theorem 2.10, at least as far the generators of U are concerned. We thank Yves F'elix for this example. Example 3.2. Let W = (U, V ) = (u1, u2, u3, u4, u5, v1, v2) be a two-stage minimal model with degrees and differential as follows: |u1| = |u2| = |u3| = |u* *4| = 3, |u5| = 7, |v1| = 9, |v2| = 11, dui= 0 for each i, dv1 = u1u5 and dv2 = u1u2u* *3u4. Then dim V = 2. We show that rk0 W 3. First we describe an extension A W, D, with A = . Set D = 0 on {a1, a2, a3, u1, u2, u3}, and Du4 = -a21, Du5 = a1u1u2, Dv1 = u1u5 + a52and Dv2 = u1u2u3u4 + u3u5a1 + a63. A straightforward check shows that D is a differential. We argue exactly as in * *the latter part of Example 3.1 to show that H( A W, D) is finite-dimensional. It follows that rk0 W 3. Once again, Conjecture 1.1 does hold for Example 3.2, although we are unable to conclude this from our preceding results. To see this, use [AP85b , Th.4.6]* * to conclude rk0 W 5. (In Example 3.2, the differential dv1 = u1u5 means that u1 and u5 correspond to non-central generators.) Moroever, the fibration with u1 as base has fibre (u2, u3, u4, u5, v1, v2) with zero differential, and (u2, u* *3, u4, u5) 14 BARRY JESSUP AND GREGORY LUPTON is clearly in the kernel of the associated Wang derivation `*. As in the proof* * of Theorem 2.1, we conclude that dimH( W ) = 2 dimker`* 2.24 = 25 2rk0( W). Another example in which rk0(X) > dimV - dimUevenis given in Example 3.3. On the other hand, we currently have no example of a two stage minimal model for which rk0(X) < dimV -dim Ueven. This leads us to wonder whether the hypotheses of Lemma 2.12 might be relaxed considerably. Note that in certain cases, the inequality rk0(X) dimV - dimUevencan be combined with other information to identify the toral rank, and we have seen instances of this in Corollary 2.13. * * As another example, whenever this latter inequality holds in a pure case, it ident* *ifies the toral rank as equal to -Øß (cf. Remarks 2.14). 3.2. Products. Now we consider the question of how the toral rank behaves with respect to products. Although our previous results concerned the two-stage case, the comments here are not restricted to that case. It is easy to see that, in g* *eneral, we have the inequality rk0(X x Y ) rk0(X) + rk0(Y ). The following example shows that we may sometimes have strict inequality. Example 3.3. Consider two-stage, elliptic minimal models M = (x, y) with |x| = 12, |y| = 23, and dy = x2; and N = (u1, u2, u3, u4, u5, u6, w, v), with |u1| = |u2| = |u3| = |u4| = 3, |u5| = 5, |u6| = 19, |w| = 18, |v| = 35, and dv = w2 + u1u2u3u4u5u6, all other differentials being zero. The homotopy Euler characteristic bound yields rk0(M) = 0. We will show that rk0(N ) = 0 by arguing that any extension of the form ( a N , D) cannot have finite-dimensional cohomology. L Let U0 = and denote {i1,...,ik}U0 := km=1 imU0. For degree reasons, any such D satisfies Dui = ffia2, for i = 1, . .,.4, Du5 = ff5a3, Du6 = ff6a10+ ~ aw + , and (10) Dv = w2 + u1u2u3u4u5u6 + w + u6 + , where the ffi and ~ are scalars, , Dw 2 + a + U0, 2 3a {1,3}U0 and , 2 + a U0. Our strategy is to show that each ffi= 0, and then to follow a similar argume* *nt as in previous examples. Applying D to (10) and equating terms that contain u6, we find X4 0 = ffia2u1. .^.ui.u.4.u5 + u1. .u.4ff5a3 - D . i=1 Since D 2 5a U0, we immediately find ffi = 0 for i = 0, . .,.5, and so DU0 = 0. Now, 0 = D2u6 = ~aDw. So either ~ = 0 or Dw = 0. Moreover, the w component of D2v being zero implies that 0 = 2Dw - ~au1. .u.5+ ~a . If ~ = 0, then this equation implies Dw = 0. On the other hand, suppose ~ 6= 0. Since 2 3a {1,3}U, by multiplying this last equation by ~a, we conclude that in fact ~ = 0, and hence that Dw = 0. The equation D2v = 0 now implies (11) 0 = -ff6a10u1. .u.5+ (ff6a10+ ) . Suppose ff6 6= 0. Then, upon considering the component of (11) in + a 1U0, * *we find that 2 + a 3U. Since 2 + a 2U0, this shows that 2 + a 5U0. Thus, (11) now yields = 0 and so ff6 = 0 in any case. FREE TORUS ACTIONS AND TWO-STAGE SPACES 15 The argument so far has shown ~ = ffi = 0 for i = 1, . .,.6. Therefore, for a* *ny minimal model ( a N , D), the corresponding pure model satisfies ( a N , Doe) ~= ( (u1, u2, u3, u4, u5, u6), 0) ( (a, w, v), Doe). Since ( (a, w, v), Doe) has* * homotopy Euler characteristic Øß = +1, it cannot have finite-dimensional cohomology, and hence neither can ( a N , Doe). As before, it follows that ( a N , D) does * *not have finite-dimensional cohomology, and hence that rk0(N ) = 0. Now consider the product M N . We show that rk0(M N ) 1, by dis- playing an extension ( a M N , D), in which H( a M N , D) is finite- dimensional. The differential is as follows: Dx = u1u2u3a2, Dy = x2 + 2u1u6a, Du1 = Du2 = Du3 = Du4 = 0, Du5 = a3, Du6 = u2u3xa, Dw = 0 and Dv = w2 + u1u2u3u4u5u6+ xau4u6+ ya2u2u3u4. A careful check reveals that this defines a differential. The associated pure model is now ( a M N , Doe) with the only non-trivial differentials Doe(y) = x2, Doe(u5) = a3 and Doe(v) = w2. T* *his is easily seen to have finite-dimensional cohomology, and hence so does the extens* *ion ( a M N , D). Therefore, we have rk0(M N ) 1 > rk0(M) + rk0(N ). Discussion 3.4. Recall that rk(X) denotes the actual toral rank (rather than its rational counterpart) for a finite complex X. It is easy to see that the inequ* *al- ity rk(X x Y ) rk(X) + rk(Y ) holds in general_one takes the obvious prod- uct action. The minimal models of Example 3.3 can be realised as the mod- els of honest `geometrical' spaces. M = (x, y) is the minimal model of S12. N = (u1, u2, u3, u4, u5, u6, w, v) is the minimal model of a sphere bundle ove* *r a product of odd spheres. We sketch this: Take S = S3 x S3 x S3 x S3 x S5 x S19, so that S has minimal model (u1, u2, u3, u4, u5, u6) with zero differential. S* * has dimension 36 and by pinching the 35-skeleton to a point, we obtain a non-trivial map q :S ! S36. Composing this with a suitable map p: S36 ! BSO(19) to the universal bundle BSO(19), we obtain a non-trivial map S ! BSO(19). Pulling back the universal real oriented line bundle to one over S, via this map, we ob* *tain a real oriented line bundle R19 ! E ! S. Finally, taking the unit sphere bundle gives a sphere bundle S18 ! Y ! S. We assert that this sphere bundle Y has minimal model N , as in Example 3.3. Our assertion can be justified by a consideration of the possible forms that the mi* *nimal model of Y can take, bearing in mind the construction of Y . Our main point here is simply to indicate how N corresponds to the minimal model of a reasonable space. As in Example 3.3, rk0(X) = rk0(Y ) = 0 and this is sufficient to conclu* *de rk(X) = rk(Y ) = 0. On the other hand, we have rk0(X x Y ) 1, but this is not sufficient to conclude rk(X x Y ) 1. We are left with the following intrigui* *ng question: Does this space X x Y admit an almost-free circle action? Note that by [Hal85, Prop.4.2], there is a simply-connected, finite complex that admits a* * free circle action and which has the rational homotopy type of X x Y . The preceding examples and discussion give rise to a number of interesting qu* *es- tions. Generally, it would be useful to have conditions under which the equali* *ty rk0(X x Y ) = rk0(X) + rk0(Y ) either holds, or does not hold. Various special * *cases are also of interest. For instance, we can ask when does rk0(X xS2n+1) = rk0(X)+ 1? We note that Halperin has an example in which rk0(X x S2n+1) > rk0(X) + 1. As another special case we could ask whether, for an n-fold product of a space * *with itself, we have rk0(Xn) = n rk0(X)? At present, we know of no example where 16 BARRY JESSUP AND GREGORY LUPTON equality does not hold. Finally, we note that, as in the discussion above, it * *is reasonable to ask all these questions in the integral setting, too. 3.3. The Gottlieb group. An interesting suggestion arising from Theorem 2.3 is a connection between rk0(X) and the dimension of the Gottlieb group. We recall that the nthGottlieb group is the subgroup of ßn(X) of elements ff = [f] such t* *hat (f, id) : Sn _ X ! X extends to a continuous map Sn x X ! X. In terms of a minimal model ( W, d), this corresponds [FHT01 , p.392] to the subspace ( ) fi 2 (W n)* | fi extends to a derivation` of degree- n of( W, d) Gn(X) = n such thatd` - (-1) `d = 0 In the two-stage case, with all generators of odd degree, the minimal model can* * be written so that dimV = dimG*(X). Thus we have the following observation: Corollary 3.5. Suppose X has minimal model that is two-stage with odd-degree generators only. Suppose that G*(X) denotes the rational Gottlieb group of X. Then dim H(X) 2dimG*(X). An examination of Example 3.2 shows that we can have rk0(X) > dimG*(X). Next, we illustrate that the dimension of the Gottlieb group can exceed the tor* *al rank by an arbitrary amount. Example 3.6. For each n 1, we describe an (n + 1)-stage model Mn. Set Mn = (x1, x2, y1, z1, y2, z2, . .,.yn-1, zn-1, yn). For degrees we have |xi| =* * 3 and |zi| = 3 for each i. The degrees of the yi are chosen to be compatible with the differential, which is as follows: d(xi) = 0 and d(zi) = 0 for each i, then d(y1)= x1x2 d(y2)= x1x2y1z1 d(y3)= x1x2y1z1y2z2 .. . d(yn)= x1x2y1z1y2z2. .y.n-1zn-1 For each n, one can show that dimG*(X) = n, but rk0(X) = 1. Further, a direct computation shows that dimH(X) = (1=3)4n+1 + 2=3. These considerations suggest the following question: Question 3.7. Let X be a finite complex with rational homotopy groups non-zero in odd degrees only. When is dimH(X) 2dimG*(X)? In spite of the apparent difficulty in establishing Conjecture 1.1, it would * *appear that the conjectured lower bound of 2rk0(X)underestimates the dimension of of t* *he cohomology, in many cases quite seriously. This is supported by examples such as Example 3.6, [AP85b , Ex. 4.5] and analogous computations of the cohomology of nilpotent Lie algebras, such as [ACJ97 ]. It is possible that the lower bou* *nd suggested in Question 3.7 might give a closer estimate in some cases. References [ACJ97]G. Armstrong, G. Cairns, and B. Jessup, Explicit Betti Numbers for a Fam* *ily of Nilpo- tent Lie Algebras, Proc. A. M. S. 125 (1997), 381-385. [AH78] C. Allday and S. Halperin, Lie Group Actions on Spaces of Finite Rank, Q* *uart. J. of Math. 28 (1978), 63-76. FREE TORUS ACTIONS AND TWO-STAGE SPACES 17 [AP85a]C. Allday and V. Puppe, Bounds on the Torus Rank, Transformation Groups,* * Pozna'n 1985, Lecture Notes in Math., vol. 1217, Springer, 1985, pp. 1-10. [AP85b]C. Allday and V. Puppe, On the Localization Theorem at the Cochain Level* * and Free Torus Actions, Algebraic Topology, Göttingen 1984, Springer, 1985, pp. 1* *-16. [DS88] C. Deninger and W. Singhof, On the Cohomology of Nilpotent Lie Algebras,* * Bull. Soc. Math. France 116 (1988), 3-14. [FHT01]Y. F'elix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Grad* *uate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. [Hal77]S. Halperin, Finiteness in the Minimal Models of Sullivan, Transactions * *A. M. S. 230 (1977), 173-199. [Hal85]S. Halperin, Rational Homotopy and Torus Actions, Aspects of Topology, L* *ondon Math. Soc. Lecture Notes, vol. 93, Cambridge Univ. Press, 1985, pp. 293-306. [Jes90]B. Jessup, L-S Category and Homogeneous Spaces, Journal of Pure and Appl* *ied Algebra 65 (1990), 45-56. [Whi78]G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathemat* *ics, vol. 61, Springer-Verlag, 1978. Department of Mathematics, University of Ottawa, Ottawa Canada K1N 6N5 E-mail address: Bjessup@sciences.uottawa.ca Department of Mathematics, Cleveland State University, Cleveland OH 44115 U.S* *.A. E-mail address: Lupton@math.csuohio.edu