The Sectional Category of a Map

Transcription

1 arxiv:math/ v1 [math.t] 23 Mar 2004 The Sectional Category o a Map M. rkowitz and J. Strom bstract We study a generalization o the Svarc genus o a iber map. For an arbitrary collection E o spaces and a map : X Y, we deine a numerical invariant, the E-sectional category o, in terms o open covers o Y. We obtain several basic properties o E-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple axioms which characterize the E-sectional category. In the inal section we obtain inequalities or the E-sectional category o a composition and inequalities relating the E-sectional category to the Fadell-Hussein category o a map and the Clapp-Puppe category o a map. MSC Classiication Primary: 55M30, Secondary: 55P99 Keywords genus, sectional category, Lusternik-Schnirelmann category, category o a map, homotopy pushout 1 Introduction The sectional category o a iber map : X Y, denoted secat(), is one less than the number o sets in the smallest open cover o Y such that admits a cross-section over each member o the cover. This simple and natural numerical invariant o iber maps was developed and studied extensively by Svarc [21] who called it the genus o. Subsequently Fet [11] and Berstein-Ganea [2] extended the deinition to arbitrary maps and related it to the Lusternik-Schnirelmann category. There have been many applications o sectional category to questions o classiication o bundles, embeddings o spaces and existence o regular maps [21] as well as applications outside o algebraic and dierential topology [7, 10]. However, since Svarc s papers, the 1

2 actual study o sectional category has been sporadic and has oten appeared as subsidiary results within a larger work [2, 2], [15, 8], [3, 4], [6, 9.3]. (n exception to this is the paper o Stanley [20] which deals with sectional category in the context o rational homotopy theory.) Recently there has been considerable interest in the study o numerical homotopy invariants o spaces and o maps. Classically the Lusternik-Schnirelmann category and cone length o a space have been studied in [16, 12, 13, 4] and, more recently, the cone length [17], Clapp-Puppe category [3] and Fadell-Husseini category o a map [9, 5] have been investigated. Together with D. Stanley, we gave a uniied axiomatic development o many o these invariants in the paper [1]. For the category o spaces and maps and a ixed collection E o objects, these axioms were o two basic types, namely, those dealing with numerical unctions relative to E deined on the objects o the category and those dealing with numerical unctions relative to E deined on the morphisms o the category. By specializing the collection E, we obtained some o the previously studied invariants as well as several new invariants. For instance, by setting E = {all spaces}, our axioms or morphisms yield the Fadell-Husseini category o a map. One invariant that was not dealt with in [1] was the sectional category. Whereas the various versions o category and cone length have been deined in numerous homotopy-invariant ways, this is not the case or sectional category. The act that sectional category does not it nicely into the general ramework that so neatly encapsulates the category and cone length o spaces and maps may account or its marginal status in the decades since its introduction. We begin in Section 3 by deining a generalization o the sectional category o a map with respect to a collection E o spaces using open covers o the target o the map. This is a straightorward extension o the classical deinition. We derive several simple basic properties o this invariant. In particular, when E = {all spaces}, we see that E secat() = secat(). This treatment o sectional category leads to new invariants obtained by varying the collection E o spaces. In Section 4 we bring sectional category o maps in line with the other invariants studied in [1]. This is done by considering maps as objects in the category whose objects are maps o spaces and whose morphisms are given by commutative squares. In this category, we apply the axiomatic approach or invariants o objects, relative to the 2

3 collection o all maps with sections which actor through a space in E. The unique numerical invariant obtained rom the axioms is then proved to be E secat. This is the content o Theorem 4.7. nother case o this axiomatic approach is obtained by working in the category o maps, relative to the the collection o all maps which actor through some space in E. The unique invariant obtained rom the axioms can then be shown to be the E Clapp-Puppe category o a map (see Remark 4.8). We hope to return to this in the uture. ll invariants that all into our axiomatic scheme will o course share certain basic properties that ollow ormally rom the axioms. major interest, however, is in the new questions which arise regarding E sectional category. Many o these are considered in Section 5 and take the orm o inequalities. There we concentrate on the ollowing: (1) How is E secat() related to the domain and the target o? (2) How does E sectional category behave with respect to composition o maps? (3) What is the relation between the E sectional category, the Clapp-Puppe category and the Fadell-Husseini category? 2 Preliminaries In this section we establish our notation and recall some basic deinitions and results which we shall use. ll spaces are to have the homotopy type o connected CW-complexes. We do not assume that spaces have base points and hence maps are not base point preserving. For spaces X and Y, X Y denotes that X and Y have the same homotopy type. We let denote a space with one point and we also write : X Y or any constant map rom X to Y. We denote by id X or id the identity map o X. I,g : X Y are two maps, then g signiies that and g are homotopic. Given maps : X Y and g : Y X, i g id Y, we say that g is a section o or that Y is a retract o X. For maps : X Y and : X Y, i there is a homotopy commutative diagram X i X r X Y j Y s Y such that ri id and sj id, then we say that dominates or that is a retract o. I, in addition, ir id and js id, then and are called equivalent, and we write. 3

4 We will also use certain basic constructions in homotopy theory. The pushout o a diagram o the orm C g B is the quotient space o B C by the equivalence relation which sets (a) equivalent to g(a), or every a. The homotopy pushout H o the diagram is the quotient space o B ( [0,1]) C by the equivalence relation which sets (a,0) equivalent to (a) and (a,1) equivalent to g(a), or every a. The pushout and homotopy pushout constuctions are clearly unctors rom the category o given diagrams to the category o spaces. There are two homotopy pushouts o special interest. The irst is the homotopy pushout o B which is the mapping cone o and is denoted B C. The second is the homotopy pushout o id B which is the mapping cylinder o and is denoted M. We note or later use that the homotopy pushout o C g B is homeomorphic to the pushout o the associated mapping cylinder diagram M. M g We next consider two versions o the category o a map. Given : X Y, we irst deine the (reduced) Fadell-Husseini category o, denoted cat FH () [9]. We note that is equivalent to the inclusion o X into M, and so we regard as an inclusion. Then cat FH () is the smallest n such that there is an open cover {U 0,U 1,...,U n } o Y with the ollowing properties: (1) i j i : U i Y is the inclusion, then j i or i = 1,...,n, (2) X U 0 and (3) there is a map r : U 0 X and a homotopy o pairs j 0 r : (U 0,X) (Y,X). It ollows that cat FH ( Y) is just the (reduced) Lusternik-Schnirelmann category cat(y) o Y. 4

5 By a collection E we mean any collection o spaces which contains the one point space. The second notion o category o a map : X Y that we consider is the (reduced) E Clapp-Puppe category o, denoted E cat CP () [3]. This is the smallest nonnegative integer n with the ollowing properties: (1) there exists an open cover {U 0,U 1,...,U n } o X, (2) there exists spaces E i E and maps u i : U i E i and v i : E i Y, or i = 0,1,...,n and (3) U i v i u i or each i. For a space X, the E Clapp-Puppe category o X is deined by E cat CP (X) = E cat CP (id X ). I E = { }, the collection consisting o a one point space, then E cat CP () is the category o the map, as discussed by Berstein-Ganea [2] and others, which we will denote by cat BG (). We note that cat BG (id X ) = cat(x). 3 Deinition and Basic Properties In this section we deine the sectional category o a map relative to a collection E o spaces. We then establish basic properties o this invariant. O particular importance are the Domination Proposition (Prop. 3.6) and the Homotopy Pushout Theorem (Thm. 3.9). Deinition I E is a collection, : X Y a map and i : U Y the inclusion map, then U is E-section-categorical (with respect to ) i there is space E E and maps u : U E and v : E X such that the ollowing diagram is homotopycommutative: v E X u U The map vu : U X is called an E section o over U. 2. For a map : X Y, the (reduced) E-sectional category o, written E secat(), is the smallest integer n such that there exists an open cover o Y by n+1 subsets, each o which is E- section-categorical. I no such integer exists, then E secat() =. IE is thecollection o all spaces, then we writesecat() or E secat() and note that secat() is just the sectional category o as deined in [2, De. 2.1]. i Y. 5

6 The ollowing result lists a number o useul properties o E secat. The proos are all straighorward, and we omit them. Proposition 3.2 Let E be a collection. 1. I, then E secat() = E secat( ). 2. I E = { }, then E secat() = cat(y), the category o Y. 3. I E and F are two collections and F E, then E secat() F secat(). 4. I : X Y is a constant map, then E secat( ) = cat(y). ssertions (2) and (4) show that the Lusternik-Schnirelmann category o Y can be obtained rom the E-sectional category o in two ways: either by making E trivial or by making trivial. Since { } E {all spaces}, we obtain rom (3) the ollowing. Corollary 3.3 For any map : X Y and collection E, secat() E secat() cat(y). Next we consider some basic properties o the identity map. Proposition 3.4 Let E be a collection and X be a space. 1. E secat(id X ) = E cat CP (X). 2. X is a retract o a space in E i and only i E secat(id X ) = 0. In particular, secat(id X ) = 0 or every space X. The ollowing examples show that the inequalities in Proposition 3.2 and Corollary 3.3 can be strict. Example For the inequality in 3.2, let E be the collection o all spaces and F a collection which does not contain a space having X as a retract. Then E secat(id X ) = 0 < F secat(id X ). For a speciic example, let F = { } and take any X. 2. For the inequality in 3.3, consider : X Y and the inclusion j : Y Y CX. Then secat(j) 1 by Lemma 5.3 (below). I cat(y CX) > 1, then secat(j) 1 < cat(y CX). This is the case, or example, when : S 2n+1 CP n is the Hop map. 6

7 We next establish a basic property o E-sectional category. In the next section this will be shown to be one o three properties which characterize E secat. Proposition 3.6 (Domination) I : X Y is dominated by : X Y, then E secat() E secat( ). Proo We are given j : Y Y and r : X X as in the deinition o domination in 2. Let E secat( ) = n with E-section-categorical cover {U 0,U 1,...,U n} o Y and maps U i u i E i v i X such that v i u i j i : U i Y, where E i E and j i is the inclusion. I U i = j 1 (U i ), then {U 0,U 1,...,U n } is an open cover o Y and U i j U i U i u i E i v i X r X is the desired E section o over U i. n immediate corollary o Proposition 3.6 is that E secat is an invariant o homotopy equivalence o maps. Corollary 3.7 I, then E secat() = E secat( ). It is well-known that every map is homotopy equivalent to a iber map [18, p. 48]. Corollary 3.7 implies that the E-sectional category o an arbitrary map is equal to the E-sectional category o the equivalent iber map. Thereore Svarc s deinition o sectional category [21], which applies only to iber maps, is equivalent to Deinition 3.1 in the special case E = {all spaces}. We next prove a result about the E sectional category o the maps o one homotopy pushout into another. To establish notation, let C g B c g C a b B be a commutative diagram, let D and D be the homotopy pushouts o the top and bottom rows, respectively, and let d : D D be the induced map. We begin with a lemma. 7

8 Lemma 3.8 I E secat(b) = n, then there exists E-section-categorical open sets N 0,N 1,...N n o D with respect to d such that M N i. Proo Let d : M M be the map d with restricted domain and target. Then d b, so E secat( d) = E secat(b) = n by Corollary 3.7. Tocompletetheproo, observethatiu M ise-section-categorical with respect to d, then it is also E-section-categorical with respect to d. Now we can prove the Homotopy Pushout Theorem, which is another basic property which we use in 4 to characterize E sectional category. Theorem 3.9 (Homotopy Pushout) With the notation above, E secat(d) E secat(b)+e secat(c)+1. Proo Let E secat(b) = n and E secat(c) = m. Let {U 0,U 1,...,U n } be a minimal open E-section-categorical cover o B with respect to b. Then there are E-section-categorical open sets N 0,N 1,...,N n o D with respect to d which cover M by Lemma 3.8. Similarly, i {V 0,V 1,...,V m } is a minimal open E-section-categorical cover o C with respect to c, then there are E-section-categorical open sets M 0,M 1,...,M m o D with respect to d which cover M g. It ollows that {N 0,...,N n,m 0,...,M m } is an E-section-categorical cover o D with respect to d. Thus E secat(d) E secat(b)+e secat(c)+1. We conclude this section with a simple application o Theorem 3.9 to the maps in a homotopy pushout square. Corollary 3.10 I is a homotopy pushout square, then g C s B D r E secat(r) E secat(g)+e cat CP (B)+1. 8

9 Proo The map o homotopy pushouts obtained rom the commutative diagram C g g isequivalent tor : B D. Now applytheorem3.9, usingproposition 3.4(1). B B 4 xioms In this section we characterize the E sectional category by simple axioms. For reasons given in Remark 4.8 we state our axioms in greater generality than is needed or sectional category. Deinition 4.1 We denote by S a non-empty collection o maps. n S-length unction is a unction γ = γ S which assigns to every map an integer 0 γ() such that 1. I S, then γ() = Let C c g g C a B b B be a commutative diagram with induced map d : D D o the homotopy pushout o the irst row into the homotopy pushout o the second row. I c S, then γ(d) γ(b) I is dominated by, then γ() γ( ). We call (1) (3) the axioms or S-length unctions. It is an immediate consequence o xiom (3) that i, then γ() = γ( ). Then, since homotopic maps are equivalent ( 2), it ollows that implies that γ() = γ( ). Remark 4.2 These axioms are analogous to the axioms or the - category o a space given in [1, Prop. 5.6(2)]. The ollowing comments are made to elucidate the analogy. In Deinition 4.1 we deine the S- length unction on the objects in the category o maps o spaces. In [1, 9

10 p.26] an S-length unction was deined on the objects in the category o spaces in the case where S is the collection o contractible spaces, though we could have taken an arbitrary collection S in the deinition. This is why the analog o xiom (3) in [1] deals with mapping cone sequences instead o homotopy pushouts. In addition, the axioms in[1, p.26] were made with respect to an arbitrary collection o spaces. It is possible to incorporate an arbitrary collection o maps into Deinition 4.1, but we have not given the deinition in this generality. We are led to the ollowing deinition. Deinition 4.3 The maximal S-length unction Γ S is deined by Γ S () = max{γ S () or everys length unctionγ S }. Note that Γ S satisies the axioms or S-length unctions. We wish to show that E secat equals Γ S or some S. For this we need to choose a collection S o maps that is closely related to E. Deinition 4.4 For a collection o spaces E, let S(E) = { : X Y there is an E section o over Y}. The main result o this section is that Γ S(E) = E secat. In order to provethis, weneedtoshowthatwecanreplaceopencovers withcovers by simplicial subcomplexes in the deinition o E sectional category. This is crucial in dealing with maps that are deined by homotopy pushouts. We require the ollowing lemma. Lemma 4.5 Let : X Y be a map and B Y an E-sectioncategorical subset. I B, then is E-section-categorical. I B C is a deormation retract, then C is E-section-categorical. The proo is elementary, and we omit it. Proposition 4.6 Let Y be a simplicial complex and : X Y a map. Then E secat() n there is a simplicial structure on Y such that Y can be covered by n + 1 E-section-categorical subcomplexes L 0,L 1,...,L n. 10

11 Proo ( ) I N i is the second regular neighborhood o L i in Y, then L i is a deormation retract o the closure N i [8, p.72]. Thus N i is E-section-categorical, and hence so is the open set N i by Lemma 4.5. ( ) Consider an E-section-categorical cover {U 0,U 1,...,U n } o Y. It is known that there is a simplicial structure on Y with subcomplexes L 0,...,L n which cover Y such that L i U i or each i [14, Lem. 2.3]. The result ollows rom Lemma 4.5. Theorem 4.7 I E is a collection o spaces and : X Y is a map, then Γ S(E) () = E secat(). Proo In the proo we write Γ or Γ S(E). To show E secat() Γ(), it suices to show that E secat satisies the axioms o Deinition 4.1 or an S(E) length unction. But this ollows rom 3. Next we show Γ() E secat(). We suppose that E secat() < and prove the result by induction on E secat(). I E secat() = 0, then S(E), and so Γ() = 0. Suppose next that the inequality holds or all maps g with E secat(g) < n and let : X Y with E secat() = n. Now X and Y have the homotopy type o simplicial complexes [19, Thm.2], so wemay assumethatx andy are simplicial complexes. By the Simplicial pproximation Theorem, we can take to be a simplicial map. By Proposition 4.6, there is an E-sectioncategorical cover {L 0,L 1,...,L n } o Y by subcomplexes with respect to. We set K i = 1 (L i ) so that {K 0,K 1,...,K n } is a cover o X by subcomplexes. Thus the ollowing diagram is commutative K n K n (K 0... K n 1 ) K 0... K n L n L n (L 0... L n 1 ) L 0... L n 1, where each i is induced by. Clearly 1 S(E). Furthermore, E secat( 3 ) n 1, and hence Γ( 3 ) n 1 by our inductive hypothesis. But the pushout o the rows o the diagram yields the map : X Y. However this is homotopy equivalent to the homotopy pushout since all horizontal maps are simplicial inclusions and hence coibrations [18, p.78]. By Deinition 4.1, Γ() Γ( 3 )+1 n. This completes the proo. 3 11

12 Remark For a collection E o spaces, the E Clapp-Puppe category o its nicely into our axiomatic ramework. Precisely, let T(E) be the collection o all maps g : X Y such that g vu, where u : X E and v : E Y, or some E E. It can be shown that E cat CP () = Γ T (E) () or all maps. 2. The axioms o Deinition 4.1 can be dualized in the sense o Eckmann-Hilton. This just consists o replacing xiom 3 with an appropriate homotopy pull-back axiom. s in Deinition 4.3, we set Λ S equal to the maximum all unctions which satisy the dual axioms. I E is any collection o spaces, we deine S (E) = { g : X Y maps h 1 : Y E and h 2 : E X with E E and h 2 h 1 g id X Then Λ S (E)(), which we can denote by E cosecat(), is the dual o the E sectional category o. It would be interesting to investigate this invariant. 5 Inequalities In this section we consider the questions raised in the introduction dealing with the E-sectional category o a composition and the relationship between the various invariants. The main theorem o this section is Theorem Composition o Maps The composition results in this subsection are proved by elementary covering arguments. Proposition 5.1 For any maps : X Y and g : Y Z, E secat(g) E secat(g). The proo is obvious, and hence omitted. The next result deals with maps that have sections. Proposition 5.2 I : X Y and g : Y Z, then 1. I has a section, then E secat(g) = E secat(g). }. 12

13 2. I g has a section, then E secat(g) E secat(). Proo First assume that has a section s. Let U Z be E-sectioncategorical with respect to g. We claim that U is also E-sectioncategorical with respect to g. This ollows easily rom the ollowing homotopy-commutative diagram X sv v E Y u U This proves E secat(g) E secat(g), and Proposition 5.1 completes the proo o (1). When g has a section t, one proves (2) by showing that i U Y is E-section-categorical with respect to, then t 1 (U) is E-section-categorical with respect to g. Next we derive additional results on the sectional category o a composition. We begin with a lemma. Lemma 5.3 I Y j Y C is a mapping cone sequence and : X Y is a map, then Proo Consider the diagram g Z. E secat(j) E secat()+1. X Y which induces the map j : X Y C o homotopy pushouts. The result now ollows rom Theorem 3.9. Our main result gives an upper bound or the E-sectional category o a composition. Theorem 5.4 I : X Y and g : Y Z, then E secat(g) cat FH (g)+e secat(). 13

14 Proo Suppose cat FH (g) = n. Then we can choose a decomposition o g [1, 3] Y 0 j 0 Y 1 j 1 jn 2 Y n 1 j n 1 Y n Y g g n s n Z where (1) g g n j n 1 j 1 j 0, (2) s n g j n 1 j 1 j 0, (3) g n s n id and (4) there is a mapping cone sequence i Y i j i Y i+1. Since g is dominated by j n 1 j 1 j 0, it ollows that g is dominated by j n 1 j 1 j 0. By Lemma 5.3, E seccat(g) E secat(j n 1 j 1 j 0 ) E secat(j n 2 j 1 j 0 )+1. E secat()+n. We conclude this subsection by showing that the inequality in Theorem 5.4 can be strict. Example 5.5 Let : X Y be any map with E secat() > 0, let Z = and let g : Y Z be the constant map. Then E secat(g) = cat( ) = 0 < cat FH (g)+e secat(). 5.2 Relations Between Invariants We begin this subsection with an inequality which provides a lower bound or the Clapp-Puppe category o a composition. Proposition 5.6 I : X Y and g : Y Z, then E cat CP (g)+1 (secat()+1)(e cat CP (g)+1). Proo Write secat() = n and E cat CP (g) = m. Then we have a cover {U 0,U 1,...,U n } o Y and sections s i : U i X or each i. Now E cat CP (g Ui ) = E cat CP (gs i ) E cat CP (g) = m, 14

15 so U i = V i0 V im with each g V ij actoring through some member o E. Then Y = {V ij i = 0,1,2,...,n; j = 0,1,2,...,m} is desired cover o Y by (n+1)(m+1) E-categorical subsets. I is a constant map, then clearly E cat CP ( ) = 0. Thus i E = { } and g, then Proposition 5.6 yields the inequality cat BG (g) secat() o Berstein-Ganea [2, Prop 2.6]. More generally, i g, then E cat CP (g) secat() or any collection E by Proposition 5.6. This gives E cat CP (g) E secat() i g. However this inequality holds without the latter assumption. Proposition 5.7 Let : X Y and g : Y Z. Then E cat CP (g) E cat CP (Y) E secat(). Proo The let inequality ollows immediately. The right inequality ollows rom the observation that i {U 0,U 1,...,U n } is an open cover o Y which is E-section-categorical or, then it is an open cover or the E Clapp-Puppe category o id Y. Next we turn to a corollary to Theorem 5.4. Corollary 5.8 For any map : X Y, E secat() cat FH ()+E cat CP (X). In particular, secat() cat FH (). Proo pply Theorem 5.4 to the composition X id X Y and use Proposition 3.4(1). This corollary is used to prove the ollowing relationships between the invariants E secat, E cat CP and E cat FH. Proposition 5.9 Let E be a collection and : X Y and g : Y Z be maps. Then 1. E cat CP (g) cat FH ()+E cat CP (X). 15

16 2. I X E, then E cat CP (g) cat FH (). 3. I g, then cat BG (g) cat FH (). Proo By Proposition 5.7 and Corollary 5.8, we have E cat CP (g) E secat() cat FH ()+E cat CP (X). lso (2) ollows rom (1) since E cat CP (X) = 0. For (3) we specialize to E = {all spaces} in Corollary 5.8 and use the Berstein-Ganea inequality cat BG (g) secat() mentioned above. In the special case E = { }, Proposition 5.9(1) reduces to cat BG (g) cat FH ()+cat(x). Next we relate sectional category and the Clapp-Puppe category. Proposition 5.10 For any collection E and any map : X Y, E cat CP () E secat(). Proo Let E secat() = n and let {U 0,U 1,...,U n } be a E-sectioncategorical cover o Y. Then there are spaces E i E and maps u i : U i E i and v i : E i X such that v i u i j i, where j i : U i Y is the inclusion map. We set V i = 1 (U i ). Then {V 0,V 1,...,V n } is an open cover o X and the maps V i Vi U i u i E i v i X Y show that E cat CP () n. Remark 5.11 This proposition does not show that cat BG () secat(). This is because E cat CP () = cat BG () when E = { } and E secat() = secat() when E = {all spaces}. In a sense, E cat CP and E secat are dual to each other. Reerences [1] M. rkowitz, D. Stanley and J. Strom, The -category and -cone length o a map, Lusternik-Schnirelmann Category and Related Topics, 15 33, Contemp. Math., 316, MS, Providence, RI,

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18 Dartmouth College Hanover, NH Western Michigan University Kalamazoo, MI