EVALUATION MAPS IN RATIONAL HOMOTOPY YVES F'ELIX AND GREGORY LUPTON Abstract.Let E be an H-space acting on a based space X. Then we re- fer to w:E ! X, the map obtained by acting on the base point of X, as a "generalized evaluation map" (see Definition 1.1 for a precise definitio* *n). We establish several fundamental results about the rational homotopy behavi* *our of a generalized evaluation map, all of which apply to the usual evaluat* *ion map Map(X, X; 1) ! X. With mild hypotheses on X, we show that a gen- eralized evaluation map w factors, up to rational homotopy, through a map w :Sw ! X where Sw is a (relatively small) finite product of odd-dimens* *ional spheres and ss# ( w) Q is injective. This result has strong consequenc* *es: if the image in rational homotopy groups of w is trivial, then the generali* *zed eval- uation map is null-homotopic after rationalization; unless X satisfies a* * very strong splitting condition, any generalized evaluation map induces the t* *rivial homomorphism in rational cohomology; the map w is rationally a homotopy monomorphism and a generalized evaluation map may be written as a com- position of a homotopy epimorphism and this homotopy monomorphism. We include illustrative examples and prove numerous subsidiary results of i* *nterest. 1.Introduction Let X be a based space and let Map (X, X) be the space of unbased, or free, m* *aps from X to itself. In general Map (X, X) is disconnected; we denote by Map (X, X* *; 1) its identity component, that is, the path component that consists of self maps that are (freely) homotopic to the identity. Then we have the evaluation map ! :Map (X, X; 1) ! X defined by evaluation at the basepoint of X. This map occupies a central place in the homotopy theory of fibrations (cf. [5, 6, 7, 8]* *). The evaluation map ! and its rationalization will play a distinguished role in this paper. However, we find that our methods and results apply equally well to other contexts in which one has an "evaluation map." For example, it is often of interest to consider the space Top(X, X) of self-homeomorphisms of X and the corresponding evaluation map w: Top(X, X; 1) ! X. Here, Top(X, X; 1) denotes the component of Top(X, X) that consists of self-homeomorphisms homotopic (via self-homeomorphisms) to the identity. Likewise, if X is a smooth manifold, then one may replace Top(X, X) with Diff(X, X), and so-forth. A further example of an "evaluation map" to which our methods apply concerns configuration spaces. Let F (X, k) denote the configuration space that consists of ordered k-tuples of di* *stinct points in a space X, and let (p1, . .,.pk) be a choice of basepoint in F (X, k)* *. Then we have a map ` :Top(X, X; 1) ! F (X, k) ____________ Date: September 28, 2005. 2000 Mathematics Subject Classification. 55P62, 55Q05. Key words and phrases. Evaluation map, H-space, action of an H-space, Gottli* *eb group, rational homotopy, rational cohomology, homotopy monomorphism, minimal models. 1 2 YVES F'ELIX AND GREGORY LUPTON given by `(ff) = (ff(p1), . .,.ff(pk)). Actually, here we have ` = w O , where : Top(X, X; 1) ! Top F (X, k), F (X, k); 1 is the natural injection defined by (ff)(q1, . .,.qk) = (ff(q1), . .,.ff(qk)),* * and w: Top F (X, k), F (X, k); 1 ! F (X, k) is an evaluation map for F (X, k) in the preceding sense, with "Top " replacing "Map ." Motivated by the preceding examples, we now make a formal definition of the evaluation maps that we consider. Recall that an H-space is a pair (E, ~) with E a based space and multiplication ~: E x E ! E a based map that satisfies ~ O J ~ r: E _ E ! E. Here, r: E _ E ! E denotes the folding map and J :E _ E ! E x E the obvious inclusion. We say that the multiplication has stri* *ct identity if ~ O J = r (equals, not just homotopic). Note that Map (X, X; 1) is * *an H-space with strict identity. Now let i1: E ! E x X and i2: X ! E x X denote the inclusions. By an action of E on X we mean a map A: E x X ! X that satisfies A O i2 = 1: X ! X. We say that the action is associative if in additi* *on we have A O (~ x 1) = A O (1 x A). Definition 1.1. A generalized evaluation map is any (based) map w: E ! X, from a connected H-space with strict identity E to a space X, for which there exists an associative action A: E xX ! X that restricts to w, that is, that sat* *isfies A O i1 = w: E ! X. Examples 1.2. (1) The action A: Map(X, X; 1)xX ! X given by A(f, x) = f(x) makes ! :Map (X, X; 1) ! X a generalized evaluation map according to Defini- tion 1.1. Similarly for all the other examples mentioned above. (2) Suppose G is a connected topological group and A: GxX ! X is a group ac- tion in the usual sense. Then the orbit map of the action is a generalized eval* *uation map G ! X. (3) More generally, suppose given a fibration X ! Y ! B. Then the connecting map @ : B ! X is a generalized evaluation map. This follows from the usual action of B on the fibre X. Note, however, that we must take Moore loops in X to obtain an H-space with strict identity. Revert now to the ordinary evaluation map ! :Map (X, X; 1) ! X. For the remainder of the paper, we assume that X is a nilpotent, finite complex. Since * *X is finite, a result of Milnor [17] implies that Map (X, X; 1) is a CW complex. Sin* *ce X is nilpotent, we may choose and fix a rationalization e: X ! XQ. Now results of [1* *2] imply that e*: Map(X, X; 1) ! Map (X, XQ; e) is a rationalization. Thus, the map !Q :Map (X, XQ; e) ! XQ, also defined by evaluation at the basepoint of X, may be taken to be the rationalization of !. We refer to !Q as the rationalized eva* *luation map. Recall that the nth Gottlieb group of X, denoted Gn(X), is the subgroup of ssn(X) defined as the image of !# :ssn Map (X, X; 1) ! ssn(X) [7]. The subgroup of ssn(XQ) defined as the image of (!Q)# :ssn Map (X, XQ; e) ! ssn(XQ) is call* *ed the nth rationalized Gottlieb group of X and denoted by Gn(XQ). By a theorem of Lang [13], we have Gn(XQ) ~=Gn(X) Q under our assumption that X is finite. The rationalized Gottlieb groups have played an important role in some of the major developments of rational homotopy theory (cf. [1, 11]). Our results in th* *is paper show that the rationalized Gottlieb groups exercise a very strong determi* *ning effect on the rationalized evaluation map. EVALUATION MAPS IN RATIONAL HOMOTOPY 3 A result of F'elix-Halperin ([1, Th.III]) implies that G2i(XQ) = 0 for all i * *and G2i+1(XQ) is non-zero for only finitely many i. Suppose {ff1, ff2, . .,.ffr} is* * a basis of G*(XQ) = Godd(XQ) with ffi2 Gni(XQ) (here we regard an element of ssn(XQ) as represented by a map ff: SnQ! XQ). For each ffi, we may choose a fii 2 ssni Map (X, XQ; e) such that !Q Ofii= ffi. The adjoint of fiigives a map Fi:S* *niQx X ! XQ that extends the map (ffi| e): SniQ_X ! XQ. Denote by SX the product of odd-dimensional rational spheres Sn1Qx . .x.SnrQwhose factors correspond to the domains of the basis of G*(XQ). Then we form a map F :SX x X ! XQ as the composition F = F1 O (1 x F2) O . .O.(1 x . .x.1 x Fr). Now set X = F O i: SX ! XQ, where i denotes the inclusion of the product of spheres as the first r factors. We refer to X as a total Gottlieb element of X* *Q. By taking the adjoint of F , we obtain a lift e X:SX ! Map (X, XQ; e) of X through the rationalized evaluation map !Q. We prove the following result: Theorem 1.3. Let X be any nilpotent, finite complex. The rationalized evaluation map !Q :Map (X, XQ; e) ! XQ factors up to homotopy through the total Gottlieb element X :SX ! XQ. In fact, there is a retraction r of e X, that is, a map r :Map(X, XQ; e) ! SX with r O e X= 1, such that !Q = X O r. This basic result has several strong consequences. An immediate one is the fo* *l- lowing striking illustration of the effect that the homomorphism induced on rat* *ional homotopy groups has on the rationalized evaluation map. Corollary 1.4. The evaluation map ! :Map (X, X; 1) ! X is rationally null- homotopic if and only if G*(XQ) = 0. Now the evaluation map ! may be viewed as a "universal connecting map" for fibrations with fibre X, in that any connecting map B ! X of a fibration X ! E ! B factors through ! [6]. A further immediate consequence of Theorem 1.3, therefore, is the following result. Corollary 1.5. Let X ! E ! B be any fibration with fibre X a nilpotent, finite space. If G*(XQ) = 0, then the connecting map @ : B ! X is rationally null- homotopic. There are many spaces to which these corollaries may be applied. For instance, any suspension that is not rationally equivalent to a sphere has trivial ration* *alized Gottlieb groups. Roughly speaking, a typical wedge or connected sum of spaces h* *as trivial Gottlieb groups, as do many non-elliptic, coformal spaces. More precise* *ly, a space whose rational homotopy Lie algebra has trivial centre has trivial ration* *alized Gottlieb group. Therefore, by Corollary 1.4, the rationalized evaluation map is* * null- homotopic in all such cases. The preceding discussion of ! and the Gottlieb groups extends naturally to generalized evaluation maps. Suppose given w: E ! X any generalized evaluation map. In Section 2 we construct a map w :Sw ! XQ such that im( w)# Q = im(w)# Q. As with SX above, Sw is a product of a relatively small number of odd-dimensional rational spheres. We refer to w as a total Gottlieb element of* * XQ with respect to w. Furthermore, w admits a lift through wQ, the rationalization 4 YVES F'ELIX AND GREGORY LUPTON of w. That is, there exists a map e w:Sw ! EQ that satisfies wQ O e w= w. Then we have generalizations of Theorem 1.3 and Corollary 1.4 as follows. Theorem 1.6. Let w: E ! X be any generalized evaluation map with X a nilpo- tent, finite complex. Suppose that w :Sw ! XQ is a total Gottlieb element of XQ with respect to w. Then wQ factors up to homotopy through w. More precisely, suppose that e w:Sw ! EQ is a lift of w through wQ. Then there is a retraction r :EQ ! Sw of e wsuch that wQ = w O r. Corollary 1.7. Let w: E ! X be any generalized evaluation map. Then w# Q = 0: ss*(EQ) ! ss*(XQ) if and only if w: E ! X is rationally null-homotopic. We continue with a theorem related to the homotopy behaviour of the maps w :Sw ! XQ. Recall that a map f :X ! Y is a homotopy monomorphism if, for any A, the induced map of homotopy sets f*: [A, X] ! [A, Y ] is injecti* *ve [3]. In general it is a difficult problem to identify when a map is a homotopy monomorphism. We say that a map of nilpotent spaces f :X ! Y is a homotopy monomorphism in the nilpotent category if f*: [A, X] ! [A, Y ] is injective whe* *never A is a nilpotent space. Theorem 1.8. Let X be a nilpotent, finite complex and w: E ! X be any gener- alized evaluation map. Then w :Sw ! XQ is a homotopy monomorphism in the nilpotent category. Theorem 1.8 is proved towards the end of Section 3. As a consequence, together with Theorem 1.6 we find that, after rationalization, a generalized evaluation * *map may be written as a composition wQ = w O r of a homotopy epimorphism and a homotopy monomorphism in the nilpotent category (Corollary 3.6). We also note the following immediate consequence of Theorem 1.8. Corollary 1.9. Let X be a nilpotent, finite complex and let ff: Sn ! XQ be any * *ra- tionalized Gottlieb element. Then ff is a homotopy monomorphism in the nilpotent category. In particular, this implies that the rationalized Hopf maps are homotopy mono* *mor- phisms in the nilpotent category. By contrast, the Hopf map j :S7 ! S4 is not a homotopy monomorphism [3]. A further consequence of Theorem 1.8 is the classification up to rational hom* *o- topy of cyclic maps. A map f :A ! X is called cyclic if (f | 1): A _ X ! X extends to a map A x X ! X [20]. Denote by G(A, X) the set of homotopy classes of cyclic maps from A into X. This is a generalization of the nth Gottlieb grou* *p of X, which we obtain by taking A = Sn. Upon rationalizing a cyclic map, we obtain a map fQ :A ! XQ in G(A, XQ). Theorem 1.10. Let X be a nilpotent, finite complex and let A be any nilpotent space. Then there is a bijection of sets G(A, XQ) ~=[A, SX ] ~= rHom (Hr(A; Q), Gr(XQ)) . This classification allows us, for instance, to easily identify situations in w* *hich G(A, XQ) is trivial and, hence, G(A, X) is finite. In Theorem 3.8, we extend th* *is result to apply to any generalized evaluation map. Our last topic is the (co)homological behaviour of generalized evaluation map* *s. For the ordinary evaluation map ! :Map (X, X; 1) ! X, this behaviour has been studied by Gottlieb [9] and Oprea [18, 19]. From [18] we have the following res* *ult: EVALUATION MAPS IN RATIONAL HOMOTOPY 5 Theorem 1.11 (Oprea). Let F ! E ! B be a fibration with connecting map @ : B ! F . Suppose that B is 1-connected and that B and F have finite type rational homology. Then there is a splitting, up to rational homotopy, F 'Q S x* * Y with S a product of Eilenberg-Mac Lanespaces and dimss*(S) Q = dim Image(hF O @# :ss*( B) Q ! H*(F ; Q)) . Here, hF :ss*(F ) Q ! H*(F ; Q) denotes the rational Hurewicz homomorphism. Oprea's result may be applied to the evaluation map ! by considering it as the connecting map in the universal fibration for fibrations with fibre X. Our main result about the homological behaviour of a generalized evaluation map is the following composite theorem, which gives a complete description for rational coefficients. Theorem 1.12. Let w: E ! X be any generalized evaluation map with X a nilpotent, finite complex. Then we have: (1) He*(w; Q) 6= 0: eH*(E; Q) ! He*(X; Q) if and only if hX O (w# Q) 6= 0: ss*(E) Q ! ss*(X) Q ! H*(X; Q); (2) if hX O(w# Q) has image in H*(X; Q) of dimension r > 0, then H*(w; Q) has image in H*(X; Q) of dimension 2r and there is a rational homotopy equivalence X 'Q S x Y , with S a product of odd-dimensional spheres such that H*(S; Q) ~=ImageH*(w; Q) and ss*(S) Q ~=ImagehX O (w# Q); (3) if X 'Q S2n+1 x Y , then He*(!; Q) 6= 0, where ! :Map (X, X; 1) ! X is the ordinary evaluation map. Our treatment here extends Oprea's theorem to a generalized evaluation map. Theorem 1.12 shows that, in most cases, the rank of H*(w; Q) is relatively smal* *l. We also deduce that H*(w; Q) is surjective only when X is an H0-space. Theorem 1.12 has various interesting corollaries, such as the following sharpening of a resu* *lt of Gottlieb [9, Th.3] for rational coefficients. Corollary 1.13. Suppose that O(X) 6= 0. Then for every generalized evaluation map w: E ! X, we have eH*(w; Q)) = 0: eH*(E; Q) ! eH*(X; Q). A further consequence is the following result: Corollary 1.14. Let M be a simply connected, symplectic manifold. Then every generalized evaluation map w: E ! M is trivial on rational homology, that is, eH*(w; Q) = 0: eH*(E; Q) ! eH*(M; Q). Consequently, if G is a connected Lie gro* *up and a: G ! M is the orbit map of any G-action on M, we have eH*(a; Q) = 0. These corollaries appear as Corollary 4.1 and Corollary 4.2, respectively. The text is divided into five parts. In Section 2 we present the factorizati* *on results. Section 3 contains some technical lemmas on Gottlieb groups, and the monomorphism theorem. The homological behaviour of generalized evaluation maps is discussed in Section 4. Section 5 is a brief, concluding section in whi* *ch we mention several problems that suggest directions for future work. We finish this introduction with some terminology and notation. We work in the homotopy category, and so we often do not distinguish between a map and the homotopy class it represents. We use ' to denote that two spaces are homotopy equivalent, or that a map is a homotopy equivalence. If f :A ! B is a map, then* * f* denotes pre-composition by f and f* denotes post-composition by f. We use H*(f) 6 YVES F'ELIX AND GREGORY LUPTON 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. Likewise, eH*(f) and eH*(f) denote reduced (co)homology. We denote the rationalization of* * a space X by XQ and of a map f by fQ (cf. [12]). By an H0-space, we mean a space whose rationalization is an H-space. We say that maps f, g :X ! Y are rationally homotopic if their rationalizations are homotopic. We denote this relation eith* *er by f =Q g :X ! Y or by fQ = gQ :XQ ! YQ. We reserve ! to denote the evaluation map ! :Map (X, X; 1) ! X. Generalized evaluation maps will be denoted with a generic w. For the remainder of the paper, we will usually drop the "generalize* *d" and refer simply to an evaluation map. We assume familiarity with rational homotopy theory and use the standard no- tation and terminology for minimal models as presented in [2]. The basic facts * *that we use are as follows: Each nilpotent space X has a unique Sullivan minimal mod* *el (MX , dX ) in the category of commutative DG (differential graded) algebras over Q. This DG algebra (MX , dX ) is of the form MX = ^V , a free graded commu- tative algebra generated by a positively graded vector space V of finite type. * *The differential dX is decomposable, in that dX (V ) ^ 2V , and V admits a basis * *{vff} indexed by a well ordered set such that dX (vff) 2 ^({vfi}fi>" """" |pQ| "" fflffl| Sp___p_//XQ in which p is both injective and onto the image of p in rational homotopy grou* *ps. Definition 2.3. Suppose given any map p: E ! X from an H0-space E to a nilpotent, finite space X. A total Gottlieb element for XQ with respect to p i* *s a map p: Sp ! XQ that admits a lift e p:Sp ! EQ through pQ, where (1) Sp is a product of rational Eilenberg-Mac Lanespaces with homotopy iso- morphic to im(pQ)# :ss*(EQ) ! ss*(XQ); and (2) p is injective in (rational) homotopy groups. In general, there may be many choices of total Gottlieb elements with respect* * to p and different lifts of each. By the above discussion, we see that such always* * exist. We keep the notation X :SX ! XQ for a total Gottlieb element with respect to the ordinary evaluation fibration ! :Map (X, X; 1) ! X. Theorem 2.4. Let j p F _____//E____//_X be a fibration sequence of nilpotent spaces in which F and E are H0-spaces and X is a nilpotent, finite space. Let p: Sp ! XQ be any total Gottlieb element f* *or XQ with respect to p and e pany lift of of p through pQ. Assume there is an action A: FQ x EQ ! EQ of FQ on EQ that satisfies A O i1 = jQ and pQ O A = pQ O p2: FQ x EQ ! XQ. Then there is a retraction r :EQ ! Sp of e psuch that pQ = p O r :EQ ! XQ. EVALUATION MAPS IN RATIONAL HOMOTOPY 9 Proof.From Proposition 2.1, we assume an identification EQ ' Y x Z, with Y and Z rational H-spaces, together with maps i: Y ! EQ and OE: Y ! FQ with i# an injection onto im(jQ)# and jQ O OE = i. Now consider the following commutative diagram: Y x Sp[_p2__//Sp[ ___ AO(OExe p)'||H____||p fflffl|__ fflffl| EQ __pQ__//_XQ Observe that A O (OE x e p) O i1 = i: Y ! EQ. Furthermore, from the long exact sequence in homotopy of the fibration, we find that A O (OE x e p) O i2: Sp ! EQ has image in homotopy that is complementary to im (jQ)# . Hence A O (OE x e p) induces an isomorphism in rational homotopy and thus is a homotopy equivalence. Consequently, there is an inverse (rational) homotopy equivalence H :EQ ! Y xSp as indicated in the diagram. Now set r = p2O H :EQ ! Sp. Then we have r O e p= p2O H O A O (OE x e p) O i2 = 1: Sp ! Sp, so that r is a retraction of e p. Fur* *thermore, since p O A(' x e p) = p O p2, we have p O r = p O p2O H = pQ :EQ ! XQ, which gives the desired factorization. We obtain Theorem 1.3 by specializing as follows: Proof of Theorem 1.3.The action A: Map*(X, X; 1) x Map(X, X; 1) ! Map (X, X; 1), defined by A(f, g) = g Of, restricts to the inclusion Map *(X, X; 1) ! Map (X, * *X; 1) and satisfies the hypothesis of Theorem 2.4. Therefore, we may apply the result* * to the evaluation fibration sequence Map *(X, X; 1) ! Map (X, X; 1) !!X and the total Gottlieb element for this evaluation map constructed from the Got* *tlieb groups as in the introduction. By the same argument, we obtain Theorem 1.3 for each of the evaluation fibra- tions in which Top, Diff, and so-forth, replaces Map , as in the introduction. The following observation allows us to strengthen Theorem 1.3 in certain cir- cumstances. We will also use it in Section 4. Roughly speaking, we may say that* * if X decomposes up to homotopy equivalence as a product, then the evaluation map decomposes as a corresponding product of evaluation maps. More precisely, suppose we have a homotopy equivalence h: X ! A x B. Then we have homotopy equivalences h*: Map(X, X; 1) ! Map (X, A x B; h) and h*: Map(A x B, A x B; 1) ! Map (X, A x B; h) . Let p1: A x B ! A and p2: A x B ! B denote the projections, and i1: A ! A x B and i2: B ! A x B the inclusions. We write h = (h1, h2), with h1 = p1 O h and h2 = p2 O h, and define evaluation maps !1: Map(X, A; h1) ! A and !2: Map(X, B; h2) ! B by evaluation at the basepoint of X. Let I :Map (A x 10 YVES F'ELIX AND GREGORY LUPTON B, A x B; 1) ! Map (A x B, A; p1) x Map (A x B, B; p2) be the standard homeo- morphism. One checks easily that h O !X = (!1 x !2) O I O (h*)-1 O h*: Map(X, X; 1) ! A x B, and thus we may identify !X with h-1 O (!1 x !2) O I O h*. Furthermore, we observe that !1 factors through !A as !1 = !A O (i1)* and likewise !2 = !B O (i* *2)*. Therefore, we have the following commutative diagram: IO(h*)-1Oh* (1) Map (X, X; 1)____'_____//Map(A x B, A; p1) x Map(A x B, B; p2) | | |(i1)*x(i2)* | fflffl|| !X || Map(A, A; 1A ) x Map(B, B; 1B ) | | | | |!Ax!B fflffl| ' fflffl| X ______________h______________//_A x B. This discussion leads to the following result, which should be compared with * *the well-known fact that G*(A x B) ~=G*(A) G*(B) [7]. Theorem 2.5. Suppose that we have a homotopy equivalence X ' A x B. Then the evaluation map !X factors through the product of evaluation maps !A x !B . We now continue with the main results. In order to study generalized evaluati* *on maps w: E ! X, we first present a global structure result concerning maps betwe* *en H0-spaces. Proposition 2.6. Let f :X ! Y be a map between H0-spaces. (a) The map f admits a Sullivan minimal model of the form ': (^(V R), 0) ! (^(V S), 0) with '(v) = v for v 2 V and such that '(R) 2 ^ 2(V S) \ ^V ^+ (S). (b) If fQ is an H-map then f admits a model of the form ': (^(V K), 0) ! (^(V S), 0) with '(v) = v for v 2 V and '(K) = 0. Proof.Let ': (^T, 0) ! (^W, 0) be any model of f. We will use standard tricks from rational homotopy to change generators in ^T and ^W so that, with respect to the new generators, the minimal model of f has the desired form. (a) We denote by V a maximal subspace of T such that Q('): V ! W is injective. Denote by R T a complement of V and by S W a complement of imQ(') in W . Let {vi}i2Ibe a graded basis for V . Then the elements '(vi) are linearly independent indecomposable elements in ^W . Denote by {rj}j2J a graded basis for R and {sk}k2K a graded basis for S. With respect to the generators {vi, rj} for ^T and {v0i= '(vi), sk} for ^W , the map ' satisfies '(vi) = v0iand '(R) ^ 2(W ). We can thus suppose '(v) = v and that '(R) is decomposable. We now change generators in R so that '(R) also belongs to the ideal generated * *by S. Suppose that this is true for R j. Other than this, however, we have very little control over how the `iextend. This point is the main source of the technicalities. We denote by V the vector space generated by the vi, and Z a choice of complement in W . Thus the minimal model of X is (^(V Z), dX ) with V = corresponding to the Gottlieb group, accompanying derivations `1, . .,.`r, and Z a complement to V in W . EVALUATION MAPS IN RATIONAL HOMOTOPY 15 Lemma 3.1. With notation as above, we may choose Z, V , and the `i such that `i(Z ) ^V ^+ Z. In particular the ideal I(Z) is `i-stable* * for each i. Proof.Let L denote the Lie algebra of derivations of MX generated by the deriva- tions `1, . .,.`r. We prove by induction on k that we may choose Z and V for wh* *ich we have `(W ) Q (^ kV + ^V ^+ Z) for any ` 2 L, for all k. Since ^V is finite dimensional, taking k > r establishes_the result. For k = 1, we choose Z = \ri=1ker(`i:W ! Q). We have directly that `i(Z) ^+ (V Z), and hence `(Z) ^+ (V Z) for any ` 2 L. Now suppose that, for some k 1, we have `(W ) Q (^kV + ^V ^+ Z) for any ` 2 L. For each generating derivation `j, and for z 2 W a basis element, wi* *th `(z) 62 Q, we write X (i ,i ,...,i ) `j(z) ~j 1 2 vi1kvi2. .v.ik i1 j, then `j O `n(z) = fl + ffi, with fl of length k - 1 but * *in ^(vi1, vi2, . .,.bvj, . .,.vn-1) ^+ (vn, vn+1, . .,.vr) and ffi 2 ^ rV + ^V * * ^+ Z. This shows again that [`n, `j] 2 L contradicts the induction hypothesis on L. It follows that all the coefficients ~(i1,i2,...,ik)jthat appear in (2) are zero. * *Therefore, we have `j(W ) Q (^k+1V + ^V ^+ Z) for any z 2 W , for each generating derivation `j. To complete the inductive step, we must also consider a general ` 2 L. Suppose that, for some z 2 W , we have X `(z) ~(i1,i2,...,ir)vi1vi2. .v.ik i1 j). However, we have `sd(O) = -`sd(O). Using Lemma 3.1 and the fact that `s is a derivation, we also have `s(fi) 2 ^ m V +^V ^+ Z. This contradicts our minimal length assumption. We claim that dX (W ) ^V ^ 2Z. Suppose this is not the case and let w be an element of lowest degree such that Xq dX (w) = zi!i+ ff , i=1 with zi2 Z, |z1| |z2| . . .|zq|, !i2 ^V and ff 2 ^ 2Z ^V . We choose then an element vs of highest degree such that !q = vsfl+ffi, fl 6= 0, fl, ffi 2 ^(v* *1, . .,.vs-1). Then `qdX (w) = zqvsmod ^ V (^ 2Z + Z<|zg|+ (z1, . .,.zq-1)) . Since `qdX (w) = dX `q(w), there exists an element w0 2 W with |w0| < |w| such that dX (w0) 62 ^V 2 Z. This is impossible by our assumption. (2) We will define a map OE: MX ! MSX MX whose composition with the projection onto the first factor (1 . ffl) O OE: MX ! MSX MX ! MSX is surjective and satisfies (1.ffl)OOE(Z) = 0, and whose composition with the p* *rojection onto the second factor is the identity, (ffl . 1) O OE = 1: MX ! MX . Translating this into topological terms, OE is the minimal model of a map F :* *SX x XQ ! XQ such that F Oi1: SX ! XQ is injective in rational homotopy and F Oi2 = 1: XQ ! XQ. In other words, we may choose F O i1 as a total Gottlieb element (the corresponding lift through !Q is given by the adjoint of F ). Furthermore,* * the model of F O i1 is (1 . ffl) O OE by construction, which satisfies (1 . ffl) O * *OE(Z) = 0. So as to avoid confusion, we write MSX MX as ^V 0 ^V ^Z, with V 0= . First, define a sequence of maps OE1, . .,.OEr: MX ! MSX MX * *by OE1(O) = O + v01`1(O), and OEs(O) = OEs-1(O) + v0s`s OEs-1(O) EVALUATION MAPS IN RATIONAL HOMOTOPY 17 for s = 2, . .,.r. Then we set OE = OEr. An inductive argument shows that OE so defined is a DG algebra map. For it is straightforward to check that OE1 is a DG algebra map. Supposing inductively that OEs-1 is a DG algebra map, the computation 0 OEs-1(O1)OEs-1(O2)= (OEs-1(O1) + v0s`s OEs-1(O1) )(OEs-1(O2) + vs`s OEs-1(O2)* * ) = OEs-1(O1)OEs-1(O2) + v0s`s OEs-1(O1) OEs-1(O2) + (-1)|O1|v0sOEs-1(O1)`s OEs-1(O2) = OEs-1(O1O2) + v0s`s OEs-1(O1O2) shows that OEs is an algebra map. A similar computation, using that OEs-1 and `s commute with dX , and also that d(V 0) = 0, shows that OEs also commutes with d* *X , and hence is a DG algebra map. Thus, each OE1, . .,.OEr is a DG algebra map and in particular so is OE = OEr. Next, we show the following: That OE(v1) = v1 + v01and, for i = 2, . .,.r, OE(vi) = vi+ v0i+ I(v01, . .,.v0i-1). This we do by induction on s. Suppose inductively that we have OEs(v1) = v1 + v* *01 and ( 0+ I(v0, . .,.v0 i)f i = 2, . .,.s OEs(vi) = vi+ vi 1 i-1 vi+ I(v01, . .,.v0i-1)if i = s + 1, . .,.r Induction starts with s = 1, where the formulas OE1(v1) = v1 + v01 and OE1(vi) = vi+ v01`1(vi) give the result. For the inductive step, we compute as follows: OEs+1(v1) = OEs* *(v1) + v0s+1`s+1(v1) = v1 + v01, since 1 < s + 1 and hence `s+1(v1) = 0. For i = 2, . * *.,.s, we have OEs+1(vi)= OEs+1(vi) + v0s+1`s+1 OEs(vi) 0 0 0 = vi+ v0i+ I(v01, . .,.v0i-1) + v0s+1`s+1 vi+ vi+ I(v1, . .,.vi-1) = vi+ v0i+ I(v01, . .,.v0i-1) since i < s + 1 and thus `s+1(vi) = 0, and also the ideal I(v01, . .,.v0i-1) is* * `s+1- stable, as `s+1(v0i) = 0. Further, OEs+1(vs+1) = OEs+1(vs+1) + v0s+1`s+1 OEs(vs* *+1) = vs+1+ I(v01, . .,.v0s) + v0s+1`s+1 vs+1+ I(v01, . .,.v0s) = vs+1+ v0s+1+ I(v01* *, . .,.v0s). Finally, for i = s + 2, . .,.r, we have OEs+1(vi)= OEs(vi) + v0s+1`s+1 OEs(vi) 0 0 = vi+ I(v01, . .,.v0i-1) + v0s+1`s+1 vi+ I(v1, . .,.vi-1) = vi+ I(v01, . .,.v0i-1) since s + 1 i - 1. This completes the induction. Finally, we observe that, for any z 2 Z, we have OE(z) 2 I(Z). This follows e* *asily from the fact that Z is `i-stable for each i. From these facts, it is evident that (1 . ffl) O OE satisfies (1 . ffl) O OE(* *v1) = v01, and (1 . ffl) O OE(vi) = v0i+ I(v01, . .,.v0i-1) for i = 2, . .,.r. It follows tha* *t (1 . ffl) O OE is surjective. Furthermore, we have (1 . ffl) O OE(z) = 0. For the other projectio* *n, it is evident from the definition of OE that we have (ffl . 1) O OE = 1. We deduce the following technical proposition. 18 YVES F'ELIX AND GREGORY LUPTON Proposition 3.3. Suppose V decomposes as V = V 0 V 00, with dX (V 0) = 0 and V 00satisfying the following: For any cycle of the form v +z +O, with v 2 V , z* * 2 Z, and O 2 ^ 2(V Z), we have v 2 V 0. Suppose the complement Z has been chosen to satisfy `i(Z) ^V ^+ Z for each i. Then any cycle of ^+ (V Z) is in the ideal I(V 0, Z) generated by V 0 Z. Proof.The proof is similar to that of part (1) of Proposition 3.2. We argue by contradiction. Suppose this is not true, and that amongst cycles of the form ff* * + fi, with ff 6= 0 2 ^V 00, fi 2 I(V 0, Z), that the shortest length term in any such* * ff is m 2. Since each `i commutes with the differential, `i(v0) is a cycle for e* *ach i. Therefore, we must have that `i(V 0) ^ m V 00+ I(V 0, Z). Now adjust our notation slightly for this situation. Write V 00= for suitab* *le s r, with corresponding derivations `001, . .,.`00s. Let O be a cycle that displays * *a shortest length part in ^V 00, and suppose that t s is the highest index for which v00* *toccurs in this shortest length part. Then write O = ff0+ ff00v00t+ ff000+ fi, with ff02 ^m (v001, . .,.v00t-1), ff006= 0 2 ^m-1 (v001, . .,.v00t-1), ff0002 ^* *m+1 V 00, and fi 2 I(V 0, Z). Since `00tcommutes with the differential, `00t(O) is again a cycle. * *However, we have `00t(O) = ff00+`00t(ff000+fi) (recall that `i(vj) = 0 for i > j). Using* * Lemma 3.1 and the fact that `t is a derivation, we have `t(ff000+ fi) 2 I(^m V 00, V 0, Z* *). This contradicts our minimal length assumption. The next result is a consequence of Oprea's Theorem 1.11. In order to be self- contained we include here a short proof. Proposition 3.4. Suppose MX is written as ^(V 0 V 00 Z) as in Proposition 3.3. Then we may identify V 0with imhX O (!Q)# . Furthermore, XQ decomposes as a product XQ ' S x Y with S a product of odd-dimensional rational spheres whose minimal model is (^V 0, 0). Proof.Suppose (^V, d) is a minimal model for X, x 2 V is a cocycle of odd degree and that there is a derivation ` of ^V such that [`, d] = 0 and `(x) = 1. Write ^V = ^(x) ^W . Then by induction on the degree we can modify the choice of W in order to have d(W ) ^W , as follows. Suppose that d(W of W and by I(n)the ideal generated by the vector spaces ^2(W(n)), s(^2(W(n))), and W(0n* *). When n = 1, we have d(y1) = 0 from our hypothesis on d. Thus we have 0 = h O fl(y1) = (esd+ds(y1)) = (y1) + (y01) = g O fl(y1) + (y01) = (y01* *) , which starts the induction. Now suppose that the result is true for i < n. Then* * we have X 0 = (esd+ds(yn)) = (yn) + (y0n) + 1_ ((sd)r(yn)) . r 1r! The hypothesis on d implies that sd(yn) 2 I(n-1). A refinement of the argument * *in the previous part shows that, in fact, each I(n-1)is stable under sd. Therefore* *, we have (sd)r(yn) 2 I(n-1)for r 1. Since (I(n-1)) = 0 by our induction hypothes* *is, we have that ((sd)r(yn)) = 0 and therefore (y0n) = 0. Of course, is already zero on W and hence vanishes on both ^2(W(n)) and s(^2(W(n))). Therefore, we have (I(n)) = 0 and the induction is complete. It follows that the ideal I pos* *sesses two key properties, namely sd(I) I and (I) = 0. We now define a homotopy : (^(V ~V V 0), ~d) ! (^T, d) simply by restricting . We remark that (sd)r(v) - (sd~)r(v) 2 I for v 2 V , f* *or r 1. Therefore the homotopy ends at (esd~+d~s(v)) = (esd+ds(v)) = h(v). Furthermore, we have (v) = (v) = g(v) for v 2 V . Thus is a homotopy between g and h. The argument so far shows that f is a homotopy monomorphism in the rational category. That is, if A is any rational space, then f*: [A, X] ! [A, Y ] 20 YVES F'ELIX AND GREGORY LUPTON is one-to-one. From the universal properties of localization, it follows that f* * is a homotopy monomorphism in the nilpotent category. Proof of Theorem 1.8.For the ordinary evaluation map ! :Map (X, X; 1) ! X, we have that X :SX ! XQ is a homotopy monomorphism in the nilpotent category by Proposition 3.5 and Proposition 3.2. Now suppose that w: E ! X is any evaluation map. From Theorem 1.6, we have the following commutative diagram of solid arrows g rw k_=E|_____//_Map(X,|XQ;=e)MffM MM _ ___ | | MMM FrX= i __e | | e MMMM6 ffff__w | | X M aeae Sw wQ| !Q| SX BB | | qq BBBw | | qqqq BB | | qqq B!fflffl|! fflffl|Xxxqqq XQ ___________XQ with retractions rX and rw of e Xand e wrespectively. We define j :Sw ! SX by j = rX O g O e wand claim that this map admits a retraction. Recall that both Sw and SX are (finite) products of odd-dimensional rational spheres. Also, since w and X are both injective in rational homotopy and X O j = w, it follows that* * j is injective in rational homotopy. In terms of minimal models, then, we have a * *map Mj:(^V, d = 0) ! (^W, d = 0) with Q(Mj) surjective. But if Q(Mj) is surjective, so too is Mj. Therefore, we may choose a splitting of Mj which corresponds to a retraction of j. Since j admits a retraction, it is a homotopy monomorphism. Finally, it follows that w is a composition of homotopy monomorphisms and hence is a homotopy monomorphism. Corollary 3.6. Let w: E ! X be any evaluation map. Then wQ factors as a composition wQ = w O rw with rw a homotopy epimorphism and w a homotopy monomorphism in the nilpotent category. Proof.The discussion at the start of this section concluded that rw is a homo- topy epimorphism and the remainder follows immediately from Theorem 1.6 and Theorem 1.8. We remark that the fact that w is associated to an evaluation map is key in Theorem 1.8. In particular, we may give the following example of a map fl :S ! X from an H0-space S into X that is injective in rational homotopy but is not a homotopy monomorphism in the nilpotent category. Example 3.7. Let S = S3ax S5 and X = S3a_ S3b[ffe8, where ff is the triple Whitehead bracket [a, [a, b]]. Then fl :S ! X is an extension of (1 | [a, b]): * *S3a_ S5 ! X obtained using the fact that [a, [a, b]] = 0 in ss*(X). Consider two maps h, k :S2 x S3 ! S3ax S5. The map h is the composition p2 i1 3 5 S2 x S3_____//S3____//Sa x S and k is the composition of the inclusion S3 _ S5 ! S3 x S5 with the map that consists of collapsing the cell S2 into a point: S2 x S3 ____//_S2 x S3=S2 = S3 __S5__//S3 x S5. EVALUATION MAPS IN RATIONAL HOMOTOPY 21 Clearly hQ and kQ are not homotopic because they do not induce the same map in rational homology. However a simple computation using minimal models show that the compositions fQ O hQ and fQ O kQ are homotopic. We finish this section with the topic of cyclic maps. A cyclic map f :A ! X m* *ay be defined as a map that lifts through the evaluation map ! :Map (X, X; 1) ! X. This definition is easily seen to be equivalent to that given above Theorem 1.1* *0 via the adjoint correspondence between a map A ! Map (X, X; 1) that lifts f and a map AxX ! X that extends (f | 1). Together with Sam Smith, the second-named author has studied cyclic maps from the rational homotopy point of view in [16]. As we mentioned in the introduction, our interest in the results of this paper * *arose from that earlier work. To state Corollary 2.9 we defined the Gottlieb groups of a space relative to an evaluation map. We say that a map f :A ! X is cyclic with respect to an evaluation map w: E ! X if f lifts through the evaluation map w. Denote the set of homotopy classes of such maps by Gw(A, X). Upon rationalizing such a map, we obtain a map in GwQ(A, XQ). Theorem 3.8. Let w: E ! X be an evaluation map with X a nilpotent, finite complex and let A be a nilpotent space. Then there are bijections of sets w GwQ(A, XQ) ~=[A, Sw] ~= rHom Hr(A; Q), Gr(XQ) . Proof.The first bijection is given by ( w)*: [A, Sw] ! GwQ(A, XQ). This is a bijection by Theorem 1.6 and Theorem 1.8. Now remark that Sw hasQthe homotopy type of a product of rational Eilenberg-Mac Lanespaces, Sw = ri=1K(Q, ni). By taking cohomology classes we thus obtain a bijection ~= r n [A, Sw] -! i=1H i(A; Q) and the result follows. Thus, for instance, we retrieve [16, Th.3.2]: If A is a space with non-zero r* *ational cohomology in even degrees only, then any map g :A ! Sw must be null-homotopic, as Sw is a product of odd-dimensional rational Eilenberg-Mac Lanespaces. Conse- quently, this hypothesis on A entails the triviality of the set GwQ(A, XQ). Man* *y of the other results of [16] may be placed in context with the results of this pap* *er. If X is a suspension, or more generally a co-H0-space, then its rationalized Gottlieb groups are generally trivial. Indeed, this is the case as long as X d* *oes not have the rational homotopy type of a single sphere. Therefore, it follows f* *rom Theorem 2.10 that any cyclic map into a co-H0-space that does not have the rati* *onal homotopy type of a sphere is rationally trivial. Basic finiteness results, such* * as those of [14], follow from this. Note, however, that a general cyclic map does not factor through the product of odd spheres that corresponds to its image in rational homotopy. That is, we * *are not able to extend Theorem 1.6 to cyclic maps. In particular, we note that there exist cyclic maps that are trivial in rational homotopy and yet not null-homoto* *pic (e.g. [16, Ex.4.1]). 4. Evaluation Maps and Homology After the preparatory results of Section 3, we prove in this section the resu* *lts concerning the homomorphism induced in rational homology by an evaluation map. 22 YVES F'ELIX AND GREGORY LUPTON Proof of Theorem 1.12.Consider ! :Map (X, X; 1) ! X as a special case first. If X is an H0-space, then the multiplication of XQ provides a section of !Q, so th* *at H*(!; Q) is surjective. If we have XQ ' S2n+1QxY , then we may apply Theorem 2.* *5. As S2n+1 is an H0-space, the above observation gives that !S2n+1 is surjective on rational homology. Furthermore, the map (i1)* in diagram (1) immediately preceding Theorem 2.5 admits a section, namely (p1)*, and so it too is surjecti* *ve on rational homology. It follows that imH*(!; Q) contains at least the H*(S2n+1; Q) factor and thus is non-zero. This establishes item (3) of Theorem 1.12. Next, suppose that hX O (!Q)# = 0. We deduce from Lemma 3.1 and Proposi- tion 3.3 that a model of X is given by ~ : (^(V Z), dX ) ! (^V, 0) with all cocycles of ^(V Z) in the ideal generated by Z and ~(Z) = 0. Now Proposition 3.2 (2) shows that the total Gottlieb element X induces the trivi* *al homomorphism in rational cohomology. On the other hand, suppose that hX O(!Q)# has image of dimension r > 0. Then Proposition 3.4 implies that we have XQ ' S x Y where S is an r-fold product of rational spheres of odd dimensions that correspond to the image of hX O(!Q)# . * *Now we apply Theorem 2.5 and conclude that imH*(!; Q) contains the H*(S; Q) factor. Furthermore, we have hY O (!Y )# = 0, otherwise the image of hX O (!Q)# would be of dimension > r. Therefore, eH*(!Y ; Q) = 0 and the image of H*(!Q; Q) is prec* *isely the H*(S; Q) factor. This establishes the remaining items of Theorem 1.12 for !. Now consider a generalized evaluation map w: E ! X. We suppose that imhX O (!Q)# is of dimension r and imhX O(wQ)# is of dimension s. Since w factors thro* *ugh !, we have s r. We write XQ ' S x Y as above, and we obtain a commutative diagram g E ______//Map(X, XQ; e) wQ|| |!Q| fflffl|' fflffl| XQ ____h___//_S x Y where g is the H-map obtained from from the definition of a generalized evaluat* *ion map. By Theorem 2.5 the coordinate maps p1 O !Q and p2 O !Q factor through (!S)Q and (!Y )Q respectively. Because of this factorization, and the fact that eH*(!Y ; Q) = 0, we may make the following identifications: im H*(wQ; Q) ~=imH*(!Q O g; Q) ~=imH*(p1 O !Q O g; Q) H*(S; Q). Since the composition p1 O !Q O g :E ! S satisfies the hypotheses of Corollary * *2.8, it admits a minimal model of the form ': (^V, 0) ! (^W, 0) with '(V ) W . Then the image of p1 O !Q O g :E ! S in rational homotopy has dimension s and we may factor its minimal model ': (^V, 0) ! (^W, 0) as the composition of a surjection and an injection ^(Vs K) ! ^Vs ! ^(Vs K0), with Vs a vector space of dimension s isomorphic to the image of imhX O (wQ)# . This corresponds to a EVALUATION MAPS IN RATIONAL HOMOTOPY 23 factorization of p1 O !Q O g :E ! S as p1O!QOg E ____________//_@@S9'9S0x S00 @@ sssss q@@@OO@ssssi1 s S0 with S0a product of odd-dimensional rational spheres with minimal model (^Vs, 0* *). It is now clear that the image in homology of wQ is isomorphic to H*(S0; Q). For X a finite complex, a result of Gottlieb ([9, Th.3]) says that if O(X) 6=* * 0, then the first degree in which the homomorphism induced by the evaluation map on rational cohomology may be non-zero is even. With Theorem 1.12, we sharpen this result in a very significant way. Corollary 4.1 (Corollary 1.13). Let X be a nilpotent, finite space. Suppose that O(X) 6= 0 or, more generally, that X does not factor up to rational homotopy as XQ ' S2n+1QxY . Then for every evaluation map w: E ! X, we have eH*(w; Q)) = 0. Recall that X is called a c-symplectic space if it is an even-dimensional rat* *ional Poincar'e duality space that possesses some class x 2 H2(X; Q), some power of which is a fundamental class [15]. Corollary 4.2 (Corollary 1.14). Let X be a simply connected, c-symplectic space. Then every evaluation map w: E ! X satisfies eH*(w; Q) = 0. Proof.It is evident that the cohomology algebra structure does not allow a deco* *m- position of the form X 'Q S2n+1x X0, and so Theorem 1.12 implies the evaluation map is trivial in rational homology. At the other extreme from the situation described in these corollaries, we ha* *ve the following: Corollary 4.3. Let w: E ! X be an evaluation map with X a nilpotent, finite complex. The following are equivalent: (1) The homomorphism H*(w): H*(E; Q) ! H*(X; Q) is surjective; (2) w :Sw ! X is a rational homotopy equivalence. When (1) and (2) pertain, X is an H0-space and the evaluation map admits a section. Proof.All parts follow easily from Theorem 1.6 and Theorem 1.12. 5.Conclusion: Some Open Problems At present, we have very little information about the map w: Top(X, X; 1) ! X or the other variations on the evaluation map ! mentioned at the start of the introduction. It would be most interesting to identify Gw*(XQ), the image in ra* *tional homotopy of w, or, more generally the rational homotopy groups of Top(X, X; 1). As specific instances of this kind of problem, we offer the following. Problem 5.1. Let M be a compact smooth manifold. Is the image in (rational) homotopy of the evaluation map w: Diff(M, M; 1) ! M strictly contained in, or equal to, the (rational) Gottlieb groups of M? 24 YVES F'ELIX AND GREGORY LUPTON Problem 5.2. Let X be an H-space and recall that G*(X) = ss*(X) in this case. Let H(X, X; 1) denote the subspace of Map (X, X; 1) that consists of H-equivale* *nces. Is the evaluation map w: H(X, X; 1) ! X surjective in (rational) homotopy? Assuming that Gw*(X) and G*(X) are generally different from each other, it would be interesting to know whether there are structural results for Gw*(XQ) c* *om- parable to those of F'elix-Halperin for the ordinary Gottlieb groups. Corollary 1.5 and Corollary 2.9 may be used to give necessary conditions for certain maps to be the connecting map of a fibration (cf. Example 2.12). This suggests the following particular version of an old problem of Massey: Problem 5.3. Let p: B ! X be a map from a loop space to a nilpotent, finite complex X. When is p the connecting map of some fibration sequence X ! E ! B? It would be nice to find other situations in which the image in rational homo* *topy groups of a map led to factorizations analogous to those of Section 2. In this direction, we offer the following rather general problem: Problem 5.4. Suppose given a map f :X ! Y with Y finite-dimensional. If the image of f# in rational homotopy groups is finite-dimensional, does f factor through an elliptic space? We have restricted ourselves entirely to the rational homotopy context in this paper. But it could be feasible to investigate similar results working either i* *ntegrally or localized at different sets of primes. We end with two "moonshots" that indi* *cate how little we know outside the rational situation. Problem 5.5. Let X be a space with trivial Gottlieb groups (integrally). Is the evaluation map ! :Map (X, X; 1) ! X null-homotopic? Problem 5.6. Let X be a nilpotent, finite complex. When is a Gottlieb element Sn ! X a homotopy monomorphism, and not just a rational homotopy monomor- phism ? References [1]Y. F'elix and S. Halperin, Rational LS category and its applications, Trans* *. Amer. Math. Soc. 273 (1982), no. 1, 1-38. MR 84h:55011 [2]Y. F'elix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduat* *e Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 2002d:55014 [3]T. Ganea, On monomorphisms in homotopy theory, Topology 6 (1967), 149-152. * *MR 34 #8402 [4]S. Ghorbal, Monomorphismes et 'epimorphismes homotopiques, Ph. D. Thesis, L* *ouvain-La- Neuve, 1996. [5]D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math.* * 87 (1965), 840-856. [6]____, On fibre spaces and the evaluation map, Ann. Math. 87 (1968), 42-55. [7]____, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 72* *9-756. [8]____, Applications of bundle map theory, Trans. Amer. Math. Soc. 171 (1972)* *, 23-50. [9]____, The evaluation map and homology, Michigan Math. J. 19 (1972), 289-297* *. MR 49 #8005 [10]S. Halperin, Lectures on minimal models, M'em. Soc. Math. France (N.S.) (19* *83), no. 9-10, 261. MR 85i:55009 [11]____, Torsion gaps in the homotopy of finite complexes, Topology 27 (1988),* * no. 3, 367- 375. MR 89h:55024 EVALUATION MAPS IN RATIONAL HOMOTOPY 25 [12]P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and* * spaces, North- Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematics Studies, * *No. 15, Notas de Matem'atica, No. 55. [Notes on Mathematics, No. 55]. MR 57 #17635 [13]G. E. Lang, Localizations and evaluation subgroups, Proc. Amer. Math. Soc. * *50 (1975), 489- 494. MR 51 #4228 [14]K. L. Lim, On cyclic maps, J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, * *349-357. MR 83e:55003 [15]G. Lupton and J. Oprea, Cohomologically symplectic spaces: toral actions an* *d the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), no. 1, 261-288. MR 95f:57056 [16]G. Lupton and S. B. Smith, Cyclic maps in rational homotopy theory, Math. Z* *. 249 (2005), no. 1, 113-124. [17]J. Milnor, On spaces having the homotopy type of CW -complex, Trans. Amer. * *Math. Soc. 90 (1959), 272-280. MR 20 #6700 [18]John Oprea, Decomposition theorems in rational homotopy theory, Proc. Amer.* * Math. Soc. 96 (1986), no. 3, 505-512. MR 87h:55008 [19]____, The Samelson space of a fibration, Michigan Math. J. 34 (1987), no. 1* *, 127-141. MR 88c:55015 [20]K. Varadarajan, Generalised Gottlieb groups, J. Indian Math. Soc. 33 (1969)* *, 141-164. Institut Math'ematique, Universit'e Catholique de Louvain, B-1348 Louvain-la-* *Neuve Belgique E-mail address: felix@math.ucl.ac.be Department of Mathematics, Cleveland State University, Cleveland OH 44115 U.S* *.A. E-mail address: G.Lupton@csuohio.edu