CYCLIC MAPS IN RATIONAL HOMOTOPY THEORY GREGORY LUPTON AND SAMUEL BRUCE SMITH Abstract.The notion of a cyclic map g:A ! X is a natural generalization of a Gottlieb element in in(X). We investigate cyclic maps from a ration* *al homotopy theory point of view. We show a number of results for rationali* *zed cyclic maps which generalize well-known results on the rationalized Gott* *lieb groups. 1.Introduction Let X be a space with basepoint x0. Denote by map (X, X; 1) the component of the identity in the unbased function space of maps from X to itself, and by ! :map(X, X; 1) ! X the evaluation map defined by !(f)(x) = f(x0). In [Got69], Gottlieb introduced and studied the evaluation subgroups Gn(X) = !# ßn map (X, X; 1) ßn(X). Note that Gn(X) can alternatively be described as homotopy classes of maps f :Sn ! X such that (f | 1): Sn _X ! X admits an extension F :Sn xX ! X up to homotopy. These so-called Gottlieb groups admit a number of generalizations. One such was introduced and studied by Varadarajan. In [Var69], he defines the homotopy set G(A, X) of cyclic maps from A to X, that is, homotopy classes of maps f :A ! X such that (f | 1): A_X ! X admits an extension F :AxX ! X. The Gottlieb group occurs when A = Sn. Note that the set of cyclic maps G(A, X) can alternatively be described as the set of homotopy classes of maps from A to* * X that admit a lift through the evaluation map ! :map(X, X; 1) ! X. The rationalized Gottlieb groups Gn(X) Q have been studied extensively (see [FH82 , Tan83, Opr95]). Our purpose in this paper is to study rationalized cycl* *ic maps with an eye toward obtaining natural generalizations of known results. In [Got69] Gottlieb studied the question of when the Gottlieb group of X is contai* *ned in the kernel of the mod p or rational Hurewicz homomorphisms. Let h1 :ß*(X) ! H*(X; Q) denote the rational Hurewicz homomorphism. Then Gottlieb's main results for the rational case can be summarized as follows: Theorem 1.1 ([Got69, Th.4.1,Th.5.1]). Let X be a space with finitely generated homology. Then G2n(X) ker(h1 ) and if the Euler characteristic Ø(X) 6= 0 then G2n-1(X) ker(h1 ). Gottlieb also showed that G2(X) is a torsion group_and hence that G2(X) Q = 0_in the case that X has a finite number of non-zero rational homology groups [Got69, Th.7.1]. ____________ Date: August 11, 2003. 2000 Mathematics Subject Classification. 55P62, 55Q05. Key words and phrases. Evaluation map, Gottlieb group, function space, cycli* *c map, rational homotopy, minimal models. 1 2 GREGORY LUPTON AND SAMUEL BRUCE SMITH F'elix and Halperin significantly sharpened these results. Let X be a simply connected space. Then the rational category of X_denoted by cat0(X)_may be defined as the Lusternik-Schnirelmann category of the rationalization of X, that is, cat0(X) = cat(XQ). Further, the rationalized Gottlieb group of X is contain* *ed in the Gottlieb group of the rationalization of X, or Gn(X) Q Gn(XQ) (these agree if X is finite [Lan75]). Theorem 1.2 ([FH82 , Th.III]). LetPX be a simply connected space of finite rati* *onal category. Then G2n(XQ) = 0 and ndim G2n+1(XQ) cat0(X). In this paper, our main results are presented in Section 3. There, we give a * *gen- eralization to cyclic maps of the F'elix-Halperin result on even dimensional Go* *ttlieb groups. Corollary 3.4 shows that cyclic maps of rational spaces are trivial und* *er certain conditions. That result includes the part of Theorem 1.2 about trivial * *even dimensional Gottlieb groups of a rational space as a special case. We also generalize Gottlieb's result about odd dimensional Gottlieb groups, as follows: To say that ff 2 ßn(X) is in the kernel of the rational Hurewicz homom* *or- phism is simply to say that H*(ff) = 0: H*(Sn; Q) ! H*(X; Q). More generally, write SH*(X; Q) for the rational spherical homology of X: that is, SH*(X; Q) is* * the image of the rationalized Hurewicz homomorphism hQ : ß*(X) Q ! H*(X; Q). Note that a map f : A ! X induces a map SH*(f) : SH*(A; Q) ! SH*(X; Q). We give a vanishing result for the map induced on rational spherical homology by a cyclic map in Corollary 3.10, which extends Gottlieb's odd dimensional result. Also in Section 3, we show results about the homomorphism induced on rational homotopy groups by a cyclic map. One such result is Corollary 3.7, which gives a bound on the size of the image that a cyclic map may induce on odd dimensional rational homotopy groups. It complements the F'elix-Halperin result on odd dime* *n- sional Gottlieb groups. Many of the results in Section 3 are actually proved fo* *r a more general class of maps than cyclic maps, namely, maps that factor through an H-space. In Section 4 we give a number of examples of cyclic maps between rational spa* *ces to illustrate our results in Section 3. We also give some sample computations * *of the rationalization of the set of cyclic maps. All spaces in the sequel are assumed to be simply connected CW complexes with rational homology of finite type_with the exception of mapping spaces such as map(X, X; 1X ). We denote the set of homotopy classes of maps from X to Y by [X, Y ]. We often do not distinguish between a map and the homotopy class it re* *pre- sents. If f :A ! B is a map, then f* denotes pre-composition by f and f* denotes post-composition by f. Thus we obtain maps of homotopy sets f* :[B, X] ! [A, X] and f*: [X, A] ! [X, B]. We use H*(f) and H*(f) to denote the map induced on homology, respectively cohomology, by the map of spaces f, and f# to denote the map induced on homotopy groups. We denote the rationalization of a space X by XQ and of a map f by fQ (cf. [HMR75 ]). We assume some familiarity with rational homotopy theory as introduced by Sullivan. Our main reference for this material is [FHT01 ]. The basic facts tha* *t we use are as follows: Each space X has a unique Sullivan minimal model (MX , dX ) in the category of simply connected DG (differential graded) algebras over Q. T* *his DG algebra (MX , dX ) is of the form MX = V , a free graded commutative algebra generated by a positively graded vector space V of finite type. The differentia* *l dX is decomposable, in that dX (V ) 2V . A map f :X ! Y has a Sullivan minimal CYCLIC MAPS IN RATIONAL HOMOTOPY THEORY 3 model which is a DG algebra map Mf: MY ! MX . The Sullivan minimal model is a complete rational homotopy invariant for a space or a map. Since the minim* *al model is determined by the rational homotopy type, the minimal models of XQ and X, and more generally those of fQ and f, agree. There is a notion of homotopy for maps of Sullivan minimal models, which we refer to as DG homotopy (of maps of Sullivan minimal models). If f, g :X ! Y are maps of rational spaces, then f and g are homotopic if and only if their Sullivan minimal models Mf and Mg are DG homotopic. Rational cohomology is readily retrieved from Sullivan min- imal models: We have a natural isomorphism H(MX , dX ) ~=H*(X; Q) and this isomorphism identifies H(Mf): H(MY ) ! H(MX ) with H*(f): H*(Y ; Q) ! H*(X; Q). Rational homotopy is retrieved as follows: Let Q(MX ) ~=V be the (quotient) module of indecomposables of MX . There is a natural isomorphism Q(MX ) ~=Hom (ß*(X) Q, Q), that identifies Q(Mf): Q(MY ) ! Q(MX ) with (f# Q)*: Hom (ß*(Y ) Q, Q) ! Hom (ß*(X) Q, Q). For facts that we have used about minimal models beyond these basics, we have given specific references from [FHT01 ]. We use the standard notation and terminology for minimal models as used in [FHT01 ]. A map of DG algebras is cal* *led a quasi-isomorphism if it induces an isomorphism on cohomology. We mention that there is a discussion of Gottlieb groups, including a proof of Theorem 1.2 cited above, given in [FHT01 , sec.28(d)]. 2.Generalities on Cyclic Maps In this section, we record some basic facts about the set G(A, X). All of the* *se results are either known as stated or occur as easy extensions of known results* * on cyclic maps or evaluation subgroups. The first is a useful result from [Var69]. Theorem 2.1 ([Var69, Th.1.3]). Let ` :B ! A be any map. Then `*: [A, X] ! [B, X] restricts to a map of sets of cyclic maps `*: G(A, X) ! G(B, X). Proof.Suppose that F :A x X ! X extends (g | 1). Then G: B x X ! X defined by G(b, x) = F (`(b), x) extends (g O ` | 1). We obtain the following two immediate consequences of this. Corollary 2.2. A cyclic map g 2 G(A, X) satisfies g](ßn(A)) Gn(X). Proof.Take ` :Sn ! A. Then g# (`) = g O ` = `*(g): Sn ! X is in Gn(X). Corollary 2.3. G(X, X) contains a self-equivalence of X if and only if X is an H-space. Proof.Let g :X ! X be a cyclic self-equivalence and h: X ! X its homotopy inverse. Then, by Theorem 2.1, 1X ' gOh is cyclic as well. The map F :XxX ! X that extends 1X _ 1X is the needed multiplication. Next we give some results, and establish our notation, concerning localization (see [HMR75 ] for full details). For any set of primes P , the P -localizatio* *n map eX :X ! XP induces a map of homotopy sets (eX )*: [A, X] ! [A, XP ]. Likewise, we have an induced map (eA )*: [AP , XP ] ! [A, XP ], which is a bijection of s* *ets. Lemma 2.4. Let A and X be simply connected and let P be any set of primes. 4 GREGORY LUPTON AND SAMUEL BRUCE SMITH (1) (eX )* restricts to a map (eX )*: G(A, X) ! G(A, XP ). (2) The bijection (eA )* restricts to a bijection (eA )*: G(AP , XP ) ! G(A,* * XP ). Proof.Both parts follow easily from standard properties of localization. Suppose that f :A ! X is a cyclic map with affiliated map F :A x X ! X. Then an affiliated map for (eX )*(f) is given by FP O (eA x 1): A x XP ! XP . This shows (1). For (2), suppose ff: AP ! XP is a cyclic map. Then (eA )*(ff) 2 G(A, XP ) by Theorem 2.1. Since the restriction of an injection is injective, it only rem* *ains to show that (eA )* restricts to a surjection. For this, suppose fi 2 G(A, XP )* * has affiliated map B :A x XP ! XP . Then fi = (eA )*(fiP ), and BP :AP x XP ! XP is an affiliated map for fiP . The latter assertion uses uniqueness of localize* *d maps, and the fact that both eA _1: A_XP ! AP _XP and eA x1: AxXP ! AP xXP are P -localization maps. Because we use minimal model techniques, we obtain results about homotopy classes of cyclic maps G(AQ, XQ), or more generally about homotopy classes of maps [AQ, XQ]. Definition 2.5. Let f :A ! X be a map of simply connected spaces. We say f is a rationally cyclic map if its rationalization fQ :AQ ! XQ is a cyclic map. * *More generally, we say f is a P -local cyclic map if fP :AP ! XP is cyclic. We denote the set of homotopy classes of P -local cyclic maps from A to X by GP (A, X). F* *rom Lemma 2.4, we have G(A, X) GP (A, X) for any set of primes P . In general the inclusion (eX )* G(A, X) G(A, XP ) may be strict: See [FHT0* *1 , pp.378-380] for an (infinite) example with (eX )* Gn(X) ( Gn(XQ), but note that (eX )* Gn(X) = Gn(XP ) if X is finite, by [Lan75]. We do have the following re* *sult, however. Lemma 2.6. Let A and X be simply connected with A a finite CW complex. Then P -localization induces a finite-to-one map * -1 (eA ) O (eX )*: GP (A, X) ! G(AP , XP ). In particular, if A is finite and G(AQ, XQ) is trivial, that is, consists of a * *single element, then the set of rationally cyclic maps GQ(A, X) is finite. Proof.The finite-to-one assertion follows from [HMR75 , Cor.5.4]. 3.Maps that Factor Through an H-space In this section we show that being cyclic entails strong restrictions on a ma* *p. Most of our results flow from the following observation. Suppose that f :A ! X is a cyclic map. As we remarked in the introduction, f admits a lift through the evaluation map as follows map(X, X; 1) ~fsss99 ssss |!| ssss fflffl| A _____f___//_X. Now the identity component map (X, X; 1) is an H-space, and thus a cyclic map factors through an H-space. Indeed, since we are assuming A is simply connected, CYCLIC MAPS IN RATIONAL HOMOTOPY THEORY 5 we can lift ~fthrough the universal cover of map (X, X; 1), which is again an H- space, and is also simply connected. Thus a cyclic map factors through a simply connected H-space. Such a factorization entails numerous consequences for a cyc* *lic map. In fact, some of our results can be proved simply by assuming that the map f :A ! X factors through some H-space_not necessarily the universal cover of map(X, X; 1). This hypothesis on a map is definitely weaker than assuming the map to be cyclic. For instance, consider any map f :S3 ! S3 _ S3 that is not trivial on homotopy. Since G3(S3 _ S3) = 0, such a map is not a cyclic map. Sin* *ce S3 itself is an H-space, any such map certainly factors through an H-space. Our primary interest is in drawing conclusions about cyclic maps. However, si* *nce our methods are those of rational homotopy, we need only require that the map be a rationally cyclic map as defined in Definition 2.5. Likewise, although a c* *yclic map actually factors through an H-space, we only require such a factorization a* *fter rationalization. In what follows, we will state our results so as to place as * *weak a hypothesis as possible on the maps and spaces. We remind the reader that our spaces are assumed to be simply connected with rational homology of finite type. Our first consequence of these observations is as follows. Proposition 3.1. Let f :A ! X be a map whose rationalization factors through an H-space. Up to DG homotopy, the Sullivan minimal model Mf: MX ! MA of f has image contained in the cycles of MA . Proof.Suppose we have a factorization (1) =Y= ~f--- --- g|| -- fflffl| AQ __fQ_//XQ with Y an H-space. Passing to minimal models, we have Mf ~ Mf~O Mg: MX ! MA . Now the minimal model of an H-space has trivial differential, that is, dY * *= 0 [FHT01 , p.143]. For every element Ø 2 MX , therefore, we have that Mg(Ø), and hence Mf~O Mg(Ø), is a cycle. Proposition 3.1 gives a serendipitious justification of the terminology "cycl* *ic map,ä t least in rational homotopy (cf. the remarks on nomenclature in [Var69]* *). With additional hypotheses on either A or X, we can draw much stronger conclu- sions. Theorem 3.2. Let f :A ! X be a map whose rationalization factors through an H-space. If X is of finite rational category and Hodd(A; Q) = 0, then f is rati* *onally trivial. Proof.We prove the result by showing that the minimal model Mf: MX ! MA of f is DG homotopic to the trivial map. See [FHT01 , Sec.12(b)] for details ab* *out DG homotopy. Suppose we have a factorization as in (1), with Y an H-space. First we show t* *hat, up to DG homotopy, we can assume that Mf~is zero on all odd-degree generators of MY . For suppose that MY = ({ai}i2I, {bj}j2J), with each ai an even-degree generator and each bj an odd-degree generator. Recall that the differential dY * *is zero, and so every element in MY is a cycle. Since ~f(bi) is an odd-degree cyc* *le 6 GREGORY LUPTON AND SAMUEL BRUCE SMITH in MA , and since we are assuming that Hodd(A; Q) = 0, we have ~f(bj) = dA (jj) for some jj 2 MA . Now define a map : MY ! MA (t, dt) on generators by setting (bj) = ~f(bj) (1 - t) + jj dt, (ai) = ~f(ai) 1. As the differen* *tial in MY is trivial, this map extends to a DG homotopy that starts at ~fand ends at a map that is zero on each odd-degree generator bj. So now assume that Mf~is zero on odd-degree generators. Let I be the ideal of MY generated by the odd-degree generators. We now show that Mg has image contained in I. Clearly, for any odd-degree generator x of MX , we must have Mg(x) 2 I. To show that Mg(x) 2 I for x an even-degree generator of MX , we argue inductively over degree. Assume that Mg(y) 2 I, for * *all generators y of degree 2n-1, and that x is of degree 2n. Since X has finite r* *ational category, there exists an element j 2 MX such that dX (j) = xk+R, for some k 2 and some R in the ideal of MX generated by generators of degree 2n - 1. We justify this assertion in Lemma 3.3 below. Now write Mg(x) = P + Q, with Q 2 I and P an element of the subalgebra of MY generated by the even-degree generator* *s. Since the differential in MY is trivial, we have Mg dX (j) = dY Mg(j) = 0. k Hence, 0 = Mg(xk+R) = Mg(x) +Mg(R) = (P +Q)k+Mg(R). Since Q 2 I, and by our inductive assumption Mg(R) 2 I, it follows that P k2 I and hence that P = 0. This completes the inductive step. We can start the induction with n = 1, where the induction hypothesis is satisfied trivially. We have shown th* *at Mg has image contained in I. Since ~fis zero on odd-degree generators, we have Mf~O Mg = 0. The technical fact about minimal models used in the preceding proof is well- known amongst experts in the field. We give a statement and proof of this useful fact for the sake of completeness. Lemma 3.3. Let X be a space of finite rational category. If x 2 MX is a generat* *or of even degree 2n, then for some k 2 there is an element j 2 MX such that dX (j) = xk + R, with R in the ideal of MX generated by generators of degree 2n - 1. Proof.First notice that if the even-degree generator is a cycle, dX (x) = 0, th* *en the result is easily proved. For the assumption of finite rational category imp* *lies, in particular, that X has finite rational cup length. Hence in this case, ther* *e is some j such that dX (j) = xk with k 2. For the general case, in which x is not a cycle, we use the mapping theorem of F'elix-Halperin [FHT01 , Th.29.5]. Suppo* *se x is of degree 2n, and let MX <2n> denote the quotient DG algebra obtained by factoring_out the DG ideal of MX generated by all generators of degree 2n - 1* *._ Let dX denote the differential induced on the quotient by dX . Then (MX <2n>, d* *X) is a minimal DG algebra and the projection MX ! MX <2n> induces a surjection of the modules_of indecomposables. It follows from the mapping theorem that (MX <2n>, dX) also has finite rational category. By the_argument_at the start * *of this proof, there is some element j 2 MX <2n> such that dX(j) = xk in MX <2n>. Therefore, dX (j) = xk + R for some R in the ideal of MX generated by generators of degree 2n - 1, as asserted. Corollary 3.4 (to Theorem 3.2). If cat0(X) < 1 and Hodd(A; Q) = 0, then G(AQ, XQ) is trivial. If, further, A is a finite CW complex, then GQ(A, X), and hence G(A, X), is a finite set. CYCLIC MAPS IN RATIONAL HOMOTOPY THEORY 7 Proof.As we observed above, any cyclic map_including those in G(AQ, XQ)_ factors through an H-space. Now apply Theorem 3.2 and Lemma 2.6. Corollary 3.4 is a most satisfactory generalization to cyclic maps of the F'e* *lix- Halperin result about even dimensional rational Gottlieb groups. Unfortunately, we have not found such a satisfactory generalization of their odd dimensional r* *esult. Our next three results take a step in this direction, however, giving restricti* *ons on the homomorphism induced on homotopy groups by a cyclic map. Proposition 3.5 (cf. [FHT01 , Cor. on p.379]). Let f :A ! X be a rationally cyclic map, with X a space of finite rational category. Then f# Q is zero in * *all even-degrees and rank(f# Q) cat0(X). Proof.By assumption, fQ :AQ ! XQ is a cyclic map. Hence (fQ)# :ßn(AQ) ! ßn(XQ) has image in Gn(AQ) by Lemma 2.2. Since (fQ)# is identified with f# Q, both assertions now follow by [FH82 , Th.III]. We also have the following result that extends the first assertion of Proposi- tion 3.5, and gives a further restriction on the rank of the homomorphism induc* *ed on rational homotopy groups by a cyclic map. Recall that we defined the rational spherical homology of X to be SHn(X; Q) = hQ(ßn(X) Q) for each n, where hQ :ß*(X) Q ! H*(X; Q) is the rational- ized Hurewicz homomorphism. The vector space dual of hQ gives a homomor- phism (hQ)*: H*(X; Q) ! Hom (ß*(X) Q, Q). The subspace of H*(X; Q) dual to SH*(X; Q) is referred to as the rational spherical cohomology of X, and is deno* *ted by SH*(X; Q). Notice that (hQ)* restricts to an injection from SHn(X; Q) into Hom (ßn(X) Q, Q). Now suppose that (MX , dX ) is the minimal model of X. Since the differential dX is decomposable, it induces the trivial differential * *on the module of indecomposables Q(MX ). Therefore, by passing to cohomology from the quotient projection MX ! Q(MX ), we obtain a map iX :H(MX ) ! Q(MX ). Under the natural identifications of H(MX ) with H*(X; Q) and Q(MX ) with Hom (ß*(X) Q, Q), the map iX is naturally identified with the dual of the rat* *io- nalized Hurewicz homomorphism (hQ)* [FHT01 , p.173]. Theorem 3.6. Let f :A ! X be a map whose rationalization factors through an H-space. (1) f# Q is zero on kernelhQ :ßn(A) Q ! Hn(A; Q) , for each n. (2) If X is a space of finite rational category, then f# Q is zero in all * *even- degrees. Proof.Recall that f# Q is identified with the map induced by Mf: MX ! MA on the (quotient) modules of indecomposables. We observed in Proposition 3.1 that the minimal model Mf: MX ! MA has image in the cycles of MA . This means, in particular, that any indecomposable terms occurring in the image of Mf must be indecomposable cycles in MA . Now the vector space of indecomposable cycles in MA is isomorphic to the rational spherical cohomology of A. Assertion (1) follows. Now suppose that cat0(X) < 1 and that, as in the proof of Theorem 3.2, we have a factorization Mf ~ Mf~O Mg: MX ! MA , for Mg: MX ! MY with Y an H-space. In that proof, we showed that Mg has image contained in I, the ideal of MY generated by the odd-degree generators (we did not use any hypothesis on 8 GREGORY LUPTON AND SAMUEL BRUCE SMITH A for that part of the argument). In particular, if x is an even-degree generat* *or of MX , then Mg(x) is decomposable in MY . It follows that g, and hence f, induces zero on all even-degree rational homotopy groups. This shows assertion (2). Thus we can add the following to Proposition 3.5: Corollary 3.7. Let f :A ! X be a rationallyPcyclic map with X a space of finite rational category. Then rank(f# Q) n odddimSHn(A; Q) . Finally, we turn to our generalization of Gottlieb's odd-degree result. We fi* *rst observe a restriction on the homomorphism induced on rational cohomology by maps which factor through an H-space. Theorem 3.8. Suppose X has finite rational category. If f :A ! X is a map whose rationalization factors through an H-space, then H*(f) Heven(X; Q) H+ (A; Q). H+ (A; Q). Proof.We show the minimal model counterpart of the assertion. It follows direct* *ly from Proposition 3.1 that the image of any cohomology class of H*(MX ) that is represented by a decomposable cycle in MX is contained in H+ (MA ).H+ (MA ). To handle the case in which a cohomology class is represented by an indecomposable cycle, we use a fact established in the proof of Theorem 3.2. As in that proof, suppose we have a factorization Mf ~ Mf~OMg: MX ! MA , where Mg: MX ! MY for Y an H-space. Let I denote the ideal of MY generated by the odd-degree generators. Then we showed in Theorem 3.2 that Mg has image contained in I. From this it follows that the image under Mg of any even degree generator is decomposable in the cycles of MA . Consequently, we have H(Mf) Heven(MX ) H+ (MA ) . H+ (MA ). We do not obtain a restriction comparable to that of Theorem 3.8 for odd-degr* *ee cohomology. This is because any cohomology class of H2n+1(A; Q) is in the image of a homomorphism induced by a map f :A ! K(Q, 2n + 1), and such a map is a cyclic map. In the following result, we do not assume that X has finite rational category. Recall that SH*(f) : SH*(A, Q) ! SH*(X, Q) denotes the map induced on rational spherical homology by f : A ! X. Theorem 3.9. Let f :A ! X be a rationally cyclic map. If SH*(f) 6= 0 then X decomposes up to rational homotopy type as X 'Q X0x K(Q, n) for some simply connected space X0. Proof.The hypothesis SH*(f) 6= 0 translates into the following: There is some ff 2 ßn(A) Q whose image under the composition H*(f) O hQ :ßn(A) Q ! Hn(X; Q) is non-zero. Therefore, f# (ff) 2 ßn(X) Q gives a non-zero spherical element hQ f# (ff) 2 SHn(X; Q). But since f is rationally cyclic, we have f# (* *ff) 2 Gn(X) Q, with hQ f# (ff) 6= 0. By a theorem of Oprea (see [Hal88, Lem.1.1]), this implies the splitting X 'Q X0x S2n+1. Note that Theorem 3.9 can also be interpreted as a restriction on the homomor- phism induced on rational cohomology by a cyclic map: A cyclic map into a space not decomposable as above must have zero image in rational spherical cohomology. We are unsure whether or not the homomorphism induced on rational cohomology by a cyclic map may contain indecomposable terms in its image without incurring the splitting of Theorem 3.9. CYCLIC MAPS IN RATIONAL HOMOTOPY THEORY 9 Corollary 3.10. Suppose X has finite dimensional rational homology and that f :A ! X is a rationally cyclic map. If Ø(X) 6= 0, then SH*(f) = 0. Proof.From Theorem 3.8, the image of H*(f) in even degrees is decomposable, and hence non-spherical. Thus SH*(f) is zero in even degrees. Since Ø(X) 6= 0, the splitting of Theorem 3.9 cannot occur with n odd_recall that K(Q, 2n+1) 'Q S2n+1. Therefore, by that result, SH*(f) is zero in odd-degrees, as well. In particular, we see that under the hypotheses of Corollary 3.10, the homomo* *r- phism induced by f on rational homology is zero if all homology of H*(A; Q) is spherical. Thus, the corollary, or better, Theorem 3.9, gives a satisfactory ge* *neral- ization to cyclic maps of Gottlieb's result concerning odd-degree Gottlieb grou* *ps. From the above results, we can also extend the main result of [Lim82 ], at le* *ast in the simply connected case. In the original, very strong hypotheses are place* *d on X, namely that both H*(X) and ß*(X) are finitely generated. If, in addition, we have Ø(X) 6= 0 and A a co-H-space, then it is shown that a cyclic map f :A ! X obtains f an element of finite order in [ A, X]. Recall that a rational co-H-space is a space A such that AQ is a co-H-space. Recall that a (rationally) elliptic space is one whose rational homology and ra* *tional homotopy are both finite dimensional. A result of Halperin [Hal77] states that * *the Euler characteristic of a rationally elliptic space is non-negative. Further, t* *hat in the case of positive Euler characteristic, that is, in the non-zero case, the r* *ational cohomology is zero in odd degrees. For brevity, we refer to an elliptic space w* *ith positive Euler characteristic as an F0-space. There are many interesting exampl* *es of such spaces, including even dimensional spheres, complex projective spaces, * *and äm ximal rank pair" homogeneous spaces G=H. Products of F0-spaces, and more generally many total spaces of fibrations in which base and fibre are F0-spaces* *, are again F0-spaces. We can relax Lim's hypotheses as follows: Corollary 3.11 (cf. [Lim82 , Th.5.2]). Let A be a rational co-H-space with fini* *te dimensional rational homology and let X be an F0-space. Then any rationally cyclic map f :A ! X has rationally trivial suspension, f 'Q *: A ! X. If A is assumed finite dimensional, then the subgroup of cyclic maps G( A, X) [ A, X] is a finite group. Proof.Since cup products in eH*(A; Q) are trivial and H*(X, Q) is evenly graded we can apply Theorem 3.8 to obtain H*(f) = 0: eH*(X; Q) ! eH*(A; Q). It follows that H*( f) = 0: eH*( A; Q) ! He*( X; Q), and hence that f 'Q *. The assertion about finiteness follows from Lemma 2.6. 4.Examples We give several examples of rationally cyclic maps. Our first examples illust* *rate that most results from the previous section are sharp. We then give some comple* *te calculations of the set G(AQ, XQ). Since our results tend towards restricting the possibilities for rationally c* *yclic maps, we begin with an example that indicates rich possibilities for such remai* *n. Of course, since any map into an odd-dimensional sphere and any rational Gottli* *eb element are rationally cyclic maps, we already have many examples. 10 GREGORY LUPTON AND SAMUEL BRUCE SMITH Example 4.1. Consider the map f :S3 x S4 ! S4, obtained by composing the quotient map q :S3 x S4 ! S7 with the Hopf map j :S7 ! S4. It is well-known that f is not rationally trivial, although it induces the trivial homomorphism * *on both rational homotopy and rational cohomology. Since S7 is an H-space, f itsel* *f, and not just its rationalization, factors through an H-space. Actually, the rat* *ional- ization of j is a rational Gottlieb element in G7(S4) Q ~=ß7(S4) Q, and so * *f is a rationally cyclic map by Theorem 2.1. This shows that we need Hodd(A; Q) = 0 in Theorem 3.2, even though the map may be trivial on cohomology. This example suggests many others of the form A ! Sn ! X, with the second map a rational Gottlieb element. Example 4.2. The image of H*(f) need not be decomposable in odd degrees, as we now illustrate. Take A = S2 _ S2 [ffe5 with the 5-cell attached by the iterated Whitehead product ff = ['1, ['1, '2]] 2 ß4(S2 _ S2). Then all cup prod* *ucts in H*(A; Q) are trivial, with H5(A; Q) ~=Q non-spherical. Now the quotient map q :A ! S5 is a map into a rational H-space, and is therefore a rationally cyclic map. In degree 5, H*(f) has indecomposable, non-spherical image. Example 4.3. Inclusion of the bottom cell f :S2 ! CP 1 is a map into an H- space, and hence is a cyclic map. Evidently, we have Hodd(S2; Q) = 0 and yet f * *is not rationally trivial_it is not trivial on rational cohomology, in fact. This * *shows that we need the hypothesis of finite rational category on X in Theorem 3.2. It also shows that without cat0(X) < 1, then a cyclic map f :A ! X may have f# Q non-zero in even degrees. Indeed, here we have G2(CP 1) = ß2(CP 1). This example illustrates the splitting of Theorem 3.9 for the case n = 2. For some spaces X, including the F0-spaces mentioned above, the rational Got- tlieb groups of X coincide with the odd rational homotopyLgroups of X. In certa* *in cases, this may generalize to a bijection G(AQ, XQ) ~= n oddHn(A; ßn(X) Q) for any space A. We now give some examples. If X is an odd dimensional sphere S2n+1 then XQ ' K(Q, 2n + 1) is an H- space and therefore G(AQ, XQ) = [AQ, XQ] = H2n+1(A; ß2n+1(S2n+1) Q) = H2n+1(A; Q). When X = S2n the same result holds but the calculation is slightly more involved. The following example includes the cases in which X is an even dimensional sphere or a complex projective space. Example 4.4. Suppose X has rational cohomology algebra a truncated polynomial algebra on a single generator of even degree 2n. If H*(X; Q) ~=Q[x]=(xk+1), then G(AQ, XQ) ~=H2n(k+1)-1(A; Q). We argue as follows: First, X has minimal model (x, y) with |x| = 2n, |y| = 2n(k + 1) - 1, dX (x) = 0, and dX (y) = xk+1. Supp* *ose f :A ! X is a rationally cyclic map with minimal model Mf: MX ! MA given on generators by Mf(x) = a and Mf(y) = b, for a, b 2 MA . Now consider an affiliated map F :A x X ! X for f. Then by taking into account the relative degrees of x and y, we can write its minimal model MF :MX ! MA MA on generators as MF (x) = x + a and MF (y) = y + b + b1x + . .+.bkxk, for bi2 MA . Since MF is a DG map, we can equate the terms that occur in dA MF (y) with those that occur in MF dX (y) = (x + a)k+1. Thus we find that dA (bk) = (k + 1)* *a. Now define a DG homotopy : MX ! MA (t, dt) on generators by setting (x) = a(1 - t) + __1__kb+k1dt CYCLIC MAPS IN RATIONAL HOMOTOPY THEORY 11 and (y) = b - __1__kb+k1ak + __1__kb+k1ak(1 - t)k+1. This is easily checked to commute with the differentials, and so defines a DG h* *omo- topy that starts at Mf and ends at a map that is zero on the even-degree genera* *tor x. So far, we have argued that a cyclic map f :A ! X has minimal model given on generators by Mf(x) = 0 and Mf(y) = b, up to homotopy. For a map of this form, b must be a cycle in MA . Furthermore, two such maps are DG homotopic exactly when the cycles that y is sent to by each represent the same cohomology class of H(MA ). This latter assertion is easily justified using the approach of [AL95 ]* *, for instance. Example 4.5. Suppose X = G=T where G isLa compact, connected Lie group and T is a maximal torus. Then G(AQ, XQ) ~= n oddHn(A; ßn(X) Q) for any space A. For in this case the minimal model of X is of the form (x1, . .,.xn, y1, . * *.,.yn) with |xi| = 2, dX (xi) = 0 and the yi of odd degree with dX (yi) a decomposable polynomial in the xi. The rational cohomology of X is evenly graded and the cyc* *les xicorrespond to a space of generators. See [FHT01 , Prop.15.16 and Ex.2,p.448] * *for justification of these assertions. By Theorem 3.8, if f : A ! X is rationally c* *yclic, then H*(f) = 0 since H2(A; Q) cannot contain non-zero decomposable terms. Thus the minimal model Mf: MX ! MA of f satisfies Mf(xi) = 0 and Mf(yi) = ,i for cycles ,i2 MA and the result follows as above. References [AL95] M. Arkowitz and G. Lupton, On finiteness of subgroups of self-homotopy e* *quivalences, Cech Centennial Conference, Contemp. Math., vol. 181, Amer. Math. Soc., * *1995, pp. 1- 25. [FH82] Y. F'elix and S. Halperin, Rational L.-S. category and its applications,* * Trans. Amer. Math. Soc. 273 (1982), no. 1, 1-38. [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. [Got69]D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. * *91 (1969), 729-756. [Hal77]S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. * *Math. Soc. 230 (1977), 173-199. [Hal88]_____, Torsion gaps in the homotopy of finite complexes, Topology 27 (19* *88), 367- 375. [HMR75]P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups * *and spaces, North- Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematics Studi* *es, No. 15, Notas de Matem'atica, No. 55. [Lan75]G. Lang, Localizations and evaluation subgroups, Proc. Amer. Math. Soc. * *50 (1975), 489-494. [Lim82]K. L. Lim, On cyclic maps, J. Austral. Math. Soc. 32 (1982), 349-357. [Opr95]J. Oprea, Gottlieb groups, group actions, fixed points and rational homo* *topy, Lec- ture Notes Series, vol. 29, Seoul National University Research Institute* * of Mathematics Global Analysis Research Center, Seoul, 1995. [Tan83]D. Tanr'e, Homotopie rationnelle: mod`eles de Chen, Quillen, Sullivan, L* *ecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983. [Var69]K. Varadarajan, Generalised Gottlieb groups, J. Indian Math. Soc. 33 (19* *69), 141-164. Department of Mathematics, Cleveland State University, Cleveland OH 44115 E-mail address: G.Lupton@csuohio.edu Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131 E-mail address: smith@sju.edu