Contemporary Mathematics The A-Category and A-Cone Length of a Map Martin Arkowitz, Donald Stanley and Jeffrey Strom Abstract.For any collection A of spaces we define two numerical invari- ants of maps: LA(f), the A-category of f, and LA(f), the A-cone length of f. These invariants are defined axiomatically, and our first results* * give equivalent constructive definitions in terms of mapping cone decompositi* *ons. We show that if A is the collection of all spaces, then LA(f) is the cat* *e- gory of f as defined by Fadell and Husseini and LA(f) is the cone length* * of f as defined by Marcum. By specializing to the maps * -!X and X -! * we obtain four invariants of spaces, including catA(X) = LA(* -!X) and clA(X) = LA(* -!X). Each of these four invariants has its own axiomatic and constructive definitions. We compare catA(X) and clA(X) with similar invariants defined by Scheerer and Tanr'e. We conclude by giving lower b* *ounds for these invariants in terms of cohomology. 1. Introduction The category of a topological space was introduced in the 1930's by Lusternik and Schnirelmann [L-S ] and shown to be a lower bound for the number of critical points of a smooth function on a manifold. Fox collected and extended the known results on L-S category in his encyclopedic 1941 paper [Fo ]. Among other thing* *s, he gave an axiomatic definition of the category of a space. Later, G. Whitehead [Wh1 ] introduced a homotopy invariant definition by means of deformations of * *the diagonal map. Subsequently, Ganea [G1 ] gave a characterization of category in terms of the existence of sections of certain fibrations. Since then there has * *been a great deal of research on category (see the surveys of James [J1 , J2] and Lema* *ire [Le ]) and on the related notion of cone length [G1 , C1 , C3 ]. Several author* *s have generalized the category and cone length of a space to that of a map [Mar , C1 * *]. In addition, the category and cone length of a space with respect to a fixed colle* *ction of spaces A have also been studied [S-T1 ]. In this paper we simultaneously generalize both of these notions by defining the category and cone length of a map f relative to a collection A, denoted LA * *(f) and LA (f) and called the A-category and A-cone length of f, respectively. We begin in Section 2 by giving two sets of simple axioms which may be satisfied by integer valued functions of maps. Then LA (f) is defined to be the maximum of ____________ 1991 Mathematics Subject Classification. Primary 55M30. Key words and phrases. Lusternik-Schnirelmann category, cone length, Ganea * *fibration. Oc0000 (copyright holder) 1 2 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM all functions which satisfy one set of axioms and LA (f) is similarly defined f* *or the other set. In Section 3 we show that LA (f) and LA (f) can be characterized as the length of the shortest appropriate A-decomposition of f. Thus, for example, LA (f) is thejsmallestjintegerjn such that f has a homotopy factorization of the form X = X0- !0X1-j! . 1.-.!Xn n-1 Y , where each ji is part of a mapping cone sequence Li-! Xi-!i Xi+1 with Li 2 A. Section 4 is devoted to proving that LA (f) is the category of a map f as defined by Fadell and Husseini [Fa , F-H ]* * and Cornea [C2 ], and that LA (f) is the cone length of the map f as defined by Mar* *cum [Mar ] when A is the collection of all spaces. The latter definition of the cat* *egory of a map is different from the one given by Fox [Fo ] and studied by Berstein a* *nd Ganea in [B-G ]. By taking f to be the map * -! X, we obtain the A-category of X and the A-cone length of X, and by taking f to be X -! * we obtain two more invariants, the A-kitegory of X, which is new, and the A-killing length of X, w* *hich was considered in [A-S1 , A-M-S ]. When A is the collection of all spaces, the first two invariants coincide with the standard notions. All four of these nume* *rical invariants of spaces have characterizations in terms of suitable A-decompositio* *ns of minimal length. In addition, we give axioms which uniquely determine each of these invariants. In Section 6 we establish order relations among the A-categor* *y, A- cone length, A-kitegory, and A-killing length of a space and give examples to s* *how that the inequalities can be strict. We end the section by comparing our notions of A-category and A-cone length of a space with those introduced by Scheerer and Tanr'e [S-T1 ]. We conclude the paper in Section 7 by giving some lower bounds * *for LA (f) and LA (f) in terms of the cohomology rings of X and Y . In [S-T2 ] Scheerer and Tanr'e have given a very general approach to Lustern* *ik- Schnirelmann type invariants. Their invariants are defined for a pair of maps w* *ith the same target, and are indexed by certain constructions on maps. Our invarian* *ts LA (f) and LA (f) can be obtained as a suitable specialization of their definit* *ion. This is discussed in some detail in Section 6. In a subsequent paper [A-S-S ] we will study the A-category and A-cone length of maps with regard to homotopy pushouts, products and fibrations. For the remainder of this section, we give our notation and terminology. All topological spaces are based and connected, and have the based homotopy type of well-pointed based spaces [May , p. 56]. All maps and homotopies preserve base points. We will distinguish notationally between a map f : X -! Y and its homo- topy class ff = [f] 2 [X, Y ]. We write f ' g to indicate that f is homotopic t* *o g. We let * denote the base point of a space or a space consisting of a single poi* *nt. In addition to standard notation, we use for same homotopy type, 0 : X -! Y for the constant map and id: X -! X for the identity map. We would like to thank the referee for several very helpful comments. 2. Axioms In this section we give the axiomatic definition of the two invariants LA and LA . We begin with some basic definitions. Definition 2.1. Given maps f : X -! Y and f0 : X0-! Y 0. THE A-CATEGORY AND A-CONE LENGTH OF A MAP 3 (1) We say that f0 dominates f (or that f is a retract of f0) if there is a homotopy commutative diagram X _____i_____//X0____r____//_X f|| |f0| f|| fflffl|j fflffl|s fflffl| Y ___________//Y_0________//_Y such that ri ' idand sj ' id. (2) If in addition ir ' id and js ' id, we say that (i, j) is a homotopy equivalence from f to f0 (and that (r, s) is a homotopy equivalence from f0 to f). Then f and f0 are called homotopy equivalent. Definition 2.2. By a collection A is meant a collection of spaces such that (1) * 2 A, (2) if A 2 A and A B, then B 2 A. A collection A is closed under suspension if A = { A | A 2 A} A. Examples of collections we shall consider are the following: (i) A = {all sp* *aces}, (ii) A = S, the collection consisting of all (finite or infinite) wedges of sph* *eres (of dimension 0) and *, (iii) A = M, the collection consisting of all wedges of M* *oore spaces, (iv) A = , the collection of all suspensions and (v) A = P, the collec* *tion of wedges of finite complexes. 2.3 A-Cone Axioms Let A be a collection. A function `A which assigns to each map f an integer 0 `A (f) 1 is said to satisfy the A-cone axioms if (1) (Homotopy Axiom) If f ' g, then `A (f) = `A (g); (2) (Normalization Axiom) `A (id) = 0; (3) (Composition Axiom) `A (f O g) f`A (f) + `A (g); (4) (Mapping Cone Axiom) If A -! X -! Y is a mapping cone sequence with A 2 A, then `A (f) 1; (5) (Equivalence Axiom) If f is homotopy equivalent to g, then `A (f) = `A * *(g). 2.4 A-Category Axioms A function `A which assigns to each map f an integer 0 `A (f) 1 is said to satisfy the A-category axioms if (1), (2), (3) and (4) of the A-cone axioms and the following axiom hold (50)(Domination Axiom) If f is dominated by f0, then `A (f) `A (f0). Clearly if `A satisfies the A-category axioms, then `A satisfies the A-cone * *ax- ioms. Remark 2.5. It is possible to slightly change and reduce the number of A- cone axioms and obtain a more streamlined set of axioms. For example, we could introduce the axiom (20)If h is a homotopy equivalence, then lA (h) = 0. Then clearly the A-cone axioms are equivalent to (1), (20), (3) and (4). We have chosen the axioms as given in (2.3) to emphasize the relation between these axi* *oms and the A-category axioms. Definition 2.6. Let A be a collection. The A-cone length LA is the maxi- mum of all functions `A satisfying the A-cone axioms, i.e., LA (f) = max{`A (f) | `A satisfies(1) - (5)}. 4 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM Similarly, the A-category of f is LA (f) = max{`A (f) | `A satisfies(1) - (4) and (50)}. Clearly LA satisfies the A-cone axioms and LA satisfies the A-category axiom* *s. Since the A-category axioms are stronger than the A-cone axioms, we have (2.7) LA (f) LA (f) for any map f. Moreover, if B A, then LA (f) LB(f) and LA (f) LB(f). Thus L{all spaces}(f) LA (f) and L{all spaces}(f) LA (f) for any collection* * A and any map f. In the next section we give alternate characterizations of LA (f) and LA (f) which will be useful in computing these numbers. 3. A-Decompositions We will see that the axiomatic definitions in (2.6) are quite useful and ser* *ve to focus on the characteristic properties of LA and LA . For calculations, however* *, it is convenient to have a constructive definition for these invariants. We develo* *p this approach in this section. We begin with some definitions. Definition 3.1. Given a collection A and a map f : X -! Y . An A-cone decomposition of f of length n is a homotopy commutative diagram j0 j1 jn-2 jn-1 X0 ____//_X1____//._._._//Xn-1____//XnOO || || || fn||s || f |fflffl| X ________________________________//_Y in which fn is a homotopy equivalence with homotopy inverse s and each map jiis part of a mapping cone sequence Ai _________//_Xi__ji___//_Xi+1 with Ai2 A. Thus fnjn-1 . .j.0' f, sf ' jn-1 . .j.0, fns ' idand sfn ' id. The homotopy commutative diagram above is an A-category decomposition of f of length n if s is simply a homotopy section of fn. Explicitly, fnjn-1 . .j.0' f, sf ' jn-1 . .j.0and fns ' id, but sfn need not be homotopic to the identity. Remark 3.2. In an A-category decomposition of a map f with notation as above, f is dominated by jn-1 . .j.0. Remark 3.3. We discuss briefly the existence of A-cone decompositions (and hence of A-category decompositions) for certain collections A. (1) If (Y, X) is a finite dimensional relative CW complex [Wh2 ], then Y is obtained from X by repeatedly attaching cells. Thus we obtain an S- cone decomposition of X ,! Y . This clearly extends to arbitrary maps f : X -! Y . (2) If X and Y are simply-connected finite dimensional complexes, then we obtain an M-cone decomposition of f from the homology decomposition of f [Hil]. Lemma 3.4. If f has an A-category decomposition of length n and f dominates g, then g has an A-category decomposition of length n. THE A-CATEGORY AND A-CONE LENGTH OF A MAP 5 Proof. We have a homotopy commutative diagram X0 ____i_____//_X____r____//_X0 g|| |f| |g| fflffl| fflffl| fflffl| Y 0____j_____//Y_____s____//_Y 0 with ri ' idand sj ' id, and an A-category decomposition of f with length n: L0 L1 |ff0| |ff1| fflffl|j0 fflffl|j1 jn-1 X0 __________//_X1_________//_._._.______//_XnOO || || || fn|oe| || f fflffl|| X _______________________________________//_Y with Li2 A and fnoe ' id. Consider j0 L0 _____ff0__//_X0 = X__________//X1_ ____ id|| |r| /.-,()*+I `1______ fflffl|rff0 fflffl| fflffl___ L0 _____________//X00____j0_____//_________________________X01 0 where X00= X0 and X01is defined as the homotopy pushout of the square I with auxilliary maps j00and `1. Thus X01is the mapping cone of rff0 so we have a map* *ping j00 0 cone sequence L0 rff0//_X00//_X1. Furthermore, there is a map g1 : X01-! Y maki* *ng the following diagram commute j0 fnjn-1...j1 X0 ___________//X1__________//Y |r| /.-,()*+I`1|| s|| fflffl|j00 fflffl|g1 fflffl| X00___________//____________88_________X01//________________* *____________Y 0. ________________________________________________________* *_____________________________________________________________________________* *_______ ___________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________ g For the next step, we consider j1 L1 ____ff1___//_X1__________//X2_ ____ id|| |`1| 76540123II`2______ fflffl|`1ff1 fflffl| _fflffl__ L1 __________//_X01___j0___//______________________X02, 1 where II is a homotopy pushout which defines X02. We then obtain g2 : X02-! Y 0 such that g2j01' g1 and g2`2 ' sfnjn-1 . .j.2. We continue in this way and end up with X0nas the mapping cone of `n-1ffn-1, a map `n : Xn -! X0nand a map gn : X0n-! Y 0such that gnj0n-1' gn-1 and gn`n ' sfn. Define a section 6 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM oe0 : Y 0-! X0nas the composition Y 0-j!Y -oe!Xn `n-!X0nThen gnoe0 ' id and oe0g ' `noejg ' `noefi' `n(jn-1 . .j.0)i ' (j0n-1. .j.00)ri ' j0n-1. .j.00. Thus j00 j01 j0n-2 0 j0n-1 X00_____//X01___//_._._.//_Xn-1____//X0nOO || g || 0 || n|oe| || g fflffl|| X0 ________________________________//Y 0 is an A-category decomposition of g of length n. Lemma 3.5. Let g : X -! Y and f : Y -! Z. (1) If g has an A-category decomposition of length k and f has an A-category decomposition of length l, then f O g has an A-category decomposition of length k + l. (2) If g has an A-cone decomposition of length k and f has an A-cone de- composition of length l, then f O g has an A-cone decomposition of leng* *th k + l. Proof. We prove the first assertion. Let R0 R1 S0 S1 |ff0| |ff1| fi0|| fi1|| fflffl|ifflffl|0i1 ik-1 fflffl|jfflffl|0j1jl-1 X0 _____//X1____//._._._//XkOO and Y0_____//Y1____//._._._//YlOO || g || || || || k||oe || fl|o| || g fflffl|| || f fflffl|| X _____________________//_Y Y _____________________//Z be A-category decompositions of g and f, respectively. Consider the diagram fi0 j0 S0 __________//_Y0 =_Y__________//Y1_ ____ id|| oe|| /.-,()*+I `1______ fflffl|oefi0 fflffl| fflffl___ S0 ____________//_Xk____ik_____//________________________Xk+1 where Xk+1 is defined as the homotopy pushout in Iowitheauxilliaryfmapsiik and `1. Note that there is a mapping cone sequence S0- ! Xk ik-!Xk+1. Now de0fine ~k+1 : Xk+1- ! Z by the diagram j0 Y0 = Y____________//Y1OO | OOO oe|| /.-,()*+I `1| Ofljl-1...j1OOOOO fflffl| ik fflffl|~k+OOO''1 Xk ____________//______________88____________Xk+1//_________* *___________________Z. ________________________________________________________* *_____________________________________________________________________________* *________________ ___________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________________@ fgk Continuing, we obtain a mapping cone sequence Sl-1-! Xk+l-1ik+l-1-!Xk+l, a ho- motopy pushout square which defines Xk+land a map ~k+l: Xk+l-! Z. Induction on r shows that ~k+rik+r-1. .i.k' fgk for 1 r l. Since fgkik-1. .i.0' fg, THE A-CATEGORY AND A-CONE LENGTH OF A MAP 7 these maps all fit into a decomposition ik-1 ik ik+1 ik+l-1 X0 _i0_//_X1_i1_//._._._//Xk____//Xk+1____//_._._.//_Xk+lUU __________ || ~ |_ae_______________* *___________________ || k+l|__________________* *______ || fg fflffl|__________ X _________________________________________________//_Z, where the section ae of ~k+l is the composition Z -o!Yl-`l!Xk+l. To prove the second assertion of Lemma 3.5, we repeat the previous proof with the added assumption that oe and o are homotopy inverses for gk and flrespectiv* *ely. From the homotopy pushout square I, we have that `1 is a homotopy equivalence. We continue as before and find that each `i is a homotopy equivalence. Thus ae * *is a homotopy equivalence. This completes the proof. Definition 3.6. Let A be a collection and f : X -! Y a map. We define an integer VA (f), 0 VA (f) 1, as follows: If f is a homotopy equivalence, set VA (f) = 0. If f has no A-category decomposition, set VA (f) = 1. Otherwise, VA (f) is the integer n such that there is an A-category decomposition of lengt* *h n but none of length < n. The integer VA (f) is similarly defined with `A-category decomposition' replaced by `A-cone decomposition.' We next show that the A-category decomposition length of f, that is, the integer VA (f), is equal to the A-category length of f, and a similar result wi* *th `A-cone' replacing `A-category.' These results are in the spirit of [S-T2 , Thm* *. 2.5] in which Scheerer and Tanr'e characterize their invariant by axioms. Their axio* *ms do not specialize to our axioms (2.3) and (2.4), though they are similar. Theorem 3.7. Let A be a collection and f : X -! Y a map. Then (1) LA (f) = VA (f); that is, LA (f) is the smallest n such that f has an A-category decomposition of length n. (2) LA (f) = VA (f); that is, LA (f) is the smallest n such that f has an A* *-cone decomposition of length n. Proof. We prove the assertion about LA (f) by first showing that LA (f) VA (f). Suppose VA (f) = n < 1, and let j0 j1 jn-2 jn-1 X0 ____//_X1____//._._._//Xn-1____//XnOO || || || fn||s || f |fflffl| X ________________________________//_Y be an A-category decomposition of f with length n. By Remark 3.2, f is dominated by jn-1 . .j.0. Therefore LA (f) LA (jn-1 . .j.0) LA (jn-1) + . .+.LA (j0) n by the Domination Axiom, the Composition Axiom, and the Mapping Cone Axiom, respectively. Thus LA (f) VA (f). Next suppose LA (f) = n. Lemmas 3.5 and 3.4 imply that VA (f) satisfies the A-category axioms (2.3), and hence VA (f) n. The proof of the second part is similar. 8 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM 4.Comparison with Other Notions of Cone Length and Category Many authors have studied notions of category and cone length similar to our* *s, so we now consider the relationships of our invariants LA and LA to previously defined concepts. In the case A = {all spaces}, Marcum has used the characteriz* *a- tion of LA in Theorem 3.7 as a definition of the cone length of a map [Mar ]. T* *his approach and a variation of it has also been given by Cornea [C1 ]. We next recall the definition of cat(f), the category of a map f : X -! Y , * *as introduced by Fadell and Husseini [Fa , F-H ] and redefined in terms of sections of fibrations by Cornea in [C2 ]. We note that these authors have denoted this * *as cat(Y, X), but we prefer cat(f) since it clearly depends on the map f and not o* *nly on the spaces X and Y . This is different from the concept described in [Fo , B* *-G ] and [J1 , Sec. 7]. One inductively defines a sequence of fibrations Fn(f)____in___//_Gn(f)__pn____//Y by the following method, known as the Ganea construction [G2 ]. Let jn+1 : Cin-! Y be the obvious extension of pn to the mapping cone Cin of in, and de- fine pn+1 : Gn+1(f) -! Y to be the result of converting jn+1 to a fibration. Th* *us Gn+1 Cin and Fn+1(f) is the fiber of pn+1 (i.e., the homotopy fiber of jn+1). The induction begins with G0(f) X, p0 the usual fiber map equivalent to f and F0(f) the fiber of p (i.e., the homotopy fiber of f). Thus there is a diagram j0 j1 jn-1 jn X _______G0(f)____//G1(f)____//._._._//Gn(f)____//_. . . f || p0|| p1|| pn|| fflffl| fflffl| fflffl| fflffl| Y ________Y__________Y________. ._._____Y________. ... We denote the inclusion jn-1 . .j.0by On : X -! Gn(f) and say cat(f) n if the* *re is a section s : Y -! Gn(f) of pn such that sf ' On. Proposition 4.1. If A = {all spaces}, then LA (f) = cat(f) for every f : X -! Y . Proof. We first show that cat(f) satisfies the A-category axioms with A = {all spaces}. Clearly catis homotopy invariant and cat(id) = 0. We now establish the Composition Axiom. Consider f : X -! Y and g : Y -! Z with cat(f) = k and cat(g) = l. Thus the fiber map pk : Gk(f) -! Y has a section s and an inclusion Ok : X -! Gk(f) with sf ' Ok; likewise, the fiber map p0l: Gl(g) -! Z has a section s0and an inclusion O0l: Y -! Gl(g) such that s0g ' O0l. There is * *also a fiber map p00k: Gk(g O f) -! Z with inclusion O00k: X -! Gk(g O f). We obtain (by l applications of the Ganea construction) the fiber map bpl: Gl(p00k) -! Z * *with inclusion bOl: Gk(g O f) -! Gl(p00k). Clearly Gl(p00k) = Gk+l(g O f), and we h* *ave homotopy commutative diagrams O00k Gl(p00k)___________Gk+l(g O f) X ______________//_Gk(g O f) EE tt EEE ttt |O00 Obl| bplEE""EEzztp00k+ltttt ffk+llffl|| fflffl|| Z Gk+l(g O f)____________Gl(p00k). THE A-CATEGORY AND A-CONE LENGTH OF A MAP 9 Thus we have the diagram X QQQ_____________O00k__________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *___________________________ QQQ _______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________ |f| QOkQQQQQ ______________________________________* *______________________________________________________________________________ fflffl| s QQ(( _g __''____________________________* *__ Y ____________//_Gk(f)pkoo______//Gk(g O f) x |g| xxxx |p00k| fflffl| bOlxxxx fflffl| ZOO xx ZOO || xxx | p0ls0|| xxx |p00k+l |fflffl|_fl --xxx | Gl(g)___________//Gl(p00k)________Gk+l(g O f) where _gand __flare induced by _g X _____id____//X Y ___s_//Gk(f)____//Gk(g O f) f|| |gOf| and |g |p00k| fflffl|g fflffl| fflffl|| fflffl| Y ___________//Z Z _________id________//Z, respectively. Then a long but straightforward calculation shows0that_thefdesir* *edl section of p00k+l: Gk+l(g O f) -! Z is the composition Z -s!Gl(g) -! Gk+l(g O f* *). This establishes the Composition Axiom. f Next we prove the Mapping Cone Axiom for cat(f). Let A -i!X -! Y be a mapping cone sequence. We must show that cat(f) 1. There is a map g : A -! F0(f) such that i0g = ki: f A _______i______//X______________//Y_ ___ |g| |k| _s______ fflffl| i0 fflffl| j0 fflffl___ F0(f)__________//_G0(f)_________//G1(f) and hence a map s : Y -! G1(f) such that the above diagram commutes. We will see that a modificationfof s is the desired section of p1 : G1(f) -! Y . The ma* *pping cone sequence A -i!X -! Y yields an action of [ A, Y ] on [Y, Y ] denoted by ex* *po- nentiation [Hil, Ch. 15]. Now [p1s] 2 [Y, Y ] and f*[p1s] = [f] = f*[id]. There* *fore there is an ff 2 [ A, Y ] such that [p1s]ff= [id]. But (p1)* : [ A, G1(f)] -! [* * A, Y ] is onto since the Ganea fibration splits after looping [G2 , Prop. 1.5], and so* * ff = (p1)*fi for some fi 2 [ A, G1(f)]. Then (p1)*([s]fi) = [p1s](p1)*fi= [id], and [s]fi[f] = f*[s]fi= f*[s] = [j0]. Therefore there is a map s0 in the homotopy class [s]fiwhich is a section of p1, and s0f ' O1. Finally we establish the Domination Axiom. Suppose g dominates f, so we have the homotopy commutative diagram X _____i_____//X0____r____//_X f|| |g| f|| fflffl|j fflffl|s fflffl| Y ___________//Y_0________//_Y 10 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM with ri ' id and sj ' id. Suppose cat(g) n. Then we have a homotopy commutative diagram Gn(f) _____'____//_Gn(g)___ae____//_Gn(f)OO pn|| oe0p0n|||| |pn| fflffl| j |fflffl| s fflffl| YO______________//Y_0____________//YOOOOO f|| g|| |f| | | r | X _______i______//X0_____________//X where oe0is the section of p0nand ' and ae are induced by (i, j) and (r, s), re* *spectively. Then oe = aeoe0j is clearly a section of pn compatible with f. Thus there is a * *map s0' oe which is a section of pn with s0f ' oef ' On. Since cat(f) satisfies the A-category axioms, cat(f) LA (f) for every map * *f. For the reverse inequality, we assumejthat cat(f) = n. Thus the sequence of mapping cone sequences Fi(f) -! Gi(f) -! iGi+1(f), 0 i < n, is an A-category decomposition of f with length n. By Theorem 3.7, LA (f) n = cat(f), and this completes the proof. Remark 4.2. When A = {all spaces}, Marcum has established the Composi- tion Axiom for LA (f) [Mar ]. In this case Cornea has given the Composition and Domination Axioms as properties of LA (f) = cat(f) [C2 ]. The following is the analogue of a known result for spaces [Ta , Section 5]. Proposition 4.3. For any collection A and map f : X -! Y , LA (f) n if and only if f is dominated by a map g such that LA (g) n. Proof. If f is dominated by a map g with LA (g) n, then LA (f) LA (g) LA (g) n. Conversely, if LA (f) n, then f has an A-category decomposition of length n in-2 in-1 X0 _i0_//_X1i1__//._._._//Xn-1____//XnOO || || || fn||s || f |fflffl| X ________________________________//_Y. Let g = in-1 . .i.0: X0- ! Xn. Then the same decomposition is an A-cone decom- position of g. By Remark 3.2, g dominates f. The following corollary is useful to prove results about LA . Corollary 4.4. If LA (f) = n, then f is dominated by a map g with LA (g) = n. Furthermore, we can choose g with the same domain as f. 5. Numerical Invariants of Spaces In this section we consider LA (f) and LA (f) for special maps f in order to obtain invariants of spaces. Definition 5.1. For any collection A and space X, define (1) the A-cone length of X by clA(X) = LA (* -! X), THE A-CATEGORY AND A-CONE LENGTH OF A MAP 11 (2) the A-killing length of X by klA(X) = LA (X -! *), (3) the A-category of X by catA(X) = LA (* -! X), (4) the A-kitegory of X by kitA(X) = LA (X -! *). Remark 5.2. (1) Let A = {all spaces}. Then catA(X) is the ordinary (reduced) category of X and clA(X) is the cone length of X as studied by Ganea, Cornea, Stanley and many others, denoted cat(X) and cl(X), re- spectively. On the other hand, the mapping cone sequence X -! X -! CX, where CX * is the cone on X, with X 2 A shows that klA(X) = kitA(X) = 1 when A = {all spaces}. (2) Since LA (f) LA (f), we have catA(X) clA(X) and kitA(X) klA(X) for any collection A. We will see in the next section that if A is clos* *ed under suspension, then klA(X) clA(X) and kitA(X) catA(X). (Note: the latter inequality is the source of the notation `kit', since kitten* *s are smaller than cats. The term `kittygory' was used first by Freyd in anot* *her context [Fr].) (3) The A-killing length was considered in [A-S1 ]. Let ZA (X) [X, X] be the subsemigroup consisting of all homotopy classes ff = [f] 2 [X, X] s* *uch that f* = 0 : [A, X] ! [A, X] for every A 2 A. It was proved in [A-S1 , Thm. 3.3] that the nilpotency of the semigroup ZA (X) is bounded above by klA(X). We first show that each of the four invariants of spaces has a characterizat* *ion in terms of decompositions. Definition 5.3. Let A be a collection and consider a sequence of mapping cone sequences Ai__________//Xi___ji___//Xi+1, 0 i < n with Ai2 A. If X0 = * and Xn X, we have a clA-decomposition of X with length n. If X0 = * and X is dominated by Xn, we have a catA-decomposition of X with length n. If X0 = X and Xn *, we have a klA-decomposition of X with length n. If X0 = X and X is dominated by *, we have a kitA- decomposition of X with length n. The n mapping cone sequences in Definition 5.3 form a kitA-decomposition of X of length n if and only if X0 = X and jn-1 . .j.0' 0. Definition 5.4. The length of a minimal catA-decomposition of X is defined as follows: It is 0 if X * and is 1 if X has no catA-decomposition. Otherwise, it is the integer n such that X has a catA-decomposition of length n but none of length less than n. Similar definitions hold for clA, klAand kitA. The following proposition is an immediate consequence of Theorem 3.7. Proposition 5.5. If A is a collection and X is a space, then (1) catA(X) is the length of a minimal catA-decomposition of X; (2) clA(X) is the length of a minimal clA-decomposition of X; (3) klA(X) is the length of a minimal klA-decomposition of X; (4) kitA(X) is the length of a minimal kitA-decomposition of X. 12 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM We next show that the four invariants can be characterized by axioms. Let A be a collection and let cA be a function which assigns to every space X an integer 0 cA (X) 1. We consider the following axioms. (1) cA (*) = 0 (2) If X and Y have the same homotopy type, then cA (X) = cA (Y ). (20)If X is dominated by Y , then cA (X) cA (Y ). (3) If A -! X -! Y is a mapping cone sequence with A 2 A, then cA (Y ) cA (X) + 1. (30)If A -! X -! Y is a mapping cone sequence with A 2 A, then cA (X) cA (Y ) + 1. Proposition 5.6. (1)clA(X) is the maximum of all cA satisfying Ax- ioms (1), (2) and (3); (2) catA(X) is the maximum of all cA satisfying Axioms (1), (20) and (3); (3) klA(X) is the maximum of all cA satisfying Axioms (1), (2) and (30); (4) kitA(X) is the maximum of all cA satisfying Axioms (1), (20) and (30). The proof closely follows the arguments given in Section 3, and hence is omi* *tted. Remark 5.7. An axiomatic treatment of the category and the cone length of a space relative to a collection A has been given by Scheerer-Tanr'e [S-T1 ]. T* *heir axioms are similar to ours, but are not equivalent, as we shall see in the next* * section. The above axioms are useful in focusing on the basic properties of these inv* *ari- ants. However they can also be used to establish some elementary inequalities f* *or these invariants. We illustrate this by giving an axiomatic proof of G. Whitehe* *ad's theorem on the nilpotency of the group [X, G] when G is a group-like space. Rec* *all that if ss is a group, then nilss is the smallest integer n 1 such that all (* *n+1)-fold commutators of the form [...[[x1, x2], x3], ..., xn+1] are trivial. We set nils* *s = 0 if ss is the trivial group. The following result was originally proved in [Wh1 ]. Theorem 5.8. If X is any space and G is a group-like space, then nil[X, G] catX. Proof. By Proposition 5.6 it suffices to show that the function nil[X, G] s* *atis- fies the cataxioms, where G is a fixed group-like space. Clearly Axioms (1) and* * (20) hold. We prove (3). Let A -! X -! Y be a mapping cone sequence and consider part of the Puppe sequence [ A, G] -ffi![Y, G] -! [X, G]. It is well known that the image of ffi is in the center of [Y, G]. From this it* * follows that nil[Y, G] nil[X, G] + 1. 6. Properties of the Invariants of Spaces In the first part of this section we consider the ordering of the invariants* * catA, clA, klA and kitA. In the second part we compare catA and clAto the analogous invariants which were studied by Scheerer-Tanr'e [S-T1 ]. Proposition 6.1. For any collection A, kl A(X) clA(X) and kit A(X) catA(X). THE A-CATEGORY AND A-CONE LENGTH OF A MAP 13 Proof. We only prove the second inequality since the first is similar. Supp* *ose catA(X) = n with catA-decomposition Ai-! Xi-! Xi+1, 0 i < n, where Ai2 A, X0 = * and X is dominated by Xn. For i = 1, . .,.n, let ki: Xi-! Xn be the composition Xi-! Xi+1-! . .-.!Xn and define X(i)to be the mapping cone of ki. Then we have a mapping cone sequence Ai-! X(i)-! X(i+1). Clearly X(0)= Xn and X(n) *, and so kl A(Xn) n. Therefore kit A(X) kit A(Xn) kl A(Xn) n = catA(X). The first inequality of Proposition 6.1 was proved in [A-S1 ]; a stable vers* *ion was proved in [Ch ]. Thus if A is a collection which is closed under suspension, then by Remark 5* *.2 and Proposition 6.1 the inequalities between the four invariants can be represe* *nted by a square clA(X) s JJJ sss JJJ sss JJ sss J (6.2) catA(X)K klA(X) KKK ttt KKK ttt KKK ttt kitA(X) where the lower number is less than or equal to the higher number. We next show that each of the inequalities in (6.2) can be strict. First observe that if X is a co-H-space which is not a suspension, then cat(* *X) = 1 < 2 cl(X). More generally, Stanley [St] has given examples of spaces X(n) with cat(X(n)) = n and cl(X(n)) = n + 1. Next let S be the collection of wedges of spheres and let X = Sn1 x . .x.Snr, r 3. If k is the smallest integer such that r 2k-1, then it was shown in [A* *-M-S , Prop. 6.2] that klS(X) = k. The standard cone decomposition of X together with the fact that catS(X) cat(X) = r imply that catS(X) = clS(X) = r. Thus klS(X) = k < r = clS(X) and kitS(X) klS(X) = k < r = catS(X). Of course these inequalities hold if A = {all spaces} and X is any space with cat(X) > 1, since kitA(X) klA(X) 1 in this case. Finally we discuss the inequality kitA(X) klA(X). Consider the collection A = of all suspensions and let X be a simply-connected co-H-space which does not have the homotopy type of a suspension. Then X is a homotopy retract of some suspension X0 [Hil, p. 209] so there are maps i : X ! X0 and rj: X0 ! X such that ri ' id. Then there is a mapping cone sequence X0-r! X -! C, with C the mapping cone of r. Clearly j ' 0, sojkit(X) 1. But if kl (X) 1, there would be a mapping cone sequence A -r!X -! * with A a suspension. Hence X A, contradicting the fact that X does not have the homotopy type of a suspension. Thus kl (X) > 1, so kit(X) < kl (X). As long as all of the spaces involved are simply connected, this kind of example will work for collections A which are not closed under retracts. As mentioned in the introduction, Scheerer and Tanr'e have developed a very general framework for category type invariants [S-T2 ]. This includes the invar* *iants we studied here as well as those studied in [S-T1 ] as special cases, as we now 14 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM describe. If A is a collection of spaces_and_f : X -! Y is a map, then we may f* *orm the collection KA (f)_consisting_of all f: Cj- ! Y , where j : A -! X is a map * *with A 2 A, f O j ' 0 and f an extension of f. The correspondence f 7! K(f) is a construction in the sense of [S-T2 ], and in the notation of [S-T2 ], LA (f) ca* *n be identified LA (f) = (KA , id)- cat(f). It can be shown that this coincides with the invariant K(GA )- cat(f) in [S-T2 , Rem. 3.4]. Scheerer and Tanr'e also identify A- cat(X) from [S-T1 ] in terms of their more general setup, namely, A- cat(X) = (K(GA ), GA (X))- cat(idX, idX). Similar considerations hold for LA (f) and A- cl(X). The main theorems in [S-T2* * ] characterize these invariants by simple properties that are similar to, but not* * the same as, our axioms. No relationship between our catA(X) and their A- cat(X) seems to follow immediately from the results of [S-T1 , S-T2 ]. We obtain a rel* *a- tionship next, as well as a relation between clA(X) and A- cl(X). For any collection A and space X, A-cat(X) and A-cl(X) have been defined in [S-T1 ]. (The latter is denoted A - Cat(X) in [S-T1 ].) Let us set ( A- fcat(X) = 0A- cat(X) + 1ifXifX*6 *. and define A-ecl(X) similarly. It is proved in [S-T1 ] that -cfat(X) = cat(X) * *and -ecl(X) = cl(X). That is, if A = {all spaces}, then catA(X) = A - fcat(X) and clA(X) = A- ecl(X) by Proposition 4.1. It is reasonable to then ask if these latter equalities hold for any collect* *ion A. Proposition 6.3. If A is a collection which is closed under suspension, then catA(X) A - fcat(X) and clA(X) A- ecl(X). Proof. We prove the proposition for cat - the proof for cl is similar. Defi* *ne ( fcatA(X) = 0 ifX *; catA(X) - 1 ifX 6 *. Then we must show fcatA(X) A - cat(X). Since A - catis the maximum of all functions which satisfy the Scheerer-Tanr'e axioms for A [S-T1 , Thm. 1, Sect. 2], it suffices to show that cfatAsatisfies these axioms. That is, we mu* *st show: (1) If X 2 A, then fcatA(X) = 0. (2) If X is dominated by Y , then fcatA(X) fcatA(Y ). (3) If X ! Y ! Z is a mapping cone sequence with X 2 A, then fcatA(Z) fcatA(Y ) + 1. The proof of this is straightforward and hence omitted. It is possible to give an example of a collection A and a space X such that * *each inequality in Proposition 6.3 is strict, and so equality does not hold in gener* *al. We present an example only for cone length. Example 6.4. Let A = {Sk, Sl_ Sl| k 2n - 1, l n - 1} and X = Sn x Sn. We sketch the argument that clA(X) 2 < A - ecl(X). The two mapping cone sequences Sn-1 _ Sn-1 -! C -! Sn _ Sn and S2n-1-W! Sn _ Sn -! Sn x Sn, THE A-CATEGORY AND A-CONE LENGTH OF A MAP 15 where C is the cone on Sn-1 _ Sn-1 and W is the Whitehead product, yields clA(X) 2. Since A- ecl(X) = A- cl(X) + 1, it suffices to show A- cl(X) > 1. If A- cl(X) 1 then one can show (using an argument similar to the proof of Theorem 3.7) that there is a mapping cone sequence A -! B -! X with A, B 2 A. The exact homology sequence of the mapping cone sequence implies that Hn(B) ~=Z Z, and so B = Sn _ Sn. Again applying the homology sequence, we conclude that H2n-1(A) ~=Z, which is impossible. Therefore A- cl(X) > 1. There is however an important case when equality does hold in Proposition 6.* *3. Let S = S0 be the collection of wedges of spheres of dimension 0 and S1 be the collection of wedges of spheres of dimension 1 so that S0 = S1. We first pro* *ve a lemma. Lemma 6.5. Suppose clS0(X) n. Then there are mapping cone sequences Li-! Xi-! Xi+1, 0 i < n, such that L0 2 S0, X0 = *, Xn X and Li 2 S1 for i 1. Proof. The proof is by induction on n. When n = 0 there is nothing to prove. Suppose the lemma is true for n - 1. Since clS0(X) n, there are mapping cone sequences Mi-! X0i-! X0i+1, 0 i < n, such that Mi2 S0, X00= * and X0n X. Consider the nth mapping cone sequence f 0 0 Mn-1 = Ln-1 _ S0 _ . ._.S0_________//Xn-1_________//Xn where Ln-1 2 S1. Since X0n-1is connected, f|S0 _ . ._.S0 ' 0. Let f0 = f|Ln-1 : Ln-1 ! X0n-1. Then X0n= Cf0_S1_. ._.S1 and we have a mapping cone sequence 0 Ln-1____i1f___//X0n-1_ S1 _ . ._.S1_______//X0n, with i1 : X0n-1! X0n-1_ S1_ . ._.S1 the inclusion. But we have that clS0(X0n-1_ S1 _ . ._.S1) clS0(X0n-1) n - 1. By induction there exist mapping cone sequences Li-! Xi-! Xi+1, 0 i < n - 1, with L0 2 S0, Li 2 S1 for i 1, X0 = * and Xn-1 X0n-1_ S1 _ . ._.S1. This proves the lemma. Proposition 6.6. catS0(X) = S1- fcat(X) and clS0(X) = S1- ecl(X). Proof. We prove the result for cone length. By Proposition 6.3 it suffices * *to show S1- ecl(X) clS0(X). If X * or if clS0(X) = 1, this is clearly true. Now suppose clS0(X) = n with 0 < n < 1. We must show S1- cl(X) n - 1. By Lemma 6.5 there exist mapping cone sequences Li-! Xi-! Xi+1, 0 i < n, L0 2 S0, Li2 S1 for i 1, X0 = * and Xn X. But L0- ! X0- ! X1 shows that X1 L0 2 S1. By the Scheerer-Tanr'e axioms, S1- cl(X) = S1- cl(Xn) S1- cl(Xn-1) + 1 . . .S1- cl(X1) + n - 1 = n - 1. Remark 6.7. In [A-S-S ] we have proved that if A is any collection that is c* *losed under wedges and joins, then catA(X x Y ) catA(X) + catA(Y ). By Proposition 6.6 this implies S1- cat(X xY ) S1- cat(X)+S1- cat(Y ). This answers a questi* *on in [S-T1 , Sect. 7, Question 8]. 16 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM 7. Cohomological Lower Bounds Recall that the cup length of a space X, denoted [- length(X), is the length of the largest nonzero cup product in eH*(X; R) for any commutative ring R. The n-fold reduced diagonal b n: X -! X(n)from X to the n-fold smash product X(n) of X with itself is the composite of the diagonal map n : X -! Xn with the quotient map Xn -! X(n). The weak category of a space X, denoted wcat(X), is the greatest integer n for which b nis not homotopic to the constant map. In the special case A = {all spaces}, the following theorem is stated without proof by Cornea [C2 , p. 68], and the corresponding statement for homology can * *be found in [F-H ]. We omit the proof. Theorem 7.1. Let A be a collection and let f : X -! Y induce the homomor- phism f* : eH*(Y ; R) -! eH*(X; R) of reduced cohomology with coefficients in t* *he commutative ring R. If LA (f) n, then (ker(f*))(He*(Y ; R))n = 0, and so min{n | (ker(f*))(He*(Y ; R))n = 0} LA (f) LA (f). The following corollary is useful for applications. Corollary 7.2. If A is any collection and f : X -! Y , then [- length(Y ) [- length(X) + LA (f). Proof. Let k = [- length(X) and n = LA (f). Suppose y1, . .,.yn+k+1 2 eH*(Y ; R) are any n + k + 1 elements. We must show y1. .y.n+k+1= 0. Now f*(y1. .y.k+1) = f*(y1) . .f.*(yk+1) = 0, so y1. .y.k+12 ker(f*). On the other hand, yk+2. .y.n+k+12 (He*(Y ; R))n. By Theorem 7.1, y1. .y.n+k+1= 0. When f : * -! Y , Corollary 7.2 yields the well-known inequality [- length(Y* * ) cat(Y ). However, when f : X -! *, we obtain no information from Corollary 7.2. To obtain lower bounds for kitAand klA, we prove the following. Proposition 7.3. Let A be a collection with the property that each A 2 A has wcat(A) 1. Then [- length(X) wcat(X) 2kitA(X)- 1 2klA(X)- 1 for every space X. Proof. The first inequality is well-known [B-G ], and the last follows from kitA(X) klA(X). Therefore we only need to prove wcat(X) 2kitA(X)- 1. We proceed by induction on kitA(X). If kitA(X) = 0, then X is contractible, and the result follows. Assume that the inequality holds for all spaces with ki* *tA< m, and that kitA(X) = m. Considerjthepfirst mapping cone sequence in a minimal kitA-decomposition of X, L0- ! X -! X1. Then there is a homotopy commutative diagram b 2 L0___________//L0 ^ L0 j || |j^j| fflffl|b 2 |fflffl X ___________//X ^ X. Since L0 2 A, it follows that b 2' 0 : L0- ! L0 ^ L0, and so b 2j ' 0. Thus the* *re is a ffi : X1- ! X ^ X such that ffip ' b 2. Hence we have a homotopy commutati* *ve THE A-CATEGORY AND A-CONE LENGTH OF A MAP 17 diagram b 2 b 2m-1 m-1 X MM________//_XO^OX_________//_(X ^ X)(2 ) MM | OO MMMM | | (2m-1) pMMMMM ffi| ffi| MM&&| b 2m-1 |m-1 X1 _______________//X(21 ). m-1) Since kitA(X1) m - 1, we have b 2m-1' 0 : X1- ! X(21 . Therefore b 2m= b 2m-1b 2' 0 : X -! X2m , and so wcat(X) 2m - 1. The result wcat(X) 2kl (X)- 1 was proved in [A-S2 ] by a similar method. Since wcat(X) cat(X) catA(X), it might be conjectured that catA(X) is bounded above by 2kitA(X)- 1 for any collection A as in Proposition 7.3. The following example shows that this is not the case for A = S. Example 7.4. Let A be a 2-dimensional, noncontractible acyclic complex. Such a space may be constructed from a perfect group G, such as the one de- scribed in [Hig ], which has a presentation with the same number of generators * *as relations. If A is obtained from a wedge of circles by attaching a 2-cell for * *each relation, then ss1(A) = G and A is acyclic. For any such space A,WA ^ A *, but n cat(An) catS(An) for each n [G-H , Th. 1.1]. Write B = nAn, and conclude that catS(B) is infinite since B dominates An for each n. We claim that kitS(B) = 1. For this, it is enough to show that kitS(An) = 1 for each n. We do this by induction on the intermediate wedges Tk(An) = {(x1, . .,.xn) | at leastn - k of thexi= *} An. When k = 1, we obtain the usual wedge, and when k = n - 1 we obtain the fat wedge. W W Since AWis 2-dimensionalLthere is a mapping cone sequence S1 -! A -! S2. Since [A, S2] ~= H2(A; Z) = 0, we see, using the comment following Definition 5.3, that kitA(A) = 1, and the same is true for T1(An) for each n. We assume kitS(Tk(An)) = 1. For each k there is a mapping cone sequence ` Tk(An) -! Tk+1(An) -! A(k+1) *. The argument used to prove Theorem 3.4 in [A-S1 ] shows that kitA is subad- ditive on mapping cone sequences (see also [A-S-S ]). Thus kitS(Tk+1(An)) kitS(Tk(An)) + kitS(*) = 1. This completes the example. The space A could be constructed equally well by removing a point from a non- simply connected manifold which is a homology 3-sphere. Also, the space B is an example of a space with infinite category and weak category 1. Felix, Halperin * *and Thomas have constructed a simply-connected space with this property in [F-H-T , p. 432]. Finally we mention that even in spaces with small cup length, cohomology can sometimes reveal useful lower bounds. This result is stated in [C1 , p. 377] a* *nd [S-T1 ] for Steenrod squares. 18 MARTIN ARKOWITZ, DONALD STANLEY AND JEFFREY STROM Proposition 7.5. Suppose that there are cohomology operations `1, `2, . .,.`n and an element u 2 H*(X) (with any coefficients) such that `1 O . .O.`n(u) 6= 0. Then klS(X) > n. W Proof. We work by induction on klS(X). If klS(X) = 0, then X Snff and clearly every cohomology operation acts trivially on H*(X). Now assume the result is known for all spacesWwith klS