RATIONALIZED EVALUATION SUBGROUPS OF A MAP AND THE RATIONALIZED G-SEQUENCE GREGORY LUPTON AND SAMUEL BRUCE SMITH Abstract.Let f :X ! Y be a based map of simply connected spaces. The corresponding evaluation map !:map (X, Y ; f) ! Y induces a homomorphism of homotopy groups whose image in in(Y ) is called the nth evaluation su* *b- group of f. The nth Gottlieb group of X occur as the special case in whi* *ch X = Y and f = 1X . We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identifica* *tion allows for the characterization of the rationalization of the nth evalua* *tion sub- group of f. It also allows for the identification of several long exact * *sequences of rational homotopy groups, including the long exact sequence induced on rational homotopy groups by the evaluation fibration. As a consequence, * *we obtain an identification of the rationalization of the so-called G-seque* *nce of the map f. This is a sequence_in general not exact_of groups and homomor- phisms that includes the Gottlieb groups of X and the evaluation subgrou* *ps of f. We use these results to study the G-sequence in the context of rat* *io- nal homotopy theory. We give new examples of non-exact G-sequences and uncover a relationship between the homology of the rational G-sequence a* *nd negative derivations of rational cohomology. We also relate the splittin* *g of the rational G-sequence of a fibre inclusion to a well-known conjecture in r* *ational homotopy theory. 1.Introduction Suppose given a based map f :X ! Y of simply connected CW complexes. Denote by map(X, Y ; f) the path component of the space of (unbased) maps X ! Y consisting of those maps that are homotopic to f. Then evaluation at the basepo* *int of X gives a map ! :map(X, Y ; f) ! Y . We refer to this map as the evaluation map. We define the nth evaluation subgroup of f to be the subgroup Gn(Y, X; f) = !# (ßn(map (X, Y ; f))) of ßn(Y ). The famous Gottlieb groups G*(X) occur as the special case when X = Y and f = 1X [Got69]. The Gottlieb groups of a space have been much studied by homotopy theorists (see [Opr95 ] for a survey of resu* *lts and references). While many general results are known, explicit computation of G*(X) appears difficult and such results are limited to a small number of spora* *dic examples. One reason that accounts in part for this difficulty is the fact that* * a map of spaces f :X ! Y does not necessarily induce a corresponding homomorphism of Gottlieb groups, since in general f# (Gn(X)) 6 Gn(Y ). In particular, attempts* * to study G*(X) via a cell decomposition of X are frustrated, since it is not clear* * what effect a cell attachment may have on the Gottlieb group. One tool for studying ____________ Date: June 16, 2003. 2000 Mathematics Subject Classification. 55P62, 55D23. Key words and phrases. Evaluation map, evaluation subgroup, Gottlieb group, * *G-sequence, !-homology, function spaces, rational homotopy, minimal models, derivations. 1 2 GREGORY LUPTON AND SAMUEL BRUCE SMITH G*(X) that attempts to circumvent this problem is the so-called G-sequence of a map, as developed and studied by Lee and Woo in a series of papers [LW88a , LW88b , LW90 , LW93 ]. The G-sequence of a map f :X ! Y is a sequence . .!.Gn(X) ! Gn(Y, X; f) ! Greln(Y, X; f) ! Gn-1(X) ! . . . of groups and homomorphisms that derives from the long exact homotopy sequence of the map f. In this sequence, Gn(X) is the Gottlieb group of X, Gn(Y, X; f) is the evaluation subgroup of f introduced above, and the third term Greln(Y, X; f* *) is a suitably defined "relative" term. The G-sequence arises as follows. One has t* *he following commutative diagram of spaces: map (X, X; 1)_f*_//map(X, Y ; f) ! || |!| fflffl| fflffl| X ______f_______//_Y Now pass to the corresponding induced homomorphisms of homotopy groups. In a standard way, the induced homomorphisms of homotopy groups (f*)# and f# can be fitted into the long exact homotopy sequences of the maps f* and f respectiv* *ely. Then the evaluation maps induce maps of each term in the long exact sequences, resulting in a homotopy ladder. The G-sequence of the map f is then the image of the top long exact homotopy sequence in that of the bottom. A portion of the G-sequence is shown here: (f*)# ßn+1(f*)_______//ßn(map (X, X; 1))__//ßn(map (X, Y ; f)) !#|| |!# !#| fflfflfflffl| fflfflfflffl|| fflfflfflffl|| f# Greln+1(Y,"X;`f)______//Gn(X)"_`________//_Gn(Y,"X;`f) | | | | | | fflffl| fflffl| f# fflffl| ßn+1(f)____________//ßn(X)______________//ßn(Y ) The homomorphisms in the G-sequence are the same as those in the long exact ho- motopy sequence of the map f. Therefore, the G-sequence forms a chain complex (consecutive compositions are trivial). However, it is not necessarily exact. C* *onse- quently, it determines graded homology groups called the !-homology of f :X ! Y* * . The determination of this invariant of the homotopy type of the map f is an ex- tremely difficult problem even under restricted circumstances. Examples of non- exactness, that is, of nontrivial !-homology, are given in [PW97 , PSW98 ]. Some general conditions are known under which the G-sequence is exact. For instance, it is exact when f is a null-homotopic cellular inclusion [LW93 ], or when f is* * a homotopy monomorphism [PW01 ]. In this paper is to bring the techniques of rational homotopy theory to bear * *on problems and questions concerning evaluation subgroups (of a map) in general, a* *nd the G-sequence in particular. Our main goal is to expand the range of applicati* *on of these techniques in this area. We are primarily concerned with establishing* * a suitable framework for considering such questions. At the same time, we obtain a number of results of interest in their own right. We now outline the results* * of RATIONALIZED EVALUATION SUBGROUPS 3 the paper. Our main results are established in Section 2 and Section 3. The bas* *ic result in Section 2 is Theorem 2.1, in which we identify the map induced on rat* *ional homotopy groups by the evaluation map ! :map(X, Y ; f) ! Y . We describe this induced homomorphism as the map induced on homology by a map of complexes of derivations of the Sullivan minimal models of X and Y . Theorem 2.1 has a number of immediate corollaries. For instance, we obtain a characterization of the rationalized evaluation subgroups of a map (Corollary 2.7) that extends a w* *ell known characterization, due to F'elix and Halperin, of the (rationalized) Gottl* *ieb groups of a space in terms of derivations of its minimal model. In Section 3, we extend and amplify the basic result of Theorem 2.1. We show that the relevant long exact sequences of rational homotopy groups of function spaces are naturally expressed as homology exact sequences of derivations com- plexes of Sullivan minimal models. In particular, we obtain a description, with* *in the framework of derivation spaces, of the G-sequence of a map after rationaliz* *ation. In Section 4, we use the framework established in Section 2 and Section 3 to study the exactness and non-exactness of the rationalized G-sequence. Just as in the integral setting, the rationalized G-sequence is not exact in general. Exam* *ple 4.1 gives a simple example for which the rationalized G-sequence fails to be exact * *at each of the three types of term that occur. Since nonexactness rationally impli* *es nonexactness integrally, this example provides a new, complete example of the failure of exactness of the G-sequence. This example also shows that the framew* *ork established in Section 2 and Section 3 provides a fairly effective setting in w* *hich to carry out explicit computations. By way of contrast, in Theorem 4.3 we give one* * set of conditions under which the rationalized G-sequence is exact. In Theorem 4.5 * *we show that under certain circumstances, the rationalized G-sequence may be exact at all occurrences of one type of term, while failing to be exact at the other * *types of term. This same result establishes a relationship, albeit under rather restr* *ictive circumstances, between the (vanishing of a certain type of the) !-homology of a map f :X ! Y and the (vanishing of) negative-degree derivations of the rational cohomology of X. The last development that we present in Section 4 is a connect* *ion between the G-sequence of certain fibre inclusion maps and a well-known conject* *ure in rational homotopy theory. Let X ! E ! S2r+1be a fibration with base an odd- dimensional sphere. For certain types of fibre space X, a conjecture of Halper* *in asserts that the fibration should be rationally TNCZ (see Conjecture 2.10 below* *). In Theorem 4.9, we show that this is the case exactly when the rationalized G- sequence of the fibre inclusion X ! E reduces to a certain short exact sequence* *. In this way, we obtain an equivalent phrasing of the conjecture of Halperin, in te* *rms of the ideas studied in this paper. We now discuss existing results on rational homotopy, function spaces, and the Gottlieb groups of a space that relate to this paper. Our aim in this discussio* *n is to indicate the basic results that exist in the area. At the same time, we iden* *tify how these results relate to our work, and our points of departure from them. Fr* *om results in rational homotopy theory, we can identify three "tributary streams" * *that flow into this work precisely at Theorem 2.1_our first main result_at varying levels of generality. There is also a fourth tributary flowing into our work, * *from outside rational homotopy theory. This consists of the work on the G-sequence, already indicated above. We say a little bit more on this below. 4 GREGORY LUPTON AND SAMUEL BRUCE SMITH The first tributary of antecedent results in rational homotopy theory concern* *s the rationalized Gottlieb groups of a space. In [FH82 ], F'elix and Halperin gave a* * char- acterization of the rationalized Gottlieb groups of a space, in terms of deriva* *tions of the Sullivan minimal model. At its most specialized level, Theorem 2.1 retri* *eves this characterization (Corollary 2.6) and extends it to a similar characterizat* *ion of the rationalized evaluation subgroups of a general map (Corollary 2.7). F'elix * *and Halperin went on to prove a remarkable result concerning the rationalized Gottl* *ieb groups of a finite complex [FH82 , Th.III]. Their result significantly extends * *results of Gottlieb from [Got69], and relates the rationalized Gottlieb groups with the* * ra- tional Lusternik-Schnirelmann category. Unfortunately, no analogous result seems forthcoming for the rationalized evaluation subgroups of a map. Nonetheless, our characterization of the rationalized evaluation subgroups of a map is as effect* *ive for concrete computations as is the earlier characterization of the rationalized Go* *ttlieb groups. The second tributary concerns the rational homotopy type of B aut1(X)_the classifying space for fibrations with fibre X. Although we are not concerned wi* *th this classifying space as such, the connection arises because we have an isomor* *phism of homotopy groups ßi+1 B aut1(X) ~=ßi map(X, X; 1) , and also the Gottlieb groups of X are obtained as the image in homotopy groups of the connecting ho- momorphism of the corresponding classifying fibration [Got69, Th.2.6]. At a hig* *her level of generality than the rationalized Gottlieb groups, Theorem 2.1 includes* * an identification of the map induced on rational homotopy groups by the evaluation map ! :map(X, X; 1) ! X, and in particular it identifies the rational homotopy groups of map (X, X; 1). Whilst the identification of these groups we give is f* *amil- iar in rational homotopy theory, our proof for this_obtained by restricting that of Theorem 2.1_is the first direct and detailed one that has actually appeared in the literature. We support this assertion as follows: In [Sul78, Sec.11], Su* *llivan sketched (with no proof) a model for the rational homotopy type of B aut1(X), f* *rom which the description of the rational homotopy groups of map (X, X; 1) contained in Theorem 2.1 follows. A justification of Sullivan's model for B aut1(X) may be gleaned by collating results from a number of sources spread through the litera* *ture (e.g. [SS, Tan84, Tan83, Gat97]). But to date, no direct proof of Sullivan's mo* *del has been given. Even amongst those articles that focus specifically on the rat* *io- nal homotopy groups, either of B aut1(X) or of map (X, X; 1)_and that therefore avoid the technical problems of dealing with the rational homotopy type_we still do not find complete details. Meier [Mei82, (1.4), (2.6)] outlines the basic id* *ea, but is actually focussed on a special kind of situation in which the minimal model * *can be replaced by its cohomology. Grivel [Gri94] focusses on the same special case as Meier, and quotes Sullivan's model directly. F'elix and Thomas [FT94 , Sec.2* *.3] give exactly the description of the rational homotopy groups of map (X, X; 1) c* *on- tained in Theorem 2.1, but no details of the proof are given. As well as includ* *ing a direct and detailed proof for the case of map (X, X; 1), Theorem 2.1 extends * *this identification of the rational homotopy groups to the general case of map(X, Y * *; f). It thus provides a natural framework for the study of rational homotopy groups * *of function spaces. Finally, the third tributary from rational homotopy theory consists of a mode* *l, due to Sullivan and Haefliger, for the rational homotopy type of map (X, Y ; f)* *. At RATIONALIZED EVALUATION SUBGROUPS 5 its full level of generality, Theorem 2.1 identifies the map induced on rationa* *l ho- motopy groups by ! :map(X, Y ; f) ! Y , a general evaluation map. The precursor to the identification we give, and to our general line of proof, is the approac* *h of Thom in [Tho57 ], although obviously there is no reference to minimal models in* * his work. Pursuing the river analogy a little further, we may think of Thom's resul* *t, if not as the source, then at least as somewhere in the headwaters. We emphasize the connection between Theorem 2.1 and Thom's approach by retrieving a basic result of his in Corollary 2.9. Coming further downstream, Sullivan also descri* *bed in [Sul78, Sec.11] a model for the space of sections of a fibration homotopic t* *o a given section. By specializing to the trivial fibration X x Y ! X, this yields* * a model for the function space map (X, Y ; f)_and more generally a model for the rational homotopy type of the general evaluation map !. A detailed proof for Su* *lli- van's model in this case was given by Haefliger [Hae82]. Now this model should * *in principle determine the rational homotopy groups of the function space (see [FT* *94 ] and [MR85 ], where it is used quite effectively). However, the model in questi* *on is a (non-minimal) DG algebra model. Therefore, the homomorphism induced by ! on rational homotopy groups_which is exactly the information we require to proceed with our development_is available only indirectly, at best. By focussing on the rational homotopy groups_as opposed to the rational homotopy type, we have arrived in Theorem 2.1 at an entirely new characterization of the map in- duced on rational homotopy groups by ! :map(X, Y ; f) ! Y . Furthermore, we have been able to give a direct proof that avoids many of the technical complex* *ities of Haefliger's work and is completely independent of it. The remaining tributary flowing into our work is from outside rational homoto* *py theory, and concerns classical results on the Gottlieb groups of a space, and m* *ore recent results on evaluation subgroups of a map and the G-sequence. We have already mentioned some of the results in this area. In his original work on eva* *luation subgroups, Gottlieb observed that a map of spaces does not necessarily induce a map of Gottlieb groups, and gave conditions under which it does [Got69, Sec.1]. Gottlieb briefly mentions the evaluation subgroups of a map in [Got69], but did not study them as such. Kim and Woo established a number of basic properties of the evaluation subgroup of a map in [WK84 ]. As we mentioned earlier, Woo and Lee, and other authors, introduced and studied the G-sequence of a map. The results basically fall into one of three areas: conditions under which the G-se* *quence is exact (e.g. [LW88b , Th.12] and [LW90 ]), examples of non-exactness (e.g. [L* *W93 ] and [PW97 ]), and extensions and generalizations of evaluation subgroups and the G-sequence (e.g. [LW98 , LW01 ]). These results give the stepping-off point for* * our work in Sections 3 and 4. Using our description of the rationalized G-sequence (Theorem 3.7), we extend the known exactness results to several new cases. We a* *lso give new instances of non-exactness. More significantly, by focussing on the ra* *tional setting, our methods make the production of such examples straightforward. On the other hand, our results and examples are not restricted to amplifying previ* *ous results in this area. In Corollary 4.10, we suggest a different kind of result* * that relates properties of the G-sequence to the triviality of a fibration. Furtherm* *ore, some results in Section 3_including Theorem 3.2 and Theorem 3.10_are of interest independently of any relation to the G-sequence. We finish this introduction by setting some notation and terminology. Through- out this paper, X and Y will denote simply connected CW complexes of finite typ* *e. 6 GREGORY LUPTON AND SAMUEL BRUCE SMITH By vector space we mean a rational graded vector space. By algebra, we mean the kind of commutative graded algebras over the rationals that arise in rational h* *o- motopy. That is, they are non-negatively graded, connected (H0 = Q) and usually simply connected (H1 = 0), with cohomology of finite type. For a vector space V , we denote the free commutative graded algebra generated by V by V . We use the acronym DG to denote differential graded: Thus, DG vector space, DG algebra, and so-forth. For a DG algebra, the differential is of degree +1. In o* *ther situations, however, particularly when we consider the complex of derivations of a DG algebra, the differential is of degree -1. We will generally refer to a DG vector space whose differential is of degree -1 as a chain complex. If f :A ! B* * is a map, either topological or algebraic, then f* denotes pre-composition by f and f* denotes post-composition by f. In any setting in which it is appropriate, we* * use H(f) to denote the map induced on homology (or cohomology) by f, and f# to denote the map induced on homotopy groups by the map of spaces f. We use ! in a generic way to denote an evaluation map, and we denote the identity map of a topological space or the identity homomorphism of an algebra by 1. We denote the rationalization of a space X by XQ and of a map f by fQ (cf. [HMR75 ]). We assume that the reader is familiar with the basics of rational homotopy. In particular, a space X has a minimal model MX , which is a certain type of DG al- gebra. Namely, MX is a free algebra V with a decomposable differential, that i* *s, d(V ) 2V . Furthermore, a map of spaces f :X ! Y induces a map of minimal models Mf: MY ! MX . We refer to this induced map as the Sullivan minimal model of the map f. It is a complete rational homotopy invariant for a map, and in principle all rational homotopy theoretic information about f can be retriev* *ed from it. Passing to cohomology, for example, gives H(Mf): H*(MY ) ! H*(MX ), which corresponds to the homomorphism of rational cohomology algebras induced by f. The results of this paper illustrate how deeper information about a space* * or map may be retrieved from the minimal model by making correspondingly more sophisticated constructions with the model. Our basic reference for rational ho* *mo- topy theory is [FHT01 ]. 2. Derivation Spaces Our purpose in this and the next section is to give a unified description in * *rational homotopy theory of all the terms involved in the definition of the G-sequence. Informally stated, we show that the homology theory of derivation complexes of Sullivan minimal models provides an algebraic model for the rational homotopy theory of function spaces at the level of homotopy groups. We focus on the following commutative square that appears in the homotopy ladder from which the G-sequence arises: (f*)# (1) ßn map (X, X; id)____//_ßn map (X, Y ; f) !# || |!#| fflffl| fflffl| ßn(X) _______f#______//ßn(Y ) It turns out that identifying the rationalization of this commutative square is* * suf- ficient to arrive not only at a characterization of the rationalized evaluation* * sub- groups of f, but also at a description of the rationalization of the G-sequence. RATIONALIZED EVALUATION SUBGROUPS 7 Furthermore, our identification of the rationalization of this square allows us* * to conclude several subsidiary results of interest. We say two maps of vector spaces f :U ! V and g :U0 ! V 0are equivalent if there exists isomorphisms ff and fi which make the diagram f U _____//_V ff~=|| fi~=|| |fflffl fflffl| U0 __g__//V 0 commutative. This notion of equivalence for vector space maps extends in the obvious way to sequences of vector space maps, commutative squares of vector space maps, and any other diagram of vector space maps. Suppose a DG algebra (A, dA ) is isomorphic to Q in degree zero, that is, A0 * *~=Q. Then the map ": A ! Q that sends all elements of positive degree to zero, and is the identity in degree zero, is an augmentation. This will be the situation in* * all cases of interest to us here, and thus we refer to ": A ! Q as the augmentation. Here, as in the sequel, we regard Q as the trivial DG algebra concentrated in d* *egree zero and with trivial differential. Thus " is a DG algebra map. Given DG algebras (A, dA ) and (B, dB ) and a (fixed) DG algebra map OE: A ! * *B, define a OE-derivation of degree n to be a linear map ` :A ! B that reduces deg* *ree by n and satisfies the derivation law `(xy) = `(x)OE(y) - (-1)n|x|OE(x)`(y). We* * will only consider derivations of positive degree, that is, those that reduce degree* * by some positive integer. When n = 1 we require additionally that dB O ` = -` O dA* * . Let Dern(A, B; OE) denote the vector space of OE-derivations of degree n, for n* * > 0. Finally, define a linear map ffi :Dern(A, B; OE) ! Dern-1(A, B; OE) by ffi(`) =* * dB O ` - (-1)|`|`OdA . A standard check now shows that ffi2 = 0 and thus Der*(A, B; OE)* *, ffi is a chain complex. In order to cut down on cumbersome notation, we will usually s* *up- press the differential from our notation, and write Hn Der(A, B; OE) for the h* *omol- ogy in degree n of the chain complex Der*(A, B; OE), ffi . That is, Hn Der(A, * *B; OE) denotes the homology represented by ffi-cycles of Der*(A, B; OE) that reduce de* *gree by n. A special case of the preceding that is of interest to us is the one in which* * A = B and OE = 1B . In this case, the chain complex of derivations Der*(B, B; 1) is * *just the usual complex of derivations on the DG algebra B. Note once again that we restrict the derivations in degree 1 to the cycles and that the complex is zero* * in non-positive degrees. Pre-composition with the DG algebra map OE: A ! B thus gives a map of chain complexes OE*: Der(B, B; 1) ! Der(A, B; OE). Furthermore, post-composition by the augmentation ": B ! Q induces DG vector space maps "*: Der*(A, B; OE) ! Der*(A, Q; ") and "*: Der*(B, B; 1) ! Der*(B, Q; "). All of the above can be applied to a map of minimal models. Suppose f :X ! Y is a map of spaces, and Mf: MY ! MX is the corresponding Sullivan minimal model of the map f. Then we have a commutative square of chain complexes (Mf)* (2) Der*(MX , MX ; 1)____//Der*(MY , MX ; Mf) "*|| |"*| fflffl| "(Mf)* fflffl| Der*(MX , Q; ")_______//_Der*(MY , Q; ") 8 GREGORY LUPTON AND SAMUEL BRUCE SMITH In this square, ": MX ! Q is the augmentation. Both horizontal maps are obtained by pre-composing with the same map Mf, but in different contexts. Since we will need to distinguish between these two maps notationally in the sequel, we have used an extra decoration on the bottom one. The main result of this section is the following: Theorem 2.1. Let X and Y be simply connected CW complexes of finite type, with X finite. For n 2, the commutative square obtained by rationalizing (1) * *is equivalent to the square obtained by passing to homology in degree n from (2). * *That is, the commutative squares (f*)# Q ßn map (X, X; 1) _Q_________//ßn map (X, Y ; f) Q !# Q || !#|Q| fflffl| fflffl| ßn(X) Q________f#_Q_________//ßn(Y ) Q and H((Mf)*) Hn Der(MX , MX ; 1) ___________//Hn Der(MY , MX ; Mf) H("*)|| |H("*)| fflffl| H(("Mf)*) fflffl| Hn Der(MX , Q; ")______________//Hn Der(MY , Q; ") are equivalent for each n 2. We prove this result below. First, we comment on some ingredients of the stat* *e- ment and proof, and give some immediate consequences. Remark 2.2. In rational homotopy theory there is a standard way to identify the rational homotopy groups of a space X, and more generally the homomor- phism of rational homotopy groups induced by a map of spaces. Namely, the rational homotopy groups are identified with the dual of the vector space of in- decomposables of the minimal model, thus ß*(X) Q ~= Hom (Q(MX ), Q). As for maps, f# Q is identified with the dual of the map of vector spaces of ind* *e- composables Q(Mf): Q(MY ) ! Q(MX ) induced by the Sullivan minimal model Mf: MY ! MX of a map f :X ! Y (see [FHT01 , Sec.15(d)] for details). From the bottom maps in the two squares of Theorem 2.1, we obtain a superficially di* *ffer- ent description of f# Q. But it is easy to see that this agrees with the stand* *ard one: Note that the derivation law implies Der*(MX , Q; ") ~=Hom (Q(MX ), Q), while t* *he minimality of MX implies that ffi = 0 in the chain complex Der*(MX , Q; "). Thus we have Hom (Q(MX ), Q) ~= H* Der(MX , Q; ") . The agreement between our description of f# Q and the standard one is obvious from this isomorphism. Theorem 2.1 also contains and depends upon basic results concerning the ratio- nalization of function space components and evaluation subgroups due to several authors. We consider this material here. When X is a finite complex, the function space map (X, Y ) has the homotopy type of a CW complex by the result of Milnor [Mil59]. In fact, by Hilton-Mislin- Roitberg [HMR75 , Ch. II, Th. 2.5] the components map (X, Y ; f) are nilpote* *nt complexes. Moreover, given a rationalization eY :Y ! YQ of Y the induced map (eY )*: map(X, Y ; f) ! map (X, YQ; eY O f) is a rationalization of map (X, Y ;* * f) RATIONALIZED EVALUATION SUBGROUPS 9 [HMR75 , Ch. II, Th. 3.11]. By [Smi96, Th.2.3], rationalization in the init* *ial variable eX :X ! XQ induces a weak equivalence (eX )*: map(XQ, YQ; fQ) ! map(X, YQ; fQ O eX ). These results, together with the naturality of the vario* *us maps involved, imply the following result. Theorem 2.3. Let f :X ! Y be a map between simply connected complexes of finite type with X finite. Let fQ : XQ ! YQ denote the rationalization of f. The commutative squares (f*)# Q ßn map (X, X; 1) _Q_________//ßn map (X, Y ; f) Q !# Q || !#|Q| fflffl| fflffl| ßn(X) Q________f#_Q_________//ßn(Y ) Q and ((fQ)*)# ßn map (XQ, XQ; 1)__________//_ßn map (XQ, YQ; fQ) (!)#|| |(!)#| fflffl| fflffl| ßn(XQ) ________(f___________//_ßn(YQ) Q)# are equivalent for each n 2. The result Gn(XQ) ~=Gn(X) Q for X a simply connected finite complex due to Lang [Lan75] is an easy consequence of Theorem 2.3, as is its generalization G*(YQ, XQ; fQ) ~=G*(Y, X; f) Q (c.f. [Smi96, PW97 ]) for X, Y simply connected complexes of finite type with X finite. The corresponding localization result * *for the relative Gottlieb group can be deduced from the preceding discussion, as we* *ll. Theorem 2.4. Let f :X ! Y be a map between simply connected complexes of finite type with X finite. Then Greln(YQ, XQ; fQ) ~=Greln(Y, X; f) Q, for n * * 3. Proof.Consider, as in the introduction, the long exact homotopy sequence of the induced map f*: map(X, X; 1) ! map (X, Y ; f) and its relative homotopy group ßn(f*) for n 3. We also have (fQ)*: map(XQ, XQ; 1) ! map (XQ, YQ; fQ) and the relative group ßn((fQ)*). The results cited above and the 5-Lemma, imply the maps induced on the various function spaces involved by the rationalizations eX* * : X ! XQ and eY : Y ! YQ induce a rationalization homomorphism R: ßn(f*) ! ßn((fQ)*). The maps eX and eY also induce a rationalization r :ßn(f) ! ßn(fQ). By naturality, we obtain a commutative diagram ßn(f*)__R__//ßn((fQ)*) !]|| |!]| fflffl|r fflffl| ßn(f) _____//_ßn(fQ). Thus Gn(XQ, YQ; fQ) = !](ßn((fQ)*)= r O !](ßn(f*))~=Greln(Y, X; f) Q Combining the preceding, we obtain the following: 10 GREGORY LUPTON AND SAMUEL BRUCE SMITH Corollary 2.5. Let f :X ! Y be a map between simply connected complexes of finite type with X finite. Then the rationalization of the G-sequence of f . .!.Gn(X) Q ! Gn(Y, X; f) Q ! Greln(Y, X; f) Q ! Gn-1(X) Q ! . . . is equivalent to the G-sequence of the rationalization of f . .!.Gn(XQ) ! Gn(YQ, XQ; fQ) ! Greln(YQ, XQ; fQ) ! Gn-1(XQ) ! . . . In the next section, we identify the rational G-sequence in the context of ra* *tio- nal homotopy theory. From Theorem 2.1 we immediately retrieve minimal model descriptions of the rationalized Gottlieb groups of a space, and of the rationa* *lized evaluation subgroups of a map. The first of these is well-known: Corollary 2.6. Let X be a simply connected finite complex. The rationalized nth Gottlieb group Gn(XQ) ~=Gn(X) Q is isomorphic to the image of the induced homomorphism H("*): Hn Der(MX , MX ; 1) ! Hn Der(MX , Q; ") for n 2. This is easily translated into the standard minimal model description of the rationalized Gottlieb groups given by F'elix and Halperin (see [FHT01 , Sec.29(* *c)]). Specifically, they describe a Gottlieb element of the minimal model MX = (V ) as a linear map ` :V n! Q that extends to a derivation of MX satisfying d` = (-1)n`d. Such a derivation ` is a cycle in Der(MX , MX ; 1), and the class that* * it represents has non-zero image under H("*) precisely when the original linear map ` :V n! Q is non-zero. On recalling that Hn Der(MX , Q; ") ~=Der(MX , Q; ") ~= Hom (Q(MX ), Q), we see the two descriptions agree. Corollary 2.7. Let f :X ! Y be a map between simply connected complexes of fi- nite type with X finite. The rationalized nth evaluation subgroup Gn(YQ, XQ, fQ* *) ~= Gn(Y, X, f) Q of the map f is isomorphic to the image of the induced homomor- phism H("*): Hn Der(MY , MX ; Mf) ! Hn Der(MY , Q; ") . for n 2. This identification of the rationalized evaluation subgroups of the map f can* * be conveniently phrased in a way comparable to the F'elix-Halperin description of * *the rationalized Gottlieb groups: An evaluation subgroup element of the minimal mod* *el Mf: MY ! MX , with MY = (W ), is a linear map ` :W n! Q that extends to an Mf-derivation ` :MY ! MX satisfying dY ` = (-1)n`dX . Whenever such a ` 2 Dern(MY , Q; ") is non-zero, it is a non-zero element in the image of H("*). In view of the preceding remarks, we introduce the following vocabulary and notation. Definition 2.8. Suppose OE: A ! B is a map of DG algebras. We define the evaluation subgroup of OE as the image of the map H("*): Hn Der(A, B; OE) ! Hn Der(A, Q; ") . We denote it by Gn(A, B; OE). In the special case in which A = B and OE = 1B , * *we refer to the Gottlieb group of B, and use the notation Gn(B). From the previous discussion, we see that Gn(MY , MX ; Mf) ~=Gn(Y, X; f) Q and Gn(MX ) ~=Gn(X) Q. RATIONALIZED EVALUATION SUBGROUPS 11 Proof of Theorem 2.1.We will define vector space isomorphisms , f, X , and Y to give the following equivalence of commutative squares: H((Mf)*) Hn Der(MX8,8MX ; 1) _______//_Hn Der(MY8,8MX ; Mf) qqqq | fppppp | qqq~=qq |H("*)| pppp~=p || ß map (X, X;q1) _Q__|___//_ß map (X, Yp; f) Q H("*)|| n (f*|)# 1 n | | | | | | | |!# | || fflffl| || fflffl| !#| Hn Der(MX , Q; ")_____|______//Hn Der(MY , Q; ") | q88q H(("M|f)*) p77p | Xqqq | Yppp | qq~=q | ppp~=p fflffl|qq fflffl|pp ßn(X) Q_______f#_1_______//ßn(Y ) Q We obtain f as the rationalization of a natural homomorphism 0f:ßn(map (X, Y ; f)) ! Hn Der(MY , MX ; Mf) , which we now define. A representative of a homotopy class ff 2 ßn(map (X, Y ; f* *)) determines, via the exponential correspondence, a map F :Sn x X ! Y that satisfies F O i2 = f, where i2: X ! Sn x X is the inclusion. The map F is often called an affiliated map for (a representative of) ff. Passing to minimal model* *s, we obtain a map MF :MY ! MSn MX with Mi2OMF = Mf (equals, not just up to DG homotopy_see Proposition A.2). Now Sn is a formal space, which means there is a quasi-isomorphism of DG algebras _ :MSn ! H*(Sn; Q). In turn, this gives a quasi-isomorphism _ 1: MSn MX ! H*(Sn; Q) MX . Write H*(Sn; Q) as (sn)=(s2n) if n is even, or as (sn) if n is odd. Given Ø 2 MY , * *we may write (_ 1) O MF (Ø) = 1 Mf(Ø) + sn `G (Ø), thus defining a linear map `G :MY ! MX that reduces degree by n. A standard check_using the fact that (_ 1)OMF (Ø) is a DG algebra map_shows that `G is an Mf-derivation that is a ffiMf -cycle. Define 0f(ff) = [`G ] 2 Hn Der(MY , MX ;* * Mf) . To show 0f(ff) is well-defined, suppose that g1 ' g2: Sn ! map (X, Y ; f) are two representatives of ff with affiliated maps F, G: Sn x X ! Y respectively. T* *hen the homotopy K :Sn x I ! map(X, Y ; f) from g1 to g2 gives a homotopy H :Sn x X x I ! Y , from F to G, by setting H(s, x, t) = K(s, t)(x). Further, since K i* *s a based homotopy, the homotopy H satisfies H O i = J,where i denotes the inclusion i(x, t) = (*, x, t) and J denotes the homotopy J(x, t) = f(x) that is stationar* *y at f. Now the homotopy H gives a DG homotopy MH :MY ! MSn MX (t, dt) between minimal models for F and G. Translating the restriction on H into minim* *al model terms allows us to assume that MH has form such that X X (3)(_ 1 1)OMH (Ø) = 1 Mf(Ø) 1+ sn ffi(Ø) ti+ sn fii(Ø) tidt, i 0 i 0 for an element Ø 2 MY . This translation is intuitively plausible, but its just* *ification requires some technical details, which we provide in Proposition A.2. Since the* * DG homotopy MH is from MFPto MG , then at t = 0 we have ff0(Ø) = `F (Ø), and from t = 1, we have i 0ffi(Ø) = `G (Ø). To establish well-definedness, we must 12 GREGORY LUPTON AND SAMUEL BRUCE SMITH show these differ by a boundary in Der(MY , MX ; Mf). To this end, use (3) to write separate expressions for (_ 1 1) O MH (ØØ0) and (_ 1 1) O MH (Ø)(_ 1 1) O MH (Ø0). Since a DG homotopy is an algebra map, these expressions agre* *e. By equating them and collecting like terms we obtain equations 0| 0 fii(ØØ0) = (-1)n|Ø|Mf(Ø)fii(Ø0) + (-1)|Ø fii(Ø)Mf(Ø ) for each i 0. By substituting fli(Ø) = (-1)|Ø|fii(Ø) for Ø 2 MY , we obtain derivations fli 2 Dern+1(MY , MX ; Mf). On the other hand, use (3) to write separate expressions for (_ 1 1) O MH (dØ) and d(_ 1 1) O MH (Ø), with * *the latter obtained by applying d to both sides of (3). Since a DG homotopy respects differentials, these expressions agree. By equating them and collecting like te* *rms we obtain equations fii(dØ) = (-1)|Ø|(i + 1)ffi+1(Ø) + (-1)ndfii(Ø) for each i 0. With the previous substitution, this gives dfli(Ø)-(-1)n+1flid(* *Ø) = (-1)n+1(i + 1)ffi(Ø), that is, n+1 1 ffi+1(Ø) = ffiMf (-1) ____if+l1i(Ø) P for each i 0. It follows that the difference of derivations `G - `F = i 1ff* *i is a ffiMf -boundary in Der(MY , MX ; Mf). Hence 0fis well-defined. It is not difficult to show that 0fis a homomorphism. But since the proof requires some technical notions from rational homotopy theory, we postpone it to the appendix (Proposition A.3). As we stated before, the map of vector spaces f is now obtained as the rationalization of the (group) homomorphism 0f. The map is defined in the same way, specializing to the case in which Y = X and f = 1X . The maps X and Y are the standard minimal model identification of the rational homotopy groups of a space, as discussed in Remark 2.2. Next, we show f is surjective. Denote by [SnQx XQ, YQ]fQ the subset of the s* *et of homotopy classes of maps SnQx XQ ! YQ consisting of classes represented by a map that restricts to fQ on XQ. By Theorem 2.3, we identify ßn(map (X, Y ; f)) Q with ßn(map (XQ, YQ; fQ)), and hence with [SnQx XQ, YQ]fQ. Now suppose given [`] 2 Hn Der(MY , MX ; Mf) . Use ` to define OE(Ø) = 1 Mf(Ø) + sn `(Ø) for Ø 2 MY . Since ` is an Mf-derivation that is a cycle, this defines a DG algebra map OE: MY ! H*(Sn; Q) MX . Now lift OE through the surjective quasi- isomorphism _ 1 as in [FHT01 , Lem.12.4], to obtain a map eOE:MY ! MSn MX that satisfies (" . 1) O eOE= Mf: MY ! MX . By the standard correspondence between maps of minimal models and maps of rational spaces, this gives a map F :SnQx XQ ! YQ that satisfies i2 O F ~ fQ :XQ ! YQ. Using, for example, [FHT01 , Th.9.7], we can adjust F into a homotopic map F 0:SnQx XQ ! YQ that satisfies i2 O F 0= fQ :XQ ! YQ, so that F 0represents a class of [SnQx XQ, YQ]* *fQ. As described at the start of this paragraph, F 0corresponds to a homotopy class ff 2 ßn(map (X, Y ; f)) Q. Evidently, we have f(ff) = `. Finally, we show f is injective. Since f is a vector space homomorphism, it is sufficient to show that ff 2 ßn(map (X, Y ; f)) Q is zero whenever f(ff) * *= 0. Using the identification of the previous paragraph, let G: SnQx XQ ! YQ be an RATIONALIZED EVALUATION SUBGROUPS 13 affiliated map for ff. Suppose that `G = ffi(j) for j 2 Dern+1(MY , MX ; Mf). Using j, define a map MH :MY ! H*(Sn; Q) MX (t, dt) by (Ø) = 1 Mf(Ø) 1 + sn `G (Ø) (1 - t) + sn j(Ø) dt. A routine check verifies that is a DG algebra map. Furthermore, it is a DG homotopy from MG to the map E :MY ! H*(Sn; Q) MX given by E(Ø) = 1 Mf(Ø). Now this latter map is a Sullivan model of an affiliated map for 0 2 ßn(map (X, Y ; f)) Q. It follows that ff = 0. We observe that, strictly speakin* *g, we have not justified that the homotopy between the maps SnQxXQ ! YQ is relative to XQ, which corresponds to the homotopy between the maps Sn ! map(XQ, YQ; fQ) being based. However, a based map from a sphere is null-homotopic if and only if it is based null-homotopic (cf. [Spa89, p.27]). It follows that ff = 0, and thu* *s f is injective. Commutativity of the cube diagram follows from the naturality of the homo- morphism 0f. By naturality, we mean the following: Suppose given maps of spaces f :A ! B and g :B ! C. The we have induced maps of function spaces g*: map(A, B; f) ! map (A, C; g O f) and f* :map(B, C; g) ! map (A, C; g O f). For either case we obtain a commutative square involving gOf. Namely, we have H (Mg)* O f = gOfO(g*)# in the first case, and H (Mf)* O g = gOfO(f*)# in the second case, as is easily checked. Since f is obtained from 0fby localiza* *tion, the isomorphism f has the same naturality property. This is sufficient to conc* *lude that the top, bottom, left, and right faces of the cube commute. For the evalua* *tion map ! :map(X, Y ; f) ! Y can be identified as i*: map(X, Y ; f) ! map(x0, Y ; y* *0), where x0 2 X and y0 2 Y denote basepoints, and i: x0 ! X inclusion of the base- point. Likewise for the evaluation map ! :map(X, X; 1) ! X, and then f :X ! Y can be identified with f*: map(x0, X; x0) ! map (x0, Y ; y0) (cf. also Remark 2* *.2). Finally, the front and rear faces commute because the squares (1) and (2) are c* *om- mutative. We finish this section with two more immediate consequences of Theorem 2.1. The first retrieves the basic result of Thom, in the rational homotopy setting. Corollary 2.9. ([Tho57 , Th.2]) Let Y = K(V, m) be an Eilenberg-Mac Lane space, for V a finite dimensional (ungraded) rational vector space. If X is a finite CW complex, and f :X ! Y is any map, then ßn map (X, Y ; f) ~=Hm-n (X; V ). Proof.The minimal model for Y is V *with zero differential, where V *denotes the dual vector space of V . It follows easily that Hn Der(MY , MX ; Mf) ~= Hom (V *, Hm-n (MX )) ~=Hom (V *, Hm-n (X; Q)) ~=Hm-n (X; V ). Notice that this result_with the remarks on rationalization preceding Theo- rem 2.3_easily extends to yield the rational homotopy type of map (X, Y ; f), in case Y is a rational H-space. That is, a space whose rationalization is an H-sp* *ace. For if Y is a rational H-space, then so too is map (X, Y ; 0), which is homoto* *py equivalent_via translation by f_to map (X, Y ; f). Now a rational H-space is de- termined up to rational homotopy type by its rational homotopy groups. Further- more, a rational H-space is a product of rational Eilenberg-Mac Lane spaces. He* *nce map(X, Y ; f) has the rational homotopy type of a product of spaces map(X, Yi; * *0), for Yi a rational Eilenberg-Mac Lane space as in the corollary. We give a further consequence of Theorem 2.1 concerning the rational homotopy groups of certain function spaces. Define an F0-space to be a finite simply con* *nected 14 GREGORY LUPTON AND SAMUEL BRUCE SMITH complex with finite dimensional rational homotopy (a rationally elliptic space)* * such that Hodd(X, Q) = 0. This type of space features in the following well-known conjecture of Halperin (cf. [FHT01 , p.516]): Conjecture 2.10. Suppose X is an F0-space. Then any fibration X ! E ! B of simply connected spaces is TNCZ, that is, the fibre inclusion j :X ! E induces a surjection on rational cohomology. The rational homotopy of map (X, X; 1) for X an F0-space is directly related to Conjecture 2.10 (see [Mei82] for details). The following result extends [Mei* *82, Prop.2.6] (also compare [Gri94, Cor.4.6] and [Hau93 , Th.B]): Corollary 2.11. Let f :X ! Y be a map between F0-spaces. Then for r 1, ß2r(map (X, Y ; f)) Q ~=Der2r(H*(Y, Q), H*(X, Q); H(f)). Proof.The argument given by Grivel_for the case in which Y = X and f = 1X _ can be used word for word to show H2r(Der(MY , MX ; Mf)) ~=Der2r(H*(Y, Q), H*(X, Q); H(f)). The result now follows from Theorem 2.1. 3.Derivation Spaces and Long Exact Sequences In this section, we identify the rationalized long exact homotopy sequences of the maps f :X ! Y and f*: map(X, X; 1) ! map (X, Y ; f), and hence the ratio- nalized G-sequence of f. Our identifications flow from the following observatio* *n: Suppose given vector space homomorphisms OEn :An ! Bn for each n. Then, up to equivalence, there is a unique way to fit these into a three-term long exact se* *quence. Since we rely on this observation for our basic results, we make a formal state* *ment of the fact. Lemma 3.1. Suppose given equivalences of vector space homomorphisms OEn An _____//Bn ~=|ffn| ~=fin|| fflffl| fflffl| A0n_OE0_//B0n n for each n. Then for any two long exact sequences of vector spaces containing t* *he OEn and the OE0nthus @n+1 OEn pn . ._.___//Cn+1____//An____//_Bn____//Cn____//__. . . ___ | | _____ ~=fln+1______ffn~=|~fin=| ~=fln____ fflffl____fflffl|fflffl|@fflffl____0n+1p0 . ._.___//C0n+1___//A0nOE0//_B0n___//C0n___//.,. . n n there exist isomorphisms fln :Cn ! C0nthat make the long exact sequences equiva- lent. Proof.We define a map fln as follows: Decompose Cn and C0nas Cn ~=im(pn) Dn and C0n~=im (p0n) D0n, where Dn and D0nare complements. For x = pn(y) 2 im(pn), define fln(x) = g0nO fin(y). For x 2 Dn, set fln(x) = y0, where y0 2 D0* *nis RATIONALIZED EVALUATION SUBGROUPS 15 such that @0n(y0n) = ffn-1 O @n(x). It is straightforward to check that fln is * *a well- defined isomorphism. Indeed, it restricts to give isomorphisms im(pn) ~=im (p0* *n) and Dn ~=D0n. The required commutativity properties follow immediately. There is nothing remarkable in this observation. It is important, nonetheles* *s, since it allows us to choose descriptions of the long exact homotopy sequences * *that we need in whatever way is most convenient for our purposes. We begin with the long exact homotopy sequence of the map f*: map(X, X; 1) ! map(X, Y ; f). From Theorem 2.1, we see that the map this induces on rational homotopy groups can be identified with H (Mf)* . Since this map is a homomor- phism induced on homology by a map of chain complexes, there is a standard way to fit it into a long exact sequence, which we now describe. Given a map OE: A ! B of chain complexes, define a relative chain complex Rel*(OE) as follows: Reln(OE) = An-1 Bn, with differential ffiOEof degree -1 * *given by ffiOE(a, b) = ffiA (a), ffiB (b) - OE(a) . Now define chain maps J :Bn ! Re* *ln(OE) and P :Reln(OE) ! An-1 by J(b) = (0, b) and P (a, b) = a. On passing to homology, we obtain a long exact sequence of the following form: H(P) H(OE) H(J) . ._.__//_Hn+1(Rel(OE))__//Hn(A)____//_Hn(B)____//Hn(Rel(OE))__//. .,. in which Hn(Rel(OE)) denotes the nth homology of the chain complex Rel*(OE). We refer to this long exact sequence as the long exact homology sequence of the ma* *p OE. Furthermore, this construction is natural. For suppose given a commutative squa* *re of DG vector spaces (4) A __OE_//_B ff|| fi|| fflffl|Offlffl|E0 A0_____//B0. Then the obvious map (ff, fi): Rel*(OE) ! Rel*(OE0) is a chain map that satisfi* *es (ff, fi)J = J0fi and ffP = P 0(ff, fi). Thus we obtain a homology ladder H(P) H(OE) H(J) . ._.__//Hn+1(Rel(OE))___//_Hn(A)____//_Hn(B)____//_Hn(Rel(OE))__//. . . H(ff,fi)|| |H(ff)| |H(fi)| |H(ff,fi)| fflffl|H(P0) fflffl|H(OE0fflffl|)H(J0) fflffl| . ._.__//Hn+1(Rel(OE0))__//Hn(A0)____//Hn(B0)____//Hn(Rel(OE0))__//. . . In particular, we can apply this construction to the map of chain complexes (5) (Mf)*: Der(MX , MX ; 1) ! Der(MY , MX ; Mf) induced by the minimal model Mf: MY ! MX of the map f :X ! Y . To display the long exact sequence that appears in the following result, we c* *on- dense some of our notation as in the proof of Theorem 2.1. We use Der(MX ) to denote Der(MX , MX ; 1), and DerM(Y, X; f) to denote Der(MY , MX ; Mf). Theorem 3.2. The long exact sequence induced by f*: map(X, X; 1) ! map(X, Y ; f) 16 GREGORY LUPTON AND SAMUEL BRUCE SMITH on rational homotopy groups is equivalent to the long exact homology sequence of the map (5). Specifically, this is a long exact sequence H(J) * . ._.__________//Hn+1(Rel((Mf)B))C_ ____________H(P)______________________________ GFfflffl| H((Mf)*) H(J) Hn Der(MX ) __________//_Hn Der M(Y, X; f)___________//. . . H(J) * . ._.___________//H3(Rel((Mf)B))C_ ____________H(P)______________________________ GFfflffl| H((Mf)*) H2 Der(MX ) ___________//H2 DerM(Y, X; f) in which Rel*((Mf)*) is the relative chain complex of the map (5), as described above. Proof.By Theorem 2.3 we may assume f :X ! Y is map of rational spaces. Theorem 2.1 gives equivalences of vector space maps (f*)# ßn map (X, X; 1)________________//ßn map (X, Y ; f) 1 ~=|| ~=||f fflffl| fflffl| Hn Der(MX , MX ; 1) __________//_Hn Der(MY , MX ; Mf) , H (Mf)* for each n 2. The top horizontal maps are contained in the long exact sequence induced by f*: map(X, X; 1) ! map (X, Y ; f) on rational homotopy groups. The bottom horizontal maps are contained in the long exact homology sequence of the map (5). From Lemma 3.1, these sequences are equivalent. Remark 3.3. When we refer to the long exact homotopy sequence of a map, we mean this in the sense of [Hil65, Chaps.3,4]: Recall that given a map f :X ! Y , this sequence is as follows: . .!.ßn(X) f#-!ßn(Y ) ! ßn(f) ! ßn-1(X) ! . .!.ß2(X) f#-!ß2(Y ). If f is the inclusion of a subspace, then the groups ßn(f) are just the usual h* *omotopy groups of a pair. Generally, ßn(f) is defined as homotopy classes of pairs (g1,* * g2) such that the diagram g1 Sn-1"_`____//X | | | |f fflffl|g2 fflffl| CSn-1 _____//Y commutes. Since ß2(f) is not necessarily abelian and we are interested in rati* *o- nalizing this sequence, we stop at ß2(Y ). On the other hand, one can convert f into a fibration and use the corresponding long exact sequence in homotopy. Eit* *her approach suits our purposes and indeed the same sequence of homotopy groups RATIONALIZED EVALUATION SUBGROUPS 17 results from either. From the above, we see that if F denotes the homotopy fibre of the map f*: map(X, X; 1) ! map(X, Y ; f), then for n 2 we have ßn+1(f*) Q ~=ßn(F) Q ~=Hn+1(Rel((Mf)*)), where Rel*((Mf)*) is the relative chain complex of the map (5). The preceding result specializes to give a description of the long exact sequ* *ence induced by a general map on rational homotopy groups. The minimal model Mf: MY ! MX of the map f :X ! Y also induces a map of chain complexes (6) (Mf)*: Der(MX , Q; ") ! Der(MY , Q; "). Theorem 3.4. The long exact sequence induced by f :X ! Y on rational ho- motopy groups is equivalent to the long exact homology sequence of the map (6). Specifically, this is a long exact sequence H(J) * . ._._________//_Hn+1(Rel((Mf)B))C_ _____________H(P)______________________________ GFfflffl| H((Mf)*) H(J) Hn Der(MX , Q; ")___________//Hn Der(MY , Q; ")__________//_. . . H(J) * . ._.__________//_H3(Rel((Mf)B))C_ _____________H(P)______________________________ GFfflffl| H((Mf)*) H2 Der(MX , Q; ")___________//H2 Der(MY , Q; ") in which Rel*((Mf)*) is the relative chain complex of the map (6). Proof.Argue exactly as in the proof of Theorem 3.2, using Theorem 2.1 and Lemma 3.1. Remark 3.5. There is already a standard way to describe the long exact sequence* * in- duced by a map on rational homotopy groups, using minimal models. This uses the notion of a so-called K-S model of the map Mf: MY ! MX [FHT01 , Sec.15(d)]. The description we give above, however, is better suited to our purposes. Note * *that if F denotes the homotopy fibre of the map f :X ! Y , then for n 2 we have ßn+1(f) Q ~=ßn(F ) Q ~=Hn+1(Rel((Mf)*)), where Rel*((Mf)*) is the relative chain complex of the map (6). It is perhaps i* *nter- esting to compare the description given in Theorem 3.4 to the standard descript* *ion of the long exact sequence in rational homotopy groups of a fibration. We now turn our attention to the G-sequence, and identify it within our curre* *nt framework. Suppose given a DG algebra map OE: A ! B. Starting from this map, we can construct the following commutative square of DG vector spaces: * Der*(B, B; 1)_OE_//Der*(A, B; OE) "*|| |"*| fflffl|OcE* fflffl| Der*(B, Q; ")____//Der*(A, Q; "). 18 GREGORY LUPTON AND SAMUEL BRUCE SMITH In this diagram, " denotes the augmentation of either A or B, and we have used a decoration to distinguish the lower horizontal map from the upper. On passing to homology and using the naturality of the relative chain complex construction, we obtain the following homology ladder: H(J) H(P) H(OE*) . ._.___//Hn+1(Rel(OE*))__//Hn Der(B, B; 1)___//_Hn Der(A, B; OE) . . . |H("*,"*)| H("*)|| |H("*)| H(J) fflffl| H(Pb) fflffl| H(cOE*) fflffl| . ._.___//Hn+1(Rel(cOE*))_//Hn Der(B, Q; ")____//Hn Der(A, Q; ") . . . for n 2. We supplement Definition 2.8 with the following vocabulary. Definition 3.6. Suppose OE: A ! B is a map of DG algebras. For n 3 we define the nth relative evaluation subgroup of OE as the image of the map * H("*, "*): Hn Rel(OE ) ! Hn Rel(cOE*) . We denote it by Greln(A, B; OE). Then the image of the upper long exact sequence in the lower, of the ladder above, gives a (not necessarily exact) sequence H(Jb) . ._._______//Greln+1(A,BB;COE)_ _______H(Pb)________________ GFfflffl| H(cOE*) H(Jb) Gn(B) _____//Gn(A, B; OE)__//_Greln(A,BB;COE)_ _______H(Pb)________________ GF fflffl| H(Jb) . . . . ._._______//_Grel3(A,BB;COE)_ _______H(Pb)________________ GFfflffl| H(cOE*) G2(B) _____//G2(A, B; OE) We refer to this sequence as the G-sequence of the map OE: A ! B. All of the above can be applied to the minimal model Mf: MY ! MX of the map f :X ! Y . By doing so, and then collecting together previous results, we obtain the following result. Theorem 3.7. The rationalization of the G-sequence of the map f :X ! Y , as far as the term G2(Y, X; f), is equivalent to the G-sequence of the correspondi* *ng map of Sullivan models. Proof.Starting from the cube displayed in the proof of Theorem 2.1, we extend each of the four left-to-right maps into their respective long exact sequences.* * This is then completed into an equivalence of ladders, by defining isomorphisms fln * *and cflnto give a commutative square ßn(f*)_fln_//Hn Rel(OE*) !# || |H("*,"*)| fflffl|cfln |fflffl ßn(f) _____//Hn Rel(cOE*) RATIONALIZED EVALUATION SUBGROUPS 19 for each n 3. These isomorphisms are defined as in Lemma 3.1, using the top and bottom faces of the cube. The one commutativity relation that needs checkin* *g, namely that of the displayed square, follows easily from the commutativity of t* *he adjacent and parallel squares, together with the way in which the fln and cflna* *re defined. The result now follows, since whenever one has such an equivalence of ladders, the equivalence restricts to give an equivalence of the corresponding sequences* * of images. In particular, we obtain the companion result to Corollary 2.6 and Corollary * *2.7. Corollary 3.8. Let f :X ! Y be a map between simply connected complexes of finite type with X finite. The rationalized nth relative evaluation subgroup Greln(YQ, XQ, fQ) ~=Gn(Y, X, f) Q of the map f is isomorphic to the image of the induced homomorphism * H("*, "*): Hn Rel((Mf) ) ! Hn Rel(Md*f) for n 3. Remark 3.9. We comment on the low-end terms in the G-sequence. In Theorem 3.4 and Theorem 3.2 we terminate our long exact sequences at the terms corresponding to ß2 map (X, Y ; f) and ß2(Y ) respectively. This is because we need a simply* * con- nected hypothesis to ensure our combination of rationalization and minimal model techniques remains valid. As a result, our algebraic description of the rationa* *lized G-sequence terminates at the term corresponding to G2(X, Y ; f). Now in Theo- rem 3.4, we require X to be simply connected and finite. As is well-known, this implies G2i(X) Q = 0 for each i. Therefore, under our hypotheses, the rationali* *zed G-sequence of a map f :X ! Y should be considered as 5-term (not necessarily exact) sequences 0 ___________//_G2n(Y, X; f) __Q___//_Grel2n(Y,BX;Cf)_ Q _______________________________________ GFfflffl| G2n-1(X) Q ____//_G2n-1(Y, X; f) __Q__//Grel2n-1(Y, X; f)___Q//0 for n 2. Our algebraic description of the rationalized G-sequence given by Theorem 3.7 includes all these 5-term sequences. The "sporadic" low-end term G2(Y, X; f) Q is best computed by using its characterization given in Corol- lary 2.7. Before turning to some applications of our algebraic description of the ratio* *nal- ized G-sequence, we give one more description of a long exact homotopy sequence. Whilst not strictly necessary for our purposes, it is nonetheless interesting. We will use map*(X, Y ; f) to denote the based mapping space component. Recall that we have the evaluation fibration sequence (7) map *(X, Y ; f)___//map(X, Y ;_f)!_//Y. We will describe the long exact sequence on rational homotopy groups induced by this fibration. Recall that we have the augmentation ": A ! Q for a DG algebra A. Let eA denote the augmentation ideal, that is, the kernel of ". Given a DG algebra map 20 GREGORY LUPTON AND SAMUEL BRUCE SMITH OE: A ! B, let eOE:A ! eBbe the DG algebra map which agrees with OE in positive degrees and vanishes in degree zero. A DG algebra map OE: A ! B together with the short exact augmentation sequence 0_____//eB_i_//_B_"__//Q____//0 of DG algebras gives rise to the short exact sequence of DG vector spaces i* "* 0_____//Der*(A, eB;_eOE)//_Der*(A, B;_OE)//_Der*(A, Q;_")//_0. This in turn gives a long exact sequence on homology, in the usual way, of the * *form (8) H(i*) H("*) . ._._*_//Hn Der(A, eB; eOE)//_Hn Der(A, B; OE)__//Hn Der(A, Q; ") . . . for n 2. Call this sequence the long exact derivation homology sequence of the DG algebra map OE: A ! B. To display the sequence that appears in the following result, we have once ag* *ain condensed some of our notation. We use Der M(Y, X; f) and Der(MY , Q), to denote Der(MY , MX ; Mf), respectively Der(MY , Q; "), as in the proof of Theo- rem 2.1. In addition, here we use DerM(Y, eX; ef) to denote Der(MY , gMX; gMf). Theorem 3.10. The long exact sequence induced by the evaluation fibration (7) on rational homotopy groups is equivalent to the long exact derivation homology sequence (8) of the map Mf: MY ! MX . Specifically, this is a long exact sequen* *ce H("*) . ._.__________//_Hn+1BDer(MYC,_Q) _____________*_____________________________ |GFfflffl H(i*) H("*) Hn Der M(Y, eX; ef)___//_Hn Der M(Y, X; f)___________//. . . H("*) . ._.___________//_H3 Der(MYB,CQ)_ _____________*_____________________________ |GFfflffl H(i*) H("*) H2 DerM(Y, eX; ef)_____//H2 DerM(Y, X; f)______//H2 Der(MY , Q) . Corollary 3.11. Let X and Y be simply connected spaces with X finite. Then the rational homotopy groups of the based function space map *(X, Y ; f) are given * *by ßn map *(X, Y ; f) Q ~=Hn Der(MY , gMX; gMf) for n 2. 4. Examples, Computations, and Further Consequences We illustrate the effectiveness of the framework established in the previous * *two sections with examples. First, we give a composite example that includes specif* *ic computation of many of the ingredients of the above. Our example is one in which the G-sequence of a map fails to be exact (after rationalization) at each of th* *e three types of term that occur. RATIONALIZED EVALUATION SUBGROUPS 21 We begin with some notational conventions. Suppose that (A, dA ) and (B, dB ) are minimal algebras, with A = (W ) and B = (V ) for suitable graded vector spaces W and V . Let OE: A ! B be a fixed DG algebra map. Since any linear map W ! B extends in a unique way to a OE-derivation, we can view the space Hom *(W, B) of negative degree linear maps as a subspace of Der*(A, B; OE). (Of course, in degree 1 we must restrict to cycles!) With this view Hom *(W, B) = Der*(A, B; OE) as graded spaces although the differential depends on the deriva* *tion structure. This point of view, whereby a derivation is specified on generators * *and then extended to the whole algebra, is one that we will invariably adopt in any practical calculation. Now suppose given a basis {w1, w2, w3, . .}.for W and an element P 2 B with |P | < |wi| we will write P @wi for the OE-derivation carrying wiPto P and vani* *shing on the other wj. Thus any derivation can be expressed as a sum iPi@wi. When B = Q, we write w*jrather than 1@wj for the derivation dual to wj. Example 4.1. Let f = (f1, f2): HP 2! S8 x HP 4be the map with coordinate functions f1: HP 2! S8 obtained by pinching out the bottom cell and f2: HP 2! HP 4the inclusion. We will use our framework from above to compute various terms from the long exact sequences corresponding to Theorem 3.2 and Theorem 3.4. Denote HP 2by X and S8xHP 4by Y , thus f :X ! Y . Our computation will show, using Theorem 3.7, that the G-sequence of f is non-exact at the terms G4(Y, X), Grel8(Y, X), and G11(X). First, MX = (x4, x11), with differential given on generators by d(x4) = 0, a* *nd d(x11) = x34, and MY = (y8, y15, y4, y19) with differential d(y8) = 0, d(y15) * *= y28, d(y4) = 0, and d(y19) = y54. In both models, subscripts denote degrees. Then the Sullivan model of f, which we denote by OE: MY ! MX , is given on generators by OE(y8) = x24, OE(y15) = x4x11, OE(y4) = x4, and OE(y19) = x24x11. For degree reasons, Deri(MX , MX ; 1) = 0 unless i = 3, 4, 7 or 11. Furthermo* *re, Der*(MX , MX ; 1) is spanned by the derivations x24@x11, x*4, x4@x11, and x*11of degree 3, 4, 7, and 11 respectively. An easy computation reveals that ffi(x*4)* * = -3x24@x11, but that x4@x11 and x*11are both (non-exact) cocycles. It follows fr* *om Theorem 2.1 that ( Q if i = 7, 11 ßi map(X, X; 1) Q ~=Hi Der(MX , MX ; 1) = 0 otherwise Further, it is evident that ( Q if i = 4, 11 ßi(X) Q ~=Hi Der(MX , Q; ") = 0 otherwise with the non-zero cohomology in degrees 4 and 11 generated by cocycles x*4and x*11, respectively. Given these generators, we see that H("*): Hi Der(MX , MX ; 1) ! Hi Der(MX , Q; ") , that is, the homomorphism !# Q: ßi map(X, X; 1) Q ! ßi(X) Q induced by the evaluation map on rational homotopy groups, is an isomorphism in degree 11 and is zero in all other degrees. It follows from Theorem 2.1_see Corollary 2.6_ that Gi(MX ) = 0 other than in degree 11, where we have G11(MX ) ~=Q. Up to this point, our observations are both well-known, and also easily obtained b* *y a number of standard methods. 22 GREGORY LUPTON AND SAMUEL BRUCE SMITH We now show that the rationalized G-sequence is non-exact at the G11(MX ) term. Recall that this term of the G-sequence, together with its adjacent terms* *, is obtained from the diagram * Rel12(OE*)_P_//_Der11(MX , MX ;_1)OE//_Der11(MY , MX ; OE) |("*,"*)| |"*| "*|| fflffl|bP fflffl| OcE* fflffl| Rel12(cOE*)____//Der11(MX , Q;_")____//Der11(MY , Q; "), by passing to homology and then considering the image of the top sequence in the bottom. A brute force calculation will display the result, but we opt to ar* *gue at a more general level so as to indicate some reason for non-exactness. It is evident that H cOE*O H("*)([x*11]) = 0 2 H11 Der(MY , Q; ") _indeed, this latter term is zero, since it is isomorphic to ß11(Y ) Q = 0. The key point for non- exactness here, however, is that in the top sequence we have H(OE*)([x*11]) 6= * *0 2 H11 Der(MY , MX ; OE) . In fact, a straightforward check shows that OE*(x*11) = x4@y15+ x24@y19. Since Der12(MY , MX ; OE) = 0, there are no non-zero boundaries in degree 11 and hence H(OE*)([x*11]) 6= 0. Consequently, [x*11] cannot be in * *the image of H(P ) in the top sequence. Therefore, since H("*) is an isomorphism in degree 11, H("*)([x*11]) = [x*11] cannot be in the image of H(Pb) O H("*, "*). * * It follows from these facts that [x*11] 2 G11(MX ) is a non-zero element in the ke* *rnel of H(cOE*) and yet is not in the image of H(Pb): Grel12(MY , MX ; OE) ! G11(MX * *). Next consider the term G4(MY , MX ; OE): Before passing to homology, the rele- vant diagram is the following: * J Der 4(MX , MX ; 1)OE_//_Der4(MY , MX ;_OE)_//Rel4(OE*) |"*| "*|| |("*,"*)| fflffl| cOE* fflffl| Jb fflffl| Der4(MX , Q; ")______//_Der4(MY , Q;_")__//Rel4(cOE*), The derivation ` = y*4+ 5x4x11@y19 2 Der4(MY , MX ; OE) is a cocycle, as is eas* *ily checked. Under H("*): H4 Der(MY , MX ; OE) ! H4 Der(MY , Q; ") , we have H("*)([`]) = [y*4] 6= 0. Since [y*4] = H(cOE*)([x*4]), it follows that H(Jb)([* *y*4]) = 0. As we noted above, however, G4(MX ) = 0. Therefore, [y*4] 2 G4(MY , MX ; OE) is a non-zero element in the kernel of H(Jb): G4(MY , MX ; OE) ! Grel4(MY , MX ; O* *E) that is not in the image of H(cOE*): G4(MX ) ! G4(MY , MX ; OE). Finally, consider the term Grel8(MY , MX ; OE): Here, we begin with the follo* *wing diagram: Der 8(MY , MX ; OE)J_//_Rel8(OE*)P//_Der7(MX , MX ; 1) |"*| ("*,"*)|| "*|| fflffl| bJ fflffl|bP fflffl| Der8(MY , Q; ")____//Rel8(cOE*)___//Der7(MX , Q; ") We find that (-2x4@x11, y*8+ 2x11@y19) 2 Rel8(OE*) is a cocycle that has non- zero image in H("*, "*): H8 Rel(OE*) ! H8 Rel(cOE*) . Furthermore, it is evide* *nt that H(Pb) O H("*, "*) (-2x4@x11, y*8+ 2x11@y19) = H(Pb)([(0, y*8)]) = 0_indee* *d, RATIONALIZED EVALUATION SUBGROUPS 23 H7 Der(MX , Q; ") ~=ß7(X) Q = 0. To see that [(0, y*8)] is not in the image * *of H(Jb): G8(MY , MX ; OE) ! Grel8(MY , MX ; OE), we will compute G8(MY , MX ; OE) to be zero. A general derivation fl 2 Der8(MY , MX ; OE) can be written as fl = ~1y*8+ ~2x11@y19, for coefficients ~i2 Q. To find the cocycles of this form, we first observe tha* *t ffi(fl) must vanish on the generators y8 and y4, for degree reasons. On y15, we compute as follows: ffi(fl)(y15)= (d(fl) - (fl)d)(y15) = 0 - fl(y28) = -2fl(y8)OE(y8) = -2~1x24. Thus, if fl is a cocycle, then we must have ~1 = 0. A similar computation shows that ffi(fl)(y19) = ~2x34, and thus that there are no non-zero cocycles in Der8(MY , MX ; OE). In summary, we have computed that ß8 map (X, Y ; f) Q ~=H8 Der(MY , MX ; OE) = 0. This last part of our computation is easily confirmed using the result of Corol- lary 2.11. It follows, of course, that G8(MY , MX ; OE) = 0 and, in particular,* * that [(0, y*8)] is not in the image of H(Jb): G8(MY , MX ; OE) ! Grel8(MY , MX ; OE). Remarks 4.2. The first example of a non-exact G-sequence, given in [PW97 ], was in dimension one. A higher dimensional example was produced later in [PSW98 ]. With the approach illustrated in the above example, it is straightforward to pr* *oduce higher dimensional examples of non-exact rationalized G-sequences. Observe that non-exact ordinary G-sequences are produced as a result, since if a sequence of abelian groups is not exact after tensoring with Q then it was not exact to beg* *in with. Thus, non-exactness rationally implies non-exactness integrally. Although the G-sequence in general is non-exact, there are certain situations* * in which it is perfectly well behaved, at least after rationalization. We now ment* *ion one such situation. We say that a space X is an H0-space if its rational cohomo* *logy algebra is a free graded algebra (exterior algebra on the odd-degree generators tensored with a polynomial algebra on the even-degree generators). Equivalently, we could require that X be an H-space after rationalization, whence such a space is also referred to as a rational H-space. Recall also the definition of an F0-* *space from before Corollary 2.11. Theorem 4.3. Let f :X ! Y be any map from an F0-space X to an H0-space Y that induces the zero homomorphism on rational homotopy groups. Then the rationalized G-sequence splits into short exact sequences 0 _____//Gn+1(Y, X; f) __QJ_//Greln+1(Y, X; f)_P_Q//_Gn(X) _Q___//0 for each n 2. Proof.Our assumption that f# Q = 0 means that the long exact sequence induced by f on rational homotopy groups splits. Furthermore, since Y is an H0-space, we 24 GREGORY LUPTON AND SAMUEL BRUCE SMITH have Gn(Y ) Q = ßn(Y ) Q for each n. It follows that Gn(Y, X) Q = ßn(Y ) Q for each n. From these general considerations, we have short exact sequences (9) 0 _____//Gn+1(Y, X; f) __QJ_//ßn+1(f) Q_P__//ßn(X) Q____//0 for n 2. To sharpen this to the statement of the theorem, we work within our minimal model framework. We use some results of Halperin [Hal77], on the rational homotopy of an F0- space. These state that X has minimal model (MX , dX ) of the form MX = (x1, . .,.xn) (y1, . .,.yn) where |xi| is even, |yj| is odd, dX (xi) = 0 and * *dX (yj) 2 (x1, . .,.xn). Furthermore, the cohomology is evenly graded, and any cocycle in I(y1, . .,.yn), the ideal of MX generated by the yi, is exact. It is well-known* * that the minimal model of an H0-space takes the form MY = (z1, z2, . .)., with triv* *ial differential. The map f :X ! Y has Sullivan minimal model OE: MY ! MX that is determined by the OE(zi). Since each zi is a cocycle, it follows that * *each OE(zi) 2 MX is a cocycle. From the results of Halperin mentioned above, we can write OE(zi) = Øi+ d(,i), for suitable Øi2 (x1, . .,.xn) and ,i2 I(y1, . .,.yn* *). The assumption that f induces zero on rational homotopy groups translates into the further restriction that each Øi is decomposable. The short exact sequences (9), translated into our derivation setting, corres* *pond to short exact sequences H(Jb) H(Pb) 0 ! Gn+1(MY , MX ; OE)___//_Hn+1 Rel(cOE*)__//Hn Der(MX , Q; ") ! 0 We first prove that each map H(Pb) restricts to give a surjection H(Pb): Greln+1(MY , MX ; OE) ! Gn(MX ). The Gottlieb elements of MX are precisely the y*j, dual to the odd-degree gener* *ators of MX . This can be seen from the description of the Gottlieb elements given in Corollary 2.6, together with the facts about the minimal model of X recalled ea* *rlier. Now write OE(zi) = Øi+ d(,i) as above and, for each j, define a derivation X `j = - y*j(,i)@zi2 Der|yj|+1(MY , MX ; OE). P i Then ffi(`j) = - id(y*j(,i))@zi, since the differential in MY is trivial. On* * the other hand, we have * * OE*(y*j)(zi)= y*iOE(zi) = yi Øi+ d(,i) = 0 + yi d(,i) * = -d yi(,i) , with the last step following because y*jis a ffi-cycle. Consequently, (y*j, `j* *) 2 Rel|yj|+1(OE*) is a ffiOE*-cycle. Since H(Pb) O H("*, "*)(y*j, `j) = y*j, it f* *ollows that H(Pb) does restrict to the desired surjection. The map H(Jb) is injective on rational homotopy groups, as we have already observed, and therefore restricts to an injection in the rationalized G-Sequenc* *e. So it only remains to show exactness at the Greln+1(Y, X; f) terms. Suppose that rel H("*, "*)([(`, _)]) 2 kerH(Pb): Gn+1(MY , MX ; OE) ! Gn(MX ) , for some cocycle (`, _) 2 Reln+1(OE*). The fact that H(Pb)([("*(`), "*(_))]) =* * 0 implies that "*(`) = 0. Thus ("*, "*)(`, _) = (0, "*(_)). Now define a derivati* *on RATIONALIZED EVALUATION SUBGROUPS 25 __ __ _ 2 Dern+1(MY_, MX ; OE) by setting _ = _ on generators of MY of degree_n + 1 and___= 0 on all other generators of MY ._It_is easily seen that _ is a cycle. * *Indeed, dX __=_0 on all generators of MY , since__ has non-zero image only in degree ze* *ro, and _dY = 0 since dY = 0. Thus H("*)([_ ]) 2 Gn+1(MY , MX ; OE) satisfies __ __ H(Jb) O H("*)([_ ]) = [(0, Ö _)] = [(0, Ö _)] = H("*, "*)([(`, _)]). That is, ker H(Pb) \ Greln+1(MY , MX ; OE) H(Jb) Gn+1(MY , MX ; OE) and the rationalized G-sequence is exact at each Greln+1(Y, X; f) term. Remark 4.4. Various conditions are known, under which the G-sequence of a map f :X ! Y is exact. For instance, it is exact when f is null-homotopic [LW93 ], * *and when f is a homotopy monomorphism [PW01 ]. The hypotheses of Theorem 4.3 are well-suited for rational homotopy theory. Both types of space are well-know* *n, and it is easy to give examples to which the theorem applies. In fact, for fix* *ed X and Y , the maps to which it applies are classified up to rational homotopy by the decomposable rational cohomology of X that occurs in those (even) degrees in which Y has a generator of rational cohomology. We emphasize that the H0- space Y must be allowed to have polynomial generators in rational cohomology, a* *nd hence be infinite-dimensional, otherwise the theorem reduces to the case in whi* *ch the map f is rationally null-homotopic. Furthermore, the hypothesis that f be z* *ero on rational homotopy groups is necessary. For example, the map f :S4 ! HP 1, given by inclusion of the bottom cell, has a non-exact rationalized G-sequence,* * as is easily confirmed by computations similar to those of Example 4.1 Since the G-sequence of a map f :X ! Y is a boundary sequence, but not usually an exact sequence, it is natural to consider its homology. This gives * *the so-called !-homology of f [LW93 ]. In general, one obtains an !-homology group * *at each of the three types of term. In the following, we restrict our attention to* * the !- homology that occurs at the Gottlieb group term G*(X), denoted Ha!*(X, Y ; f) in [LW93 ]. Thus we consider the sub-quotients of the Gottlieb groups G*(X) defined by Ha!n(X, Y ; f) = ker{f#_:Gn(X)_!_Gn(Y,_X;_f)}_. im{P :Greln+1(Y, X; f) ! Gn(X)} When Y is an H0-space, the rational !-homology of f :X ! Y is related to the the negative derivations on the rational cohomology of X that are induced by derivations on the minimal model. To be precise, define a linear map of degree * *zero * * 'X :H* Der(MX , MX ; 1) ! Der*(H (X; Q), H (X; Q); 1) by the rule 'X ([`])([Ø]) = [`(Ø)], for ` a cycle in Der*(MX , MX ; 1) and Ø a * *cocycle in MX . It is straightforward to check that 'X is well-defined. (cf. [Gri94, Pr* *op.1.6]. In fact, 'X is a morphism of graded Lie algebras.) In the next result, and the example that follows it, we illustrate that the r* *ational- ized G-sequence may be exact at all occurrences of one type of term, while fail* *ing to be exact at the other types of term. In other words, the rational !-homology* * of a map may be zero at one type of term, yet non-zero at the other types of term. Theorem 4.5. Let X be a finite complex for which the map 'X defined above is trivial and let Y be an H0-space. Then Ha!*(X, Y ; f) Q = 0 for any map f :X ! Y . 26 GREGORY LUPTON AND SAMUEL BRUCE SMITH Proof.Since Y is an H0-space, its minimal model MY ~=H*(Y ; Q) has trivial differential. Let OE: H*(Y ; Q) ! MX denote the minimal model of f. For a deriv* *a- tion ` 2 Der*(H*(Y ; Q), MX ; OE), we have ffi(`) = dX `. Using this observati* *on, we obtain a map of chain complexes * * ~: Der*(H*(Y ; Q), MX ; OE) ! Der* H (Y ; Q), H (X; Q); H(OE) , defined by ~(`)(Ø) = [`(Ø)]. Using the preceding observation, together with the free-ness of H*(Y ; Q), it is straightforward to check that ~ induces an isomor* *phism on passing to homology (note that the right-hand term has trivial differential,* * and so is its own homology). Furthermore, the following diagram commutes: H(OE*) H* Der(MX , MX ; 1) _______________//_H* Der(H*(Y ; Q), MX ; OE) 'X || ~=~|| fflffl| fflffl| Der*(H*(X; Q), H*(X; Q); 1)_________*//_Der*H*(Y ; Q), H*(X; Q); H(OE) H(OE) Therefore, the assumption that 'X = 0 implies that the top map H(OE*) in the above diagram is zero. A straightforward diagram chase using the homology ladder that defines the rationalized G-sequence now gives the result. Example 4.6. Following Theorem 4.3 we remarked that the cellular inclusion S4 ! HP 1 does not have an exact rationalized G-sequence. However, it does satisfy the hypotheses of Theorem 4.5, since here X = S4 has the property that * *all derivations of the cohomology algebra are trivial. Remark 4.7. The hypothesis on X in Theorem 4.5, that 'X = 0, deserves some comment. First, we observe that the nature of the hypothesis distinguishes stru* *c- ture at the minimal model level from structure at the cohomology level. This is a distinction that is made in rational homotopy for a wide variety of structure* *s. Next, we observe that this condition is satisfied for many, if not all, F0-spac* *es X. Indeed, Conjecture 2.10_the long-standing conjecture of Halperin concerning F0- spaces_is equivalent to the assertion that all negative-degree derivations on t* *he cohomology algebra of an F0-space are trivial (see [Mei82] for details). Whenev* *er this conjecture is true_and it has been verified in many cases_obviously we have 'X = 0. Therefore, Theorem 4.5 can be compared with Theorem 4.3, as a result with weaker hypotheses, and correspondingly weaker conclusion. Finally, we note that the map 'X makes an appearance in a completely different context, in the work of Belegradek and Kapovitch [BK03 ]. Our last set of results relate directly to Conjecture 2.10. First, we observe* * that for an inclusion of a summand of a product, the G-sequence behaves in a particu* *larly nice way. Since it is no harder to do so, we state and prove this result in the* * integral setting. Proposition 4.8. Suppose that i1: X ! X x B is the inclusion into the first summand. Then the G-sequence of i1 is exact, and furthermore reduces to split short exact sequences _(i2)#_________________________________* *_______________________________ (i1)# uu_______________________________________* *______________________________________________ 0_____//Gn(X)_____//Gn(X x B, X; i2)(p2)#//_ßn(B)//_0, RATIONALIZED EVALUATION SUBGROUPS 27 where p2: X x B ! B is projection onto the second summand and the splitting is induced by inclusion into the second summand i2: B ! X x B. Proof.This follows from results in [LW88b ] (see also [Woo97 ]), but we give a * *brief argument here. First, let X !j E !p B be any fibre sequence. Hilton's excision homomorphism for relative homotopy groups gives an isomorphism ß*(j) ~=ß*(B) [Hil65, Chap.3]. Thus we may view Grel*(E, X; j) as a subgroup of ß*(B) and the G-sequence of the fibre inclusion j as a subsequence the long exact homotopy sequence of the fibration. Now apply this remark to the trivial fibration X i1!X x B p2!B. The inclusion i2: B ! X x B induces a splitting of the long exact homotopy sequence of this trivial fibration in the usual way. The result now follows from the observation* * that (i2)# ßn(B) Gn(X x B, X), as is easily established from the definitions. Of course, this result and its proof can be rationalized, and it is in the ra* *tional setting that we will use it. Conjecture 2.10 concerns fibrations X ! E ! B with fibre an F0-space and arbitrary base. However, it is well-known how to reduce t* *he conjecture to consideration of such fibrations with base an odd-dimensional sph* *ere [Mei82]. Furthermore, in [Lup98 ] it is pointed out that, for such fibrations w* *ith base an odd-dimensional sphere, Halperin's conjecture actually asserts that the fibr* *ation should be trivial. From these remarks, we see that a necessary condition for Conjecture 2.10 to * *be true is that any fibration X ! E ! S2r+1 with X an F0-space must have a fibre inclusion whose G-sequence reduces to the split short exact sequences correspon* *ding via Proposition 4.8 to the inclusion i1: X ! X x S2r+1. Perhaps surprisingly, t* *he converse is true. Theorem 4.9. Let X !j E !p S2r+1 be any fibration with X an F0-space. The following are equivalent: (1) The fibration is rationally TNCZ, that is, j*: H*(E; Q) ! H*(X; Q) is surjective. (2) The rationalized G-sequence of the fibre inclusion reduces to split shor* *t exact sequences _(i2)#_________________________________* *_______________________________ (j)# tt________________________________________* *_______________________________________ 0 ____//_Gn(X) Q____//Gn(E, X; j) (Qp)#//_ßn(S2r+1) __Q_//_0 for each n 2. Proof.The implication (1) ) (2) follows by the remarks preceding the enunciatio* *n. We prove (2) ) (1). Suppose the fibration X ! E ! S2r+1 has minimal model (u)__i__// (u) V, Dß__//_( V, d), where i denotes the inclusion i(u) = u 1 and ß is the projection. The hypothe* *sis that p# :G2r+1(E, X; j) Q ! ß2r+1(S2r+1) Q is onto_included in (2), when translated into our derivation setting, gives the existence of a ß-derivation _* * 2 Der2r+1( (u) V, V ; ß) that is a cocycle, and that satisfies _(u) = 1. Usi* *ng this _, define a linear map : (u) V ! (u) V by setting (a + ub) = a + ub + u_(a) for a typical element a + ub 2 (u) V . We will check that * *is 28 GREGORY LUPTON AND SAMUEL BRUCE SMITH actually a DG algebra isomorphism ( (u) V, D) ! ( (u) V, d). First, is an algebra map. This follows from the fact that _ is a derivation. Suppose given two elements a + ub, a0+ ub02 (u) V . Then we have 0 0 0 |a| 0 0 (a + ub)(a + ub )= aa + u((-1) ab + ba ) = aa0+ u((-1)|a|ab0+ ba0) + u_(aa0). On the other hand, we have 0 0 0 (a + ub) (a0+ ub0)= a + ub + u_(a) a + ub + u_(a ) |a| 0 0 |a| 0 0 = aa0+ u (-1) ab + ba + (-1) a_(a ) + _(a)a . These agree, since _(aa0) = _(a)a0+ (-1)|a|a_(a0). Next, we check that com- mutes with differentials, that is, that D = d . This will follow from the fact that _ is a cocycle. First observe that d (u) = 0 = D(u). It is thus sufficient to check that D(Ø) = d (Ø) for a typical element Ø 2 V . To this end, write D(Ø) = d(Ø) + u`(Ø). Now we calculate DØ = d(Ø) + u`(Ø) = d(Ø) + u`(Ø) + u_(dØ) and d Ø = d Ø + u_(Ø) = d(Ø) - ud_(Ø). These agree since _ is a cocycle, whence -d_(Ø) = _D(Ø) = _ d(Ø) + u`(Ø) . It is evident that is an isomorphism, since we have (u) = u and ß O = ß. Therefore, : ( (u) V, D) ! ( (u) V, d) is a DG isomorphism and the fibration is rationally trivial. This leads to the following equivalent phrasing of Conjecture 2.10: Corollary 4.10. Let X be an F0-space. Then X satisfies Conjecture 2.10 if and only if the G-sequence of the fibre inclusion in every fibration of the form X ! E ! S2n+1 decomposes into split short exact sequences as in (2) of Theorem 4.9. Proof.In [Mei82], Meier showed that Halperin's conjecture for X is equivalent to the collapsing of the rational Serre spectral sequence for all fibrations with * *fibre X and base an odd sphere. By [Lup98 , Th.2.3] this latter condition is equivale* *nt to the rational homotopy triviality of the fibration. The result now follows f* *rom Theorem 4.9. Remark 4.11. It is possible to develop the ideas leading to Theorem 4.9 substan* *tially beyond the application given above. Namely, one can use the splitting of the G- sequence of the fibre inclusion as a measure of how close the fibration is to b* *eing trivial. Now in the above results, we see that for fibrations of the form X ! E* * ! S2n+1, with X an F0-space, this notion coincides with the notion of the fibrati* *on being TNCZ. Furthermore, in this closely restricted setting, both coincide with* * the fibration actually being trivial. In general, however, this G-sequence point of* * view gives a new way of measuring how close a fibration is to being trivial. We inte* *nd to develop these ideas in a subsequent paper. RATIONALIZED EVALUATION SUBGROUPS 29 Appendix A In this appendix, we give careful proofs of the results from DG algebra homot* *opy theory that are used to establish Theorem 2.1. We rely heavily on the notion of pullback in the DG algebra setting. By this, we mean the following. Suppose giv* *en DG algebra maps f :A ! C and g :B ! C. Then we form the (DG algebra) pullback (or fibre product, as it is called in [FHT01 ]) as A C B = {(x, y) 2 * *A B | f(x) = g(y)}. Here A B denotes the direct sum of DG algebras. Together with the projections, the pullback forms the following (strictly) commutative square* * of DG algebra maps: (10) A xC B __p1_//A p2|| |f| |fflffl fflffl| B ___g___//_C This square possesses the usual universal property of pullbacks. Namely, suppose given DG algebra maps ff: Z ! A and fi :Z ! B that satisfy f O ff = g O fi. Then there exists a DG algebra map OE = (ff, fi): Z ! A xC B, which is the unique DG algebra map for which p1OOE = ff and p2OOE = fi. We emphasize that throughout t* *his appendix we distinguish carefully between diagrams that are strictly commutative and ones that are commutative only up to DG homotopy. Indeed, it is precisely this distinction that calls for the proofs of this appendix. The following basic property of the pullback is readily gleaned from the disc* *ussion in [FHT01 , Sec.13(a)]: Lemma A.1. Suppose that either f or g is surjective in the pullback diagram (10* *). If f is a quasi-isomorphism, then p2 is a quasi-isomorphism. We also make use of the so-called surjective trick, described in [FHT01 , Sec* *.12(b)]. Given a DG algebra map j :B ! A, this manoeuvre results in a diagram __~__// B oo___B E(A) 1." uu j || ufluuu fflffl|zzzzuuuu A in which fl is a surjection, and both 1 . ä nd ~ are quasi-isomorphisms. Some * *parts of the diagram commute, thus (1 . ") O ~ = 1 and fl O ~ = j. Other compositions result in commutativity only up to DG homotopy. Recall that the notion of DG homotopy is only defined for DG algebra maps from a minimal model. Given any map OE: M ! B E(A) from a minimal model into B E(A), we have OE ~ ~ O (1 . ") O OE, where ~ denotes DG homotopy of maps from a minimal model. In particular, we thus have j O (1 . ") O OE ~ fl O OE. Now suppose given a map f :X ! Y . We choose and fix a minimal model Mf: MY ! MX for f as follows ([FHT01 , Sec.12(c)]): Let A*(f): A*(Y ) ! A*(X) denote the map induced by f on polynomial differential forms. Let jX :MX ! A*(X) and jY :MY ! A*(Y ) denote minimal models for X and Y . As in [FHT01 , Sec.12(b)], we convert jX into a surjection flX :MX E(A*(X)) ! A*(X) and lift A*(f) O jY through the surjective quasi-isomorphism flX , using [FHT01 , Lem.12* *.4], 30 GREGORY LUPTON AND SAMUEL BRUCE SMITH to obtain OEf: MY ! MX E(A*(X)). Now set Mf = flX O OEf. All this is summarized in the following diagram. _______MX00_E(A*(X))_________________________* *______________________________________________________________@ OEf________________________________________________* *______________________________________________________________@ ___________________nnn_________________________________* *___nnn______________________________________________ffnnnnnn ___________________nnnn'__________________________________* *___nnnn___________________________________nnnn'nnnn ___________________nnnnnn____________________________________* *_________________________nnnnnn MY ____________//____wwnnfiw____________________________________* *____wnn_________________MX Mf '____flX________________________________* *_______________________________________________________ jY '|| jX '|| ________________________________________* *______ fflffl| fflwwww_______________________________________* *___________________________ffl| A*(Y )___A*(f)__//_A*(X) By construction, we have flX O OEf = A*(f) O jY , flX O ff = jX , and fi O ff =* * 1. Remaining parts of the diagram only commute up to DG homotopy, however, thus we have jX O fi ~ flX , jX O Mf ~ A*(f) O jY , and so-on. Now let ff: Sn ! map (X, Y ; f) be a representative of a homotopy class in ßn map (X, Y ; f) . Let F :Sn x X ! Y be an affiliated map for ff, that is, F (s, x) = ff(s)(x). Since ff is a based map, we have F O i = f :X ! Y , where i: X ! Sn x X denotes (based) inclusion into the second summand i(x) = (*, x). In the following result, we justify that the Sullivan minimal model of any af* *filiated map, and a DG homotopy between two such, have the restricted form that we require of them for the definition and well defined-ness of f. Proposition A.2. Suppose given maps F, G: Sn xX ! Y and H :Sn xX xI ! Y a homotopy from F to G that is stationary on X xI. Suppose that H(*, x, t) = f(* *x) for f :X ! Y and let Mf: MY ! MX be a fixed choice of minimal model for f. There is a DG homotopy MH :MY ! MSn MX (t, dt) from MF to MG , minimal models for F and G, of the form + MH (Ø) = 1 Mf(Ø) 1 + terms inMSn MX (t, dt). In particular, any map F :Sn x X ! Y that satisfies F O i = f has a minimal model MF :MY ! MSn MX of the form + MF (Ø) = 1 Mf(Ø) + terms inMSn MX . Proof.Construct the following pullback: p1 * P __________//_MX (t, dt) E(A (X x I)) p2|| ' |fl| fflffl| fflfflfflffl| A*(Sn x X x I)_____A*(i)___////_A*(X x I) Here, i: XxI ! SnxXxI denotes the inclusion i(x, t) = (*, x, t) and fl denotes * *the surjective quasi-isomorphism obtained by converting the quasi-isomorphism MX (t, dt) ! A*(X xI) to a surjection. Since i is an inclusion, the induced map A* **(i) is a surjection. From Lemma A.1, we have that p2 is a quasi-isomorphism. Now RATIONALIZED EVALUATION SUBGROUPS 31 consider the following commutative diagram: MSn MX (t,