HOMOTOPY ACTIONS, CYCLIC MAPS AND THEIR DUALS MARTIN ARKOWITZ AND GREGORY LUPTON Abstract.An action of A on X is a map F :A x X ! X such that F|X = id:X ! X. The restriction F|A :A ! X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottli* *eb groups of a space, each of which has been studied extensively. We prove * *some general results about actions and their Eckmann-Hilton duals. For instan* *ce, we classify the actions on an H-space that are compatible with the H-struct* *ure. As a corollary, we prove that if any two actions F and F0 of A on X have cyclic maps f and f0with f = f0, then F and F0 give the same action of A on X. We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equ* *al to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea cat* *egory of g is 1. We also briefly discuss the relationship between a map bein* *g cyclic and its cocategory being 1. 1.Introduction Group actions on spaces are natural objects of study in topology. Their analo* *gues in homotopy theory lead to the notion of a homotopy action of one space on anot* *her. The restriction to the basepoint of X of a group action G x X ! X yields a map G ! X, known as the orbit map of the action. These maps, and their homotopy-theoretic analogues, are frequently of interest. For instance, the nth Gottlieb group of a space X, denoted Gn(X), may be defined as homotopy classes * *of maps f :Sn ! X such that (f | id): Sn _X ! X admits an extension Sn xX ! X up to homotopy [Got69]. That is, the nth Gottlieb group is the set of homotopy classes of orbit maps of Sn-actions on X. These groups can be generalized. For example, in [Var69] the homotopy set G(A, X) of cyclic maps from A to X is defined as the homotopy classes of maps f :A ! X such that (f | id): A _ X ! X admits an extension A x X ! X. Such an extension is called an f-action, and so a cyclic map f : A ! X is just the orbit map of an f-action of A on X. Actions and cyclic maps have been well-studied in the literature. For the forme* *r, see [Gan67a , Zab76, IO03]. For the latter, see [Var69, HV75 , LW01 ]. The Gott* *lieb groups have also received much attention in rational homotopy, following the re* *sults of F'elix-Halperin in [FH82 ]. In this paper we prove some general results about actions and cyclic maps. In Section 2, we focus on actions and the operations on homotopy sets that are ind* *uced by actions. We deduce several formulas, such as one for the homomorphism induced on homotopy sets (Lemma 2.5). We also conclude that any two actions loop to the same action (Proposition 2.8), which implies that any two multiplications on an ____________ Date: September 26, 2005. 2000 Mathematics Subject Classification. 55Q05, 55M30, 55P30. Key words and phrases. Action, cyclic map, category of a map, coaction, cocy* *clic map, cocat- egory of a map. 1 2 MARTIN ARKOWITZ AND GREGORY LUPTON H-space loop to the same multiplication (Corollary 2.9). Our results dualize, a* *nd we summarize the dual results at the end of Section 2. In Section 3, we concent* *rate on the dual of a cyclic map, called a cocyclic map. We introduce a new version * *of the Lusternik-Schnirelmann category of a map and show that the maps of category 1 are precisely the cocyclic ones (Theorem 3.7). Since there has been increas* *ing interest in Lusternik-Schnirelmann category recently (see [CLOT03 ]), this giv* *es motivation for the study of cocyclic maps. Furthermore, we clarify the relation between cocyclic maps and connecting maps of cofibration sequences (Theorem 3.16). In doing so, we fill a gap in the literature. The relation between cyc* *lic maps and connecting maps of fibration sequences has long been well-understood [Got69, HV75 ]. The dual situation, however, has remained unclear due to the la* *ck of a dual for the classifying space construction. All spaces, maps and homotopies are assumed to be based. All spaces are as- sumed to be CW complexes. We use *: X ! Y and id:X ! X to denote, respectively, the constant map and the identity map. We denote the set of homo- topy classes of maps from X to Y by [X, Y ]. However, we often do not distingui* *sh between a map and the homotopy class it represents. In Section 3 we frequently consider lifts and sections of certain maps, by which we mean lifts and sections up to homotopy. If f :A ! B is a map, then f* denotes pre-composition by f and f* denotes post-composition by f. Thus we obtain maps of homotopy sets f* :[B, X] ! [A, X] and f*: [X, A] ! [X, B]. We also use f* to denote the map i* *n- duced by f on cohomology and f# to denote the map induced on homotopy groups. The wedge of spaces X and Y is denoted X _ Y . The unit interval [0, 1] is deno* *ted I. The cone on a space X is written CX and the path space is EX. The loop space, respectively suspension, of a space X is denoted X, respectively X. We recall that the standard multiplication ~ on Y induces a group structure in [X, Y ] and the standard comultiplication on X induces a group structure in [ X, Y ]. These group structures are isomorphic via adjointness, that is, [X, Y ] ~=[ X,* * Y ]. Under this isomorphism, the identity id: X ! X has an adjoint which we de- note p: X ! X and the identity id: X ! X has an adjoint which we denote by e: X ! X. By an H-space, we mean a pair (X, m) with m: X x X ! X a (not-necessarily associative) multiplication up to homotopy. Dually, by a co-* *H- space we mean a pair (X, n) with n: X ! X _ X a (not-necessarily associative) comultiplication up to homotopy. 2.Actions on Spaces and Operations on Homotopy Sets In this section we consider the action of one space on another, and cyclic ma* *ps from one space to another. We present several general results about these notio* *ns that illustrate the relation between them. At the end of the section, we brief* *ly consider the dual notions of coactions and cocyclic maps. Let i1: A ! A x X and i2: X ! A x X denote the inclusions. Definition 2.1. By a (left) action of A on X, we mean a map F :A x X ! X that satisfies F O i2 = id:X ! X. If F O i1 = f :A ! X, then we refer to F as an f-action. Given an f-action F , we say that f is the orbit map of F . A map f :A ! X is called a cyclic map if it is the orbit map of some f-action F . We denote the set of cyclic homotopy classes G(A, X) [A, X]. HOMOTOPY ACTIONS, CYCLIC MAPS AND THEIR DUALS 3 In the literature, an f-action is often referred to as an affiliated map (of * *f). There is an alternate point of view on actions and cyclic maps that is often co* *nve- nient to adopt: Let X be a space with basepoint x0 and map (X, X; id) denote the component of the identity in the function space of unbased maps from X to itsel* *f. Let ! :map(X, X; id) ! X denote the evaluation map defined by !(h) = h(x0), where x0 is the base point of X. Then an f-action F :AxX ! X corresponds to a lift "f:A ! map(X, X; id) of f through !, that is, !Of"= f. This correspondence* * is given by "f(a)(x) = F (a, x). Then the set of cyclic maps G(A, X) may be descri* *bed as the image of the induced map of homotopy sets !*: [A, map(X, X; id)] ! [A, X* *]. It follows easily from this point of view that, if A is a suspension, then G(A,* * X) is a subgroup of [A, X] [Var69]. We illustrate these notions with some well-known examples. Examples 2.2. (1) The projection p2: A x X ! X is a *-action. (2) Let (X, m) be an H-space. Then the multiplication m: X x X ! X is a id-action. (3) Suppose given an f-action F :A x X ! X and any map h: A0! A. Then F O(hxid): A0xX ! X is an (f Oh)-action. It follows that h*: G(A, X) ! G(A0, X). By combining this and the previous observations, we obtain: If (X, m) is an H-space, then m O (f x id): A x X ! X is an f-action for any f :A ! X, and so G(A, X) = [A, X]. (4) Let ! :map(X, X; id) ! X be the evaluation map as above. Define the map W :map (X, X; id) x X ! X by W (f, x) = f(x). Then W is a (right) !-action of map (X, X; id) on X. Indeed, W is a "universal action" on X, in that every action of a space A on X factors through W . From this we see that ! is a "universal cyclic map," in the sense that it is a cyclic* * map through which every cyclic map to X factors. (5) Suppose A __j_//_Xp__//Bis a fibration. We call this a homotopy principal fibration if there exists a j-action F :A x X ! X of the fibre on the to* *tal space, such that the following diagram homotopy commutes: A x X __F__//X p2|| |p| fflffl| fflffl| X ___p__//_B, see [PT58 , Por71]. This is the obvious homotopy-theoretic analogue of t* *he situation in which A = G is a topological group and p: X ! B is a princi* *pal G-bundle. An important special case of a homotopy principal fibration is that of a fibration B ! E ! X induced from the path-space fibration over B by some map g :X ! B. An action of one space on another induces an operation on homotopy sets as follows. Suppose F :A x X ! X is an f-action and W is any space. Given a 2 [W, A] and h 2 [W, X], we define a . h 2 [W, X] as the composition axh F W _____//W x W ____//_A x X____//X, where is the diagonal map (see [Hil65, Ch.15] and [Zab76, p.56]). This gives * *an operation of the set [W, A] on the set [W, X] with the following properties: 4 MARTIN ARKOWITZ AND GREGORY LUPTON (i)* . h = h; (ii)a . * = f*(a); (iii)l*(a . h) = l*(a) . l*(h) for any map l :W 0! W ; (iv)F = p1 . p2 for the projections p1: A x X ! A and p2: A x X ! X. We now adopt the following notational convention. If (X, m) is an H-space, we write the binary operation induced on [W, X] by "+m ". On the other hand, if (W, n) is a co-H-space, we write the binary operation induced on [W, X] by " n". This is defined as follows: if a, b 2 [W, X], then a n b = (a|b) O n 2 [W, X]. Example 2.3. Let (X, m) be an H-space. If f :A ! X is any map then, as in (3) of Examples 2.2, m O (f x id) is an f-action. We denote this action by Acfm= m O (f x id): A x X ! X. The induced operation of [W, A] on [W, X] satisfies a . h = f*(a) +m h for all a 2 [W, A] and h 2 [W, X]. We now return to a general f-action F :A x X ! X and identify the operation when W is a co-H-space. Lemma 2.4. Suppose (W, n) is a co-H-space and F :A x X ! X is an f-action. Then we have a . h = f*(a) n h, for all a 2 [W, A] and h 2 [W, X]. Proof.Observe that the diagram W:_:W _a_h_//A _GX nvvvv | | GGG(f|id)G vvv | | GGG vv | | G## W HH |J| J || wX;; HHH | | www HHH | | wwFw H$$fflffl| fflffl|ww W x W axh__//A x X homotopy commutes, where J is the inclusion. Lemma 2.5. Let F :A x X ! X be an f-action and suppose given a 2 [W, A] and h 2 [W, X]. Let (U, n) be a co-H-space. Then the induced homomorphism (a . h)*: [U, W ] ! [U, X] satisfies (a . h)*(fl) = f*a*(fl) n h*(fl), for all fl 2 [U, W ]. Proof.Apply property (iii) above to (a.h)*(fl) = fl*(a.h), then use Lemma 2.4. We may specialize Lemma 2.5 to obtain a formula for the homomorphism induced on homotopy groups by a map of the form a . h, special cases of which are used * *in both [LO96 ] and [ALM01 ]. HOMOTOPY ACTIONS, CYCLIC MAPS AND THEIR DUALS 5 Definition 2.6. Let (X, m) be an H-space and F :AxX ! X an f-action. We say that the action F is m-associative if the following diagram homotopy commutes: FxidX A x X x X _____//X x X idAxm|| m|| fflffl| fflffl| A x X ____F____//X. Notice that in the case in which A = X and the action F is the multiplication m, this definition reduces to that of homotopy-associativity of m. Proposition 2.7. Let (X, m) be an H-space and f :A ! X a map. (1) If m is homotopy-associative, then Acfmis an m-associative f-action. (2) If F :A x X ! X is any m-associative f-action, then F = Acfm. Proof.The first part is clear. For the second, let i1: A ! AxX denote the inclu* *sion into the first factor and j :X ! X x X the inclusion into the second factor, so* * that idAx j = i1 x idX:A x X ! A x X x X. Then we have F = F O (idA x m) O (idA x j) = m O (F x idX) O (idA x j) f = m O (F O i1) x idX = m O (f x idX) = Acm . The following result implies that two f-actions loop to the same f-action. A similar result in a different context appears in [AOS05 ]. Let ~: A x X ! (A x X) denote the standard homeomorphism given by ~(a, b)(t) = a(t), b(t) , where t 2 I. Proposition 2.8. Let F :A x X ! X be an f-action and let ~ be the loop multi- plication on X. Then ( F ) O ~ = Ac~f : A x X ! X, and thus ( F ) O ~ is a ~-associative f-action. If, in addition, F 0:A x X ! X* * is an f0-action such that f = f0: A ! X, then F = F 0. Proof.Let p: (A x X) ! A x X be the canonical map. Then, using (iii) and (iv) of the properties listed above, we have F O p = p*(p1 . p2) = p*(p1) . p*(p2), where the last term uses the operation of [ (A x X), A] on [ (A x X), X]. By Lemma 2.4, it follows that i j i j F O p = f O p1 O p p2 O p , where in [ (A x X), X] is induced by the suspension structure on (A x X). By taking adjoints and noting that the adjoint of p is id: (A x X) ! (A x X), we have i j F = ( f) O ( p1) +~ p2. Let ss1: A x X ! A and ss2: A x X ! X be the projections. Then i j i j ( F ) O ~= ( f) O ( p1) O ~ +~ ( p2) O ~ i j = ( f) O ss1 +~ ss2 = Ac~f. The other assertions of Proposition 2.8 follow easily. 6 MARTIN ARKOWITZ AND GREGORY LUPTON The following corollary may be known, but we have not found a proof of it in the literature. It effectively answers Problem 35 from Stasheff's H-space probl* *em list in [Sig71]. Corollary 2.9. If m, m0 are two multiplications on X, then ( m) O ~ = ~ = ( m0) O ~. Consequently, m = m0. For the remainder of this section, we consider the dual of the preceding dis- cussion. Since most of this can be obtained mutatis mutandis from the previous material, we provide few details. Let p1: X _ B ! X and p2: X _ B ! B denote the projections. Definition 2.10. By a (right) coaction of B on X, we mean a map G: X ! X _B that satisfies p1 O G = id:X ! X. If p2 O G = g :X ! B, then we refer to G as a g-coaction. Given a g-coaction G, we say that g is the co-orbit map of G. A map g :X ! B is called a cocyclic map if it is the co-orbit map of some g-coaction * *G. We denote the set of cocyclic homotopy classes by G0(X, B) [X, B]. Notice that there is no natural candidate for the dual of the function space map(X, X; id). So in the coaction setting, we do not have the alternative point* * of view that a "coevaluation map" would provide. For results on cocyclic maps, see [Var69, HV75 , Lim87, LW01 ]. In the case in which B = K(ss, n), the set of coc* *yclic maps G0(X, B) has been called the nth dual Gottlieb group of X (cf. [FLT94 ]). We can dualize most of Examples 2.2; we mention two of these explicitly. Examples 2.11. (1) Suppose given a g-coaction G: X ! X _B and any map h: B ! B0. Then (id_ h) O G: X ! X _ B0 is an (h O g)-coaction. It follows that h*: G0(X, B) ! G0(X, B0). If (X, n) is a co-H-space, then (id_ g) O n: X ! X _ B is a g-coaction for any g, so G0(X, B) = [X, B]. (2) Suppose A __j_//_Xp__//_Bis a cofibration. We call this a homotopy prin- cipal cofibration if there exists a p-coaction G: X ! X _ B such that the following diagram homotopy commutes: j A _______//X j|| |i1| fflffl| |fflffl X _G__//_X _ B An important case occurs when the cofibration A ! X ! Z is induced from the cofibration Z ! CZ ! Z via a map Z ! A [Hil65, Ch.15]. Suppose G: X ! X _ B is a coaction and W is any space. Given a 2 [B, W ] and h 2 [X, W ], we define h . a 2 [X, W ] as the composition X __G__//X _ B_h_a//_W _ W_r__//_W. This gives a (right) operation of the set [B, W ] on the set [X, W ] [Hil65, Ch* *.15]. We use the same notation for this operation and the previous one. We note that properties (i)-(iv) stated earlier may be dualized. Example 2.12. Let (X, n) be a co-H-space. If g :X ! B is any map, then (id_g)On is a g-coaction which we denote by Coacng:X ! X _B. In this case, the induced operation of [B, W ] on [X, W ] satisfies h.a = h n g*(a), for all a 2 * *[B, W ] and h 2 [X, W ]. HOMOTOPY ACTIONS, CYCLIC MAPS AND THEIR DUALS 7 Lemma 2.4 dualizes in a straightforward way to obtain the following. Lemma 2.13. Suppose (W, m) is an H-space and G: X ! X _ B is a g-coaction. Then we have h . a = h +m g*(a), for all a 2 [B, W ] and h 2 [X, W ]. Definition 2.14. Let (X, n) be a co-H-space and G: X ! X _ B a g-coaction. We say that G is n-coassociative if the following diagram homotopy commutes: X ____G____//X _ B n || |n_idB| fflffl| fflffl| X _ X idX_G//_X _ X _ B. We remark that Proposition 2.7 dualizes in a straightforward way; we omit the dual statement and its proof. Let ~0: (X _ B) ! X _ B denote the standard homeomorphism. Proposition 2.15. If G: X ! X _ B is a g-coaction and is the suspension comultiplication on X, then ~0O ( G) = Coac g: X ! X _ B, and thus ~0O ( G) is a -coassociative g-coaction. In addition, if G0:X ! X _ B is a g0-coaction such that g = g0, then G = G0. Corollary 2.16. If n, n0 are two comultiplications on X, then ~0O ( n) = = ~0O ( n0), and so n = n0. We may combine the dual notions discussed above in the following interesting way. Suppose given an f-action F :AxX ! X and a g-coaction G: X ! X _A (so A = B here). These give operations of [X, A] and [A, X] on [X, X]. In particula* *r, given any f 2 [A, X], g 2 [X, A] and h 2 [X, X], we have h . f, g . h 2 [X, X] obtained using the coaction and the action, respectively. Lemma 2.17. With the above notation, h . f = g . h. This map induces (g . h)# = f# g# + h# on homotopy groups and (h . f)* = h* + g*f* on cohomology groups. Proof.Let (id| f): X _ A ! X be determined by idand f, let (id, g): X ! X x A be determined by idand g and let T :X xA ! AxX interchange factors. Consider the homotopy-commutative diagram X;_;A _h_id//_X _GA G wwww | | GGG(id|f)G ww | | GGG www | | G## X GG J|| J || ;X.; GGG | | wwww GGG | | wwFOTw (id,g)G##fflffl| fflffl|ww X x A _hxid//_X x A Then h . f= (id| f) O (h _ id) O G = F O T O (h x id) O (id, g) = F O (g, h) = g . h. 8 MARTIN ARKOWITZ AND GREGORY LUPTON The other assertions of Lemma 2.17 follow from Lemma 2.5 and its dual. A special case of this situation is considered in [ALM01 ]. 3.Lusternik-Schnirelmann Category and Coactions In this section we relate cocyclic maps to a variant of the Lusternik-Schnire* *lmann category of a map. We begin by reviewing the Ganea fiber-cofiber construction [Gan67b ]. In this section, we generally do distinguish between a map and its homotopy class. Definition 3.1. Suppose given a fibration F __i_//_Ep__//B. We define a sequence of fibrations Fn(p)_in_//_En(p)pn//_B inductively for n 0. Set F0(p)__i0_//E0(p)p0_//Bequal to the given fibra- in-1 pn-1 tion. Assume Fn-1(p)_____//En-1(p)___//Bhas been defined. Set E0n-1(p) = En-1(p) [in-1CFn-1(p), the mapping cone of in-1. Define p0n:E0n(p) ! B by p0n|En-1(p)= pn-1 and p0n|CFn-1(p)= *. Then replace p0nby a fiber map to obtain a fiber sequence Fn(p)__in_//En(p)pn_//B. Note that there is a map jn-1: En-1(p) ! En(p) such that the diagram jn-1 En-1(p)G____________//En(p) GGG yyyy pn-1GGGG yypn G## __yy B commutes. There are two special cases of this construction that we shall need. Examples 3.2. (1) Given a space B, consider the path space fibration B ! EB ! B. Applying the above Ganea construction to this fibration, we ob- tain a sequence of fibrations which we write as Fn(B) _in_//_Gn(B)pn_//B. (2) Given a map g :X ! B, we first form the fibration B ! E ! X induced via g from the path space fibration over B by the pullback construction. Then we apply the Ganea construction to this fibration to obtain a seque* *nce of fibrations which we write as Kn(g)__ln_//Hn(g)rn_//_X. Notice that the two examples above are related. From the pullback constructio* *n, there is a map "g:E ! EB such that the following diagram commutes "g E _____//EB p0|| p|| fflffl| fflffl| X __g__//_B. HOMOTOPY ACTIONS, CYCLIC MAPS AND THEIR DUALS 9 Since the Ganea construction is functorial, we get a map "gn:Hn(g) ! Gn(B) such that the following diagram is commutative Hn(g) __"gn//_Gn(B) rn|| |pn| fflffl| fflffl| X ____g____//B for each n 0. The constructions in Examples 3.2 lead to the following definitions of the ca* *te- gory of a map. Definition 3.3. Suppose given a map g :X ! B. (1) We say that catBG(g) n if g can be lifted through pn to Gn(B). That is, if there exists ^gn:X ! Gn(B) such that Gn(B)<< xx ^gnxxx|pn| xxx fflffl| X ___g___//B commutes up to homotopy. (2) We say that cat(g) n if rn :Hn(g) ! X has a section. That is, if there exists sn :X ! Hn(g) such that rn O sn ' id:X ! X. We define catBG(g) (respectively, cat(g)) to be the least n such that catBG(g* *) n (respectively, cat(g) n). Remarks 3.4. (1) catBG of a map is just the Berstein-Ganea category of a map as defined in [BG62 ]. However, the notion introduced in part (2) of Definition* * 3.3 appears to be new. (2) We note that catBG(*) = cat(*) = 0, where * : X ! Y is the constant map, and catBG(idB) = cat(idB) = cat(B), where cat(B) denotes the ordinary Lusternik-Schnirelmann category of the space B. (3) It follows from the definitions that catBG(g) cat(g). We will see later* * that they are generally different. Our main result in this section is that a map g is cocyclic if and only if it* * satisfies cat(g) 1. To prove this, we need some notation and a preliminary result. Notation 3.5. Given a map g :X ! B, define the graph of g to be = {(x, g(x)) | x 2 X} X x B. Let R1(g) denote the space E(X x B; , X _ B) of paths in X x B that start in and end in X _ B with fiber map "ss0:R1(g) ! X defined by "ss0(l) = the first coordinate of l(0). One then obtains R1(g) as the fiber * *space over X induced via (id, g): X ! X x B from the fibration over X x B obtained by converting the inclusion J :X _ B ! X x B into a fibration. Finally, if fl is a* * path in a space W and s 2 I, then fls denotes the path in W defined by fls(t) = fl(s* *t). 10 MARTIN ARKOWITZ AND GREGORY LUPTON Lemma 3.6. Suppose given a map g :X ! B. Then there is a map w :H1(g) ! R1(g) which induces an isomorphism on homology and is such that the diagram H1(g)E_____w______//_R1(g) EEE yyyy r1EEE yys"s0 E"" __yy X commutes. Proof.The proof, which we sketch here, is an adaptation of an argument given by Gilbert in [Gil, Prop.3.3]. As stated earlier, p0:E ! X is induced via g, and so we have "g:E ! EB (see the discussion after Examples 3.2). Let S = {(e, ) | e 2 E, 2 XI, p0(e) = (0)} and choose a lifting function ~: S ! EI for p0. The extension p01:E [ C B ! X of p0 is given by p01|E = p0 and p01|C B = *. Then we replace p01with a fiber map r1: H1(g) ! X, where H1(g) = {(z, ) | z 2 E [ C B, 2 XI, p01(z) = (1)} and r1(z, ) = (0). We now define w :H1(g) ! R1(g) by w (e, s), = , g"~(e, - )(1) s , for s 2 I, (e, s) 2 E [ C B CE and 2 XI with p01(e, s) = (1). One easily checks that w is well-defined, has codomain R1(g), and satisfies "ss0O w* * = r1. To prove that w is a homology isomorphism, it suffices to show that ^w, the map induced on fibers, is a homology isomorphism. This is done by factoring ^wthrou* *gh ( X x C B) [ (EX x B) EX x C B and showing, as in the argument of Gilbert, that each of the two resulting maps induces a homology isomorphism. We omit the details. Theorem 3.7. Let g :X ! B be a map of simply connected spaces. Then g is cocyclic if and only if cat(g) 1. Proof.By definition, g is cocyclic if and only if (id, g): X ! X xB factors thr* *ough X _ B. One sees easily that this is so if and only if the fibration "ss0:R1(g) * *! X has a section. Since X and B are both simply connected, w :H1(g) ! R1(g) is a homotopy equivalence by Lemma 3.6. It follows that g is cocyclic if and only * *if r1: H1(g) ! X has a section. In the case X = B and g = id, Theorem 3.7 reduces to the following well-known result: X is a co-H-space if and only if p: X ! X admits a section [Gan70 ]. The next corollary is a consequence of Remarks 3.4 (3). Corollary 3.8. If g :X ! B is a cocyclic map of simply connected spaces, then catBG(g) 1. We next give an example to illustrate that the converse of Corollary 3.8 does* * not hold. Example 3.9. We show that the projection p1: Sm xSn ! Sm has catBG(p1) 1 and yet is not cocyclic. That is, we have 1 = catBG(p1) < cat(p1). Since Sm is a suspension, there is a section of p: Sm ! Sm and hence a section of the first Ganea fibration G1(Sm ) ! Sm . Thus we have catBG(p1) 1. To complete the example, we use the following fact: if g :X ! B is cocyclic, then for any fl 2 H+ (X) and fi 2 H+ (B), we have fl[g*(fi) = 0. This fact follows easily fr* *om the factorization of (g, id) through the wedge X _ B, where such cup-products are z* *ero HOMOTOPY ACTIONS, CYCLIC MAPS AND THEIR DUALS 11 (cf. the inclusion ~G(X; Q) S(X; Q) of [FLT94 , Th.1]). Let bm 2 Hm (Sm ) and bn 2 Hn(Sn) = Hn(Sm xSn) be basic classes. Then the cup-product bn[p*1(bm ) 6= 0 2 Hm+n (Sm x Sn), so p1 is not cocyclic. We note that there is a third version of the category of a map, due to Fadell- Husseini [FH94 ] and studied by Cornea in [Cor98] as the relative category of a map. Definition 3.10. Suppose given a fibration F __i_//_Ep__//B. Apply the Ganea construction to obtain Fn(p)_in_//_En(p)pn//_Band maps jn-1: En-1(p) ! En(p) such that pn-1 = pn O jn-1. We define catFH(p) to be the smallest n such that pn has a section sn with sn O p = jn-1 O . .O.j0. 0 Given a map g :X ! B, we form the induced fibration B _____//E_p_//_Xas before. Lemma 3.11. With the above notation, we have cat(g) catFH(p0). Proof.This is immediate from the definition. Example 3.12. The inequality above will be strict whenever we find a p0:E ! X which has a section, but which is not a homotopy equivalence. For instance, let g = *: X ! B. Then E = X x B and p0is projection onto the first factor. We have established a connection between a map being cocyclic and its categor* *y. Our point of view is that the category of a map is inherently of interest, and * *that this connection motivates interest in the notion of a cocyclic map. The dual of this connection, namely, one between the familiar notion of a cyclic map and the cocategory of a map, then serves to motivate interest in the cocategory of a ma* *p. We summarize some results on the cocategory of a map before returning to the cocyclic/category side of things. The fiber-cofiber construction of Definition 3.1 may be dualised in the obvi- ous way. Starting from a cofibration X __j__//C_q_//_Z, one obtains a sequence of cofibrations X __jn_//Cn(j)qn_//Zn(j). We are interested in two cases of th* *is construction. If we start from the cofibration A __j__//CA_q__// A, we obtain a sequence of cofibrations dual to the Ganea fibrations. We denote these by 0 A _jn_//_G0n(A)qn//_Fn0(A). If we start from the cofibration X __j__//C___//_ * *A, induced via a map f :A ! X from the cofibration j :A ! CA, then we obtain a sequence of cofibrations dual to those of (2) of Examples 3.2. We denote these * *cofi- brations by X __jn_//H0n(f)qn//_K0n(f). Then we define cocatBG(f) n to mean that there exists a map "f:G0n(A) ! X so that "fO jn = f. We define cocat(f) n to mean that there exists a retraction of jn :X ! H0n(f). From the definitions,* * we have cocatBG(f) cocat(f). Now it is easy to identify the first Ganea cofibration A ! G01(A) with the canonical map e: A ! A. Furthermore, it is well-known that a space A is an H-space if and only if this map admits a retraction. As we remarked after Definition 2.1, a cyclic map factors through the H-space map (X, X; id). From t* *his it follows that a cyclic map f :A ! X factors through A, and so cocatBG(f) * *1. 12 MARTIN ARKOWITZ AND GREGORY LUPTON On the other hand, we may have a map with cocatBG(f) 1 which is not cyclic. Example 3.13. Let f :S3 ! S3 _ S3 be either i1 or i2, the inclusion into the first or second summand. Since S3 is an H-space, f factors through S3, and so cocatBG(f) 1. But if f is cyclic, then the Whitehead product [f, fi] = 0, for* * all fi 2 ss*(S3 _ S3) [Got69, Prop.2.3]. Since [i1, i2] 6= 0, then f cannot be cycl* *ic. The main question to be resolved is the following: Question 3.14. For a map f :A ! X, is the condition that f be cyclic equivalent to cocat(f) 1? We now return to cocyclic maps and discuss a result of Halbhavi-Varadarajan in the light of the above. In [HV75 , Prop.1.2], the following result is shown: Theorem 3.15. Let f :A ! X be a cyclic map. Then there is a fibration X ! E ! A such that f = @ O e, where @ : A ! X is the connecting map of the fibration and e: A ! A is the canonical map. This result makes clear the close relation between cyclic maps and connecting maps of fibration sequences. It is an extension of results of Gottlieb for the * *case in which f :Sn ! X is a Gottlieb element. The proof makes essential use of the classifying space for fibrations with given fibre. As the authors of [HV75 ] po* *int out, it is not clear if the dual of this result holds. Because there is no analogue * *of the classifying space in the setting of cofibrations, their proof cannot be dualize* *d. Theorem 3.15 implies that every cyclic map f :A ! X factors through the canonical map e: A ! A. We next give a result which is a slightly weaker version of the dual of Theorem 3.15, but which does establish a relation between cocyclic maps and connecting maps of cofibration sequences. Theorem 3.16. Let g :X ! B be any cocyclic map of simply connected spaces. Then there is a cofibre sequence B ! E ! H1(g) and a map s: X ! H1(g) such that p O @ O s = g, H1(g)O_@__//_O B s|| p|| | fflffl| X ___g___//_B, where @ is the connecting map of the cofibre sequence and p is the canonical ma* *p. Proof.We identify G1(B) B with the mapping cone of B ! EB and H1(g) with the mapping cone of B ! E. By the discussion after Examples 3.2, there is a map "g1: H1(g) ! B such that the following diagram commutes H1(g) _"g1//_ B |r1| p|| fflffl| fflffl| X ___g___//_B. Since g is cocyclic, there exists a section s: X ! H1(g) of r1 by Theorem 3.7. * *Thus it suffices to show that "g1= @ : H1(g) ! B. 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MR 43 #6926 [Zab76] Alexander Zabrodsky, Hopf spaces, North-Holland Publishing Co., Amst* *erdam, 1976, North-Holland Mathematics Studies, Vol. 22, Notas de Matem'atica,* * No. 59. MR MR0440542 (55 #13416) Department of Mathematics, Dartmouth College, Hanover NH 03755 E-mail address: M.Arkowitz@Dartmouth.edu Department of Mathematics, Cleveland State University, Cleveland OH 44115 E-mail address: G.Lupton@csuohio.edu