Essential Category Weight and Phantom Maps

Transcription

1 Essential Category Weight and Phantom Maps Jeffrey Strom Wayne State University The purpose of this paper is to study the relationship between maps with infinite essential category weight and phantom maps. Amapf : X Y has essential category weight at least N if f g whenever g : Z X and the Lusternik-Schnirelmann category of Z is at most N. We write E(f) N; observethatiff, thene(f) < cat(x). The appendix to this paper contains a brief summary of the main results on essential category weight. Amapf has infinite essential category weight if E(f) N for all N. Write E(X, Y ) to denote the set of maps f : X Y with E(f) =. It follows from Theorem 9 of the appendix that E(X, Y ) is functorial in both X and Y. Rudyak [11] has made the observation that if E(f) =, thenf must be a phantom map. Recall that a map f : X Y is a phantom map if f g whenever g : Z X and the dimension of Z is finite (this is a phantom map of the first kind in McGibbon s article [7]). Since finite dimensional spaces have finite category, this observation is immediate. Let Ph(X, Y ) denote the set of homotopy classes of phantom maps f : X Y. Thus, E(X, Y ) Ph(X, Y ). It is not hard to see that this inclusion can be proper. For example, there is a phantom map f : CP S 3 whose suspension is nontrivial [5]. Thus, Σf is a phantom map, but E(Σf) =1. For another example, if G is any one of the groups Sp(2),Sp(3),G 2,F 4,then there are stably nontrivial phantom maps ΩG K for certain finite type CW complexes K [7]. The suspensions of these maps are phantom maps with essential category weight 1. This leads us to a natural question: for what spaces is it true that E(X, Y )=Ph(X, Y )? Our first theorem is a partial answer to this question. 1

2 Theorem 1 Let X be a space whose loop space ΩX is homotopy equivalent to a connected finite dimensional CW complex. Then for any Y E(X, Y )=Ph(X, Y ). Let f : X Y be a phantom map; we need to show that E(f) =. By Theorem 10 in the appendix, it suffices to show that f BN ΩX for each N>0. This is the case because, since ΩX is homotopy equivalent to a finite dimensional CW complex, so is B N ΩX. It is trivial that E(X, Y )=Ph(X, Y )ifph(x, Y )=. But this is far from the situation in Theorem 1, as we now show. According to Gray and McGibbon [6], if ΩX is homotopy equivalent to a finite dimensional CW complex, then the universal phantom map out of X is nontrivial. Thus, we have the following corollary. Corollary 2 If ΩX is homotopy equivalent to a finite dimensional CW complex, then there is a space Y such that E(X, Y ). It follows that cat(x) =. The universal phantom map [6] is a useful tool in the study of Ph(X, Y ). There is an analagous map out of X which is weakly universal with respect to the property of having infinite essential category weight. For each CW complex X, we may form the cofiber sequence BN ΩX BΩX i X Φ ΣB N ΩX. in which i is the wedge of the inclusions of the B N ΩX. Theorem 3 Every map f : X Y with E(f) = has a factorization X f - Q QQs Φ 3 ΣBN ΩX. Y This factorization is not unique in general. 2

3 The proof is a straighforward adaptation of Gray and McGibbon s Theorem 1 in [6], in which we replace the filtration by the filtration X 0 X 1 X n X B 1 ΩX B N ΩX BΩX X and use Theorem 10 to interpret essential category weight in terms of this filtration. This filtration of X can also be used to give an algebraic computation of E(X, Y ). Theorem 4 For CW complexes X and Y, E(X, Y ) = lim 1 [ΣB N ΩX, Y ]. According to Bousfield and Kan [1], there is a short exact sequence lim 1 [ΣB N ΩX, Y ] [X, Y ] lim[b N ΩX, Y ]. Theorem 10 allows us to identify E(X, Y )withj 1 ( ). Some other results of [6] carry over as well. For example, we have the following corollary, which is analagous to Theorem 2 in [6]. Corollary 5 Let X be a CW complex; then E(X, Y )= for every Y if and only if ΣX is dominated by ΣB N ΩX. Gray and McGibbon use the universal phantom map to show that if f and g are composable phantom maps, then f g ; the same is true a fortiori if E(f) =E(g) =. However, we can say much more. Proposition 6 If g is a phantom map and E(f) > 1, then f g. This is the case, in particular, if E(g) =. j 3

4 We have the following commutative diagram g X - Y - Σ( X n ),,, j, f Since cat(σ ( X n )) = 2 and E(f) 2, we conclude that f j,which finishes the proof. We have seen how to apply the theory of phantom maps to the study of maps with E(f) =. Our final result goes in the other direction: we will use the theory of essential category weight to answer a question of McGibbon. Following Roitberg [9], McGibbon observes that if X and Y have finite type and Y is grouplike, then Ph(X, Y ) has an abelian group structure which is natural in X. He asks in Question 6 of [7] whether the assumption of finite type is necessary. The exact sequence due to Bousfield and Kan shows that it is not. McGibbon s intention was to ask for a geometric argument that shows why the commutator of two phantom maps should be trivial. This is the question we resolve in our proof of Theorem 7. Theorem 7 If X and Y are CW complexes and Y is grouplike, then Ph(X, Y ) has an abelian group structure which is natural in X. We have to show that the commutator of any two phantom maps is trivial. Let χ : Y Y Y denote the commutator map. Since χ Y Y,there is a factorization χ Y Y - Y Q QQs 3 Y Y 4

5 in which is the usual quotient map. By Theorem 8 in the appendix, E( ) > 1. Using Theorem 9 we see that E(χ) > 1. Now let f,g : X Y be phantom maps; it follows that their product f g : X X Y Y is also phantom. Since the commutator [f,g] is equal to χ (f g), the theorem follows directly from Proposition 6. Appendix: Essential Category Weight This is a summary of results from [12]. Essential category weight is homotopy invariant version of category weight, which was introduced by Fadell and Husseini in [2]. Essential category weight has been studied independently by Rudyak [11], who called it strict category weight. Many of the most important lower bounds on the category of a space X take the form of a product formula. Theorem 8 can be considered the basic product formula from which all others follow. Theorem 8 Let f : X K and g : Y L. In the diagram f g - X Y K L Q QQQ f g Qs? K L we have E( f g) E(f)+E(g). Write E(f) =p and E(g) =q, and suppose h : Z X Y with cat(z) p + q. Then we may write Z = A B with cat(a) p and cat(b) q. Thus (f h) A and (g h) B. We can use homotopy extension to lift (f g) h into K L, which shows that (f g) h and completes the proof. 5

6 When f and g represent cohomology classes (in any cohomology theory), this result becomes a stronger version of the classical cup length lower bound for cat(x). When f and g are maps to a grouplike space, we obtain a strengthening of Whitehead s theorem [15]that the nilpotence length of [X, Y ] is bounded above by the category of X. Next we give a formula for the essential category weight of a composition of two maps. Theorem 9 Let f : X Y and g : Y Z. Then E(g f) E(g) E(f). Write E(f) =p and E(g) =q and suppose h : W X with cat(w ) pq. Then we may write W = A 1 A q with cat(a i ) p. Thus(f h) Ai for each i,andsof h has a factorization h - Q QQs 3 i W X - Y Z W f g - in which W = W ( CA i ). Since cat(w ) q and E(g) =q, we conclude that g i, which proves the theorem. A very important consequence of this theorem is that E(g f) E(g) and E(g f) E(f). If f represents a cohomology class u and g represents a cohomology operation θ, we find that E(θ(u)) E(θ) E(u). Thus, Fadell and Husseini s result [2] that E(P I (u)) 2if u = e(i) can be instantly improved to E(P I (u)) 2E(u). Finally, we address the question of how to compute the essential category weight of a given map. Recall that if X is a CW complex, then we can form a filtration B 1 ΩX B N ΩX BΩX X by the usual construction of classifying spaces [8], [13], [3]. 6

7 Theorem 10 Let f : X Y be any map. Then E(f) N if and only if the composite B N ΩX BΩX X Y f is nullhomotopic. Recall from [13] that a map f : Z BG factors through a space Q with cat(q) N if and only if f factors through B N ΩX. Thus, if f BN ΩX, then E(f) N. It follows from the construction of B N ΩX that cat(b N ΩX) N; thus, if E(f) N, thenf BN ΩX. The filtration of X by B N ΩX gives rise to a Rothenberg-Steenrod spectral sequence (also called a Moore spectral sequence) [10]; the connection between this spectral sequence and cat(x) has been studied by Whitehead [16], Ginsburg [4] and Toomer [14]. Many of their results an be derived as instant corollaries of Theorem 10. Furthermore, we can see that the E term of this spectral sequence is precisely the graded module on H (X) associated to the filtration by essential category weight. References [1] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics, 304, Springer, [2] E. Fadell and S. Y. Husseini, Category weight and Steenrod operations, Boletin de la Sociedad Matemática Mexicana 37, 1992, [3] T. Ganea, Lusternik-Schnirelmann category and strong category, Ill. J. Math. 11 (1967) [4] M. Ginsburg, On the Lusternik-Schnirelmann category, Ann. Math. 77 (1963) [5] B. Gray, Spaces of the same n-type for all n, Topology 5 (1966)

8 [6] B. Gray and C. A. McGibbon, Universal phanotom maps, Topology 32 (1993) [7] C. A. McGibbon, Phantom maps, a chapter in The Handbook of Algebraic Topology, I. M. James, ed. North Holland, [8] J. Milnor, Construction of universal bundles, II Ann. Math. 63 (1956) [9] J. Roitberg, Weak identities, phantom maps, and H-spaces, Israel J. Math, 66 (1989) [10] M. Rothenberg and N. Steenrod, The cohomology of classifying spaces of H spaces, Bull. AMS, (1965), [11] Y. Rudyak, Some remarks on category weight, Topology 38 (1999) [12] J. Strom, Category Weight and Essential Category Weight, University of Wisconsin-Madison Thesis, (1997). [13] A. Svarc The genus of a fiber space, AMS Translations 55, (1966) [14] G. Toomer, Lusternik-Schnirelmann category and the Moore spectral sequence., Math. Z. 138 (1974) [15] G. W. Whitehead, On mappings into group-like spaces, Comm. Math. Helv. 28, [16] G. W. Whitehead, The homology suspension in Colloque de Topologique Algebique Tenue à Louvain, (1956)