2 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON no such integer exists, we put cat f = 1. Of course cat f reduces to cat X when X = Y and f is the iden

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1 JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 11, August 1998 THE RELATIVE LUSTERNIK-SCHNIRELMANN CATEGORY OF A SUBSET IN A SPACE WITH RESPECT TO A MAP Eun Ju Moon, Chang Kyu Hur and Yeon Soo Yoon Abstract. In this paper we shall dene a relative Lusternik-Schnirelmann category of a subset in a space with respect to a map which generalizes the category of a space, the category of a map and the relative category of a subset in a space. We shall study some properties of the relative Lusternik-Schnirelmann category of a subset in a space with respect to a map and generalize many results of the above categories. 1. Introduction The notion of category of a space was proposed by Lusternik and Schnirelmann[7] in 1934, and proved that, when X is a smooth manifold, cat X gives a lower bound for the number of critical points of a smooth function on X. The denition adopted here is due to R. H. Fox[3]. He altered the origin denition by replacing closed by opencoverings as follows The category, cat X, of a topological space X is the least integer n such that X can be covered by then open subsets each of which iscontractible in X if there is no such integer, cat X = 1. The notion of category of a space can be generalized in a number of ways. One of these is the notion of the category of a map, due to Berstein and Ganea[1]. For a map f : X! Y, the category, cat f, of a map f is the least integer n 1 with the property that X may becovered by n open subsets on each of which f is homotopic to a constant map if Received by the editors on Nov. 20, Mathematics Subject Classications : 55M30. Key words and phrases: Lusternik-Schnirelmann category. 1

2 2 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON no such integer exists, we put cat f = 1. Of course cat f reduces to cat X when X = Y and f is the identity. For a subset A of X and an inclusion i : A! X, cat i is generally written as cat X A and called the relative category, cat X A, of a subset A in X. In this paper we shall dene a Lusternik-Schnirelmann category of a subset with respect to a map which generalizes the above several Lusternik-Schnirelmann categories. We shall study some properties of the Lusternik-Schnirelmann category of a subset with respect to a map and generalize many results of the above categories. 2. The relative Lusternik-Schnirelmann Category of a subset in a space with respect to a map Definition 2.1. Let A be a subset of X and f : X! Y a map. Then the relative Lusternik-Schnirelmann category, cat Y f A,ofAin X with respect to f is the least integer n with the property that A can be covered by the n open subsets in A each of which f is homotopic to a constant map. If no such covering exists we say that the relative category of A in X with respect to f is innite. Remark 2.2. (1) If f =1 X,then cat X (1 X ) A = cat X A. (2) In fact, cat Y f A = cat (fi), where i : A! X is the inclusion. (3) If f g : X! Y,then cat Y f A = cat Y g A. (4) cat Y f A = 1 if and only if f j A : A! Y. (5) If A = X, then cat Y f A = cat f. The following Theorem 2.3 says that cat Y f A is subadditive. Theorem 2.3. If f : X! Y and A X and A = A 1 [ A 2 and A 1 A 2 be open subsets of A, then cat Y f A cat Y f A1 + cat Y f A2. Proof. Let cat Y f A1 = m and cat Y f A2 = n. Then there exist coverings fu i j U i : open in A 1 f j U i : U i! Y i = 1 mg of A 1 and fv j j V j : open in A 2 f j V j 0 : V j! Y i = 1 ng of A 2. Now we show that fu i V j j U i V j are open in A 1 A 2 respectively,

3 THE RELATIVE LUSTERNIK-SCHNIRELMANN CATEGORY 3 f j U i : U i! Y f j V j : V j! Y i = 1 m j = 1 ng is an open covering of A. Since U i is open in A 1 and A 1 is open in A, U i is open in A. Similarly, V j is open in A. Since A 1 = S i U i, A 2 = S j V j and A = A 1 [ A 2, fu i V j g is an open covering of A. Hence cat Y f A cat Y f A1 + cat Y f A2. Corollary 2.4. [1] If X = A [ B and A, B are open in X, then cat f cat f j A + cat f jb. Theorem 2.5. If A B X, thencat Y f A cat Y f B. Proof. Let fv g be a covering of B such that each V is open in B and on each V f is null homotopic. Since B S V and A is subset of B, A S (V \ A). Thus fv \ Ag is a covering of A such that each V \ A is open in A and on each V \ Afis null homotopic. This proves the theorem. Taking B = X in Theorem 2.5, we have the following corollary. Corollary 2.6. cat Y f A cat f for any subset A of X. Theorem 2.7. For any two maps f : X! Y, g : Y! Z and a subset A of X, cat Z (g f) A min fcat Y f A cat Z g f(a) g. Proof. (1) We show that cat Z (g f) A cat Y f A. Let fu g be a covering of A such that for each, U is open in A and f i : U! Y. Then gf i g = c g() : U! Z. Thus cat Z (gf) A cat Y f A. (2) We show thatcat Z (g f) A cat Z g f(a). Let fv g be a covering of f(a) such that for each, V is open in f(a) andg i 0 : V! Z. Then for each, there exists an open U in Y such thatv = f(a)\u. Since f is continuous, f ;1 (U ) is open in X and f ;1 (U ) \ A is open in A. Moreover A f ;1 f(a) = f ;1 ( S V ) = S (f ;1 (U ) \ A).

4 4 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON Consider the following commutative diagram f ;1 (U ) \ A f y V i ;;;! X yf i 0 ;;;! Y: Then g f i = g i 0 f f = 0 : f ;1 (V )! Z. Thus cat Z (g f) A cat Z g f(a). From (1) and (2), we knowthatcat Z (g f) A min fcat Y f A cat Z g f(a) g. From Theorem 2.7 and Corollary 2.6, wehave the following corollary. Corollary 2.8. [4] For any two maps f : X! Y and g : Y! Z we have cat (g f) min fcat f cat gg. In particular cat f cat X and cat f cat Y. Corollary 2.9. For any map f : X! Y and any subset A of X, cat Y f A min fcat Y f(a) cat X Ag. Proof. In Theorem 2.7, take g = i f(a) : f(a)! Y. Since cat i f(a) = cat Y f(a) and cat Y f A cat i A = cat X A, we know, from Theorem 2.7, that cat Y f A min fcat Y f(a) cat X Ag. Corollary Let h : X 0! X has a left homotopy inverse k : X! X 0. Then for a subset A 0 of X 0, cat X h A 0 = cat X 0 A 0. Proof. Since cat X h A 0 = cat hi 0 and cat X 0 A 0 = cat i 0,we know, from Corollary 2.8, that cat X h A 0 cat X 0 A 0. Thus we show thatcat X 0 A 0 cat X h A 0. Let fv g be a covering of A 0 such that for each, V is open in A 0 and hi : V! X. Thus i khi c k() : V! X 0. Therefore cat X 0 A 0 cat X h A 0. This proves the corollary. Theorem For a map f : X! Y, let h : X 0! X be a homotopy equivalence with homotopy inverse k : X! X 0. Let A 0 be a subset of X 0 such thatkh(a 0 ) A 0. Then cat Y (f h) A 0 = cat Y f h(a0 ).

5 THE RELATIVE LUSTERNIK-SCHNIRELMANN CATEGORY 5 Proof. From Theorem 2.7, we have thatcat Y (f h) A 0 cat Y f h(a0 ). Thus we show thatcat Y f h(a0 ) cat Y (f h) A 0. Let fv g be a covering of A 0 such that for each, V is open in A 0 and f h i 0 : V! Y. Then for each, there exists an open set U in X 0 such that V = A 0 \ U. Since k is continuous, k ;1 (U ) is open in X and h(a 0 ) \ k ;1 (U ) is open in h(a 0 ). Since fv g is a covering of A 0 and k h(a 0 ) A 0, h(a 0 ) k ;1 (A 0 ) k ;1 ( S V ) S [k;1 (U )]. Thus h(a 0 ) S (h(a0 )\k ;1 (U )). Since f hi 0 : V! Y, there exists a continuous function K : V I! Y such that K( 0) = f h i 0 and K( 1) = c. Dene H : h(a 0 ) \ k ;1 (U ) I! Y by H(x t) = K(k(x) t). Then since K and k are continuous maps, H is continuous map. From the following commutative diagram h V y i 0 ;;;! X 0 yh h(a 0 ) \ k ;1 (U ) i ;;;! X H(x 0) = K(k(x) 0) = f h i 0 k(x) =f i h k(x). Since 1 X hk : X! X, there exists a continuous function R : X I! X such that R( 0)=1 X R( 1) = h k. Dene G : h(a 0 ) \ k ;1 (U ) I! Y by G(x t) = ( f R(x 2t) if 0 t 1 2, H(x 2t ; 1) if 1 2 t 1. For t = 1 2, f R(x 1) = f h g(x) =f i h g(x) =H(x 0). Thus G is well-dened and continuous. Since G(x 0) = f R(x 0) = f(x) = f i (x), G(x 1) = H(x 1) = K(k(x) 1) = c, f i G c : h(a 0 ) \ k ;1 (U )! Y. Hence fh(a 0 ) \ k ;1 (U )g is an open covering of h(a 0 ) and f i : h(a 0 )\k ;1 (U )! Y. Therefore cat f h(a0 ) cat f h A 0. This proves the theorem.

6 6 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON In Theorem 2.11, taking f = 1 X : X! X and applying Corollary 2.10 and Remark 2.2(1), we have the following corollary. Corollary [4] Let h : X 0! X be a homotopy equivalence and A 0 a subset of X 0. Then cat X 0 A 0 = cat X h(a 0 ). In particular, cat X 0 = cat X. From now on we turn our attention to path connected spaces with base point. If there is an open neighborhood of whichiscontractible in a space, then we describe the space as categorically well based. In the n-fold topological product n X of a pointed space X with itself, let T n X be the subspace of n X consisting of the n-tuples (x 1 x n ) such that a least one of the x i equals. Then we have the following theorem which is a generalized result of James[4]. Theorem Suppose that A is a normal subspace of X and Y is a path-connected and categorically well based space, and f : X! Y is a continuous map. Then cat Y f A n if and only if there exist a continuous function g : A! T n Y such that the following diagram is homotopy commutative A fi A y Y g ;;;! T n Y jy ;;;! n Y: Proof. Since f i A and j g are homotopic, there exists a continuous function h t : A! n Y such thath 0 =fi A, h 1 = j g. Since Y is categorically well-based, there exist an open neighborhood N of which N is contractible in Y, that is, i : N,! Y. For all k = 1 n, let U k = h ;1 1 p ;1 (N), where k p k : n Y! Y is the projection. Then U k is open in A. Since i : N! Y, there exists a continuous map K t : N! Y such that K 0 = i and K 1 = c. Let p t = K t p k h 1 : k h 1 K U k ;! t N ;! Y. Then t : U k! Y is continuous

7 THE RELATIVE LUSTERNIK-SCHNIRELMANN CATEGORY 7 and Dene R t : U k! Y by 0 = K 0 p k h 1 = i p k h 1 = p k h 1 1 = K 1 p k h 1 = c p k h 1 = c : R t = ( p k h 2t if 0 t 1, 2 2t;1 if 1 2 t 1. By the pasting lemma, R t is continuous and R 0 (x) =p k h 0 (x) =p k ( f i A )(x) =f(x) =f i k (x) R 1 (x) = 1 (x) =c (x) = where i k : U k! X is the inclusion. Thus f i k : U k! Y. Moreover, since S U k = S [h ;1 1 p ;1 k (N)] = h;1 1 [ S p ;1 1 (N)] = h ;1 1 [(N XX) S (XN X) S S (XXN)] = h ;1 1 [ n X]= A, fu k g is a covering of A. Hence cat Y f A n. Conversely, suppose fv k jv k is open in A 1 k ng is a covering of A each of which f is homotopic to a constant map. For each k = 1 n, there exists a continuous function g k : V k I! Y such that g k ( 0) = f i k g k ( 1) = c xk where i k : V k! X is the inclusion. Since Y is path-connected, for each k = 1 n,there exists a path p k : I! Y such that p k (0) = x k p k (1) =. Let h k : V k I! Y be given by h k (x t) = ( g k (x 2t) if 0 t 1, 2 1 p k (2t ; 1) if 2 t 1. Then h k is continuous and h k ( 0) = g k ( 0) = f i k, h k ( 1) = p k (1) = c. Since A is normal, there exists a covering fa 1 A n j A k is closed in A, for all kg of A with the property that for each k =1 n,there exists an open set W k in A such that A k W k W k V k. Since A k and A ; W k are disjoint sets, by the Urysohn's lemma, there exists a

8 8 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON map t : A! I such that k (A k )=1 k (A ; W k )=0. Dene a map d k : A k I! Y by d k (a t) = ( f(a) if a 2 A ; Wk, h k (a t k (a)) if a 2 V k. Then d k is well dened and continuous. Let d : A I ;! (A I) (A I) d 1d ;! n Y Y. Then d is continuous and d(a 0) = (d 1 (a 0) d n (a 0)) = (f(a) f(a)) = f i A (a). Let a 2 A = S A k. Then there exists a subset A k0 such that a 2 A k0 and k0 (a) =1. Since a 2 A k0 V k0, d k0 (a 1) = h k0 (a k0 (a)) = h k0 (a 1) = c (a) =: Thus d(a 1) = (d 1 (a 1) d n (a 1)) 2 T n Y. Let g = d( 1) : A! T n Y. Then f i A d j g. This proves the theorem. Corollary [4] Suppose that X is normal, and Y is a pathconnected and categorically well based space, and f : X! Y is a continuous map. Then cat f n if and only if there exist a continuous function g : X! T n Y such that the following diagram is homotopy commutative X y f Y g ;;;! T n Y jy ;;;! n Y: Theorem Let F ;! i E ;! p B be a bration and B 0 B a = p ;1 (B 0 ) is a path connected categorically well based such that E 0 normal categorically well based path connected space. Then cat E E 0 cat E 0 i cat B 0 p E 0. In particular, cat E E 0 cat E 0F catb 0. Proof. Let cat B 0 p E 0 = n. Then by Theorem 2.13, there exists a map : E 0! T n B 0 such that the following diagram is homotopy

9 THE RELATIVE LUSTERNIK-SCHNIRELMANN CATEGORY 9 commutative E 0 ;;;! T n B 0 pi 0 y y j B 0 B 0 n B ;;;! 0 n B 0 where i 0 : E 0! E is the inclusion. Thus there exists a continuous function H : E 0 I! n B 0 such that H( 0) = n B 0 (p i0 )= n (p i 0 ) n E 0 H( 0) = j B 0 : Consider the following commutative diagram ;;;! E 0 E 0 E 0 n E 0 pi 0 y y n (pi) B 0 n B ;;;! 0 B 0 B 0 : Since p i 0 : E 0! B 0 is a bration, n (p i 0 ) : n E 0! n B 0 is a bration. For the commutative diagram E 0 f0g yinj E 0 I n E 0 ;;;! n E 0 y n (pi 0 ) H ;;;! n B 0 there exists acontinuous function G : E 0 I! n E 0 such that G( 0) = n E 0 n (p i 0 ) G = H: Let 0 = G( 1) : E 0 f1g! n E 0. Then n (p i 0 ) 0 = n (p i 0 ) G( 1) = H( 1) = j B 0 : E 0! n B 0. Therefore (1) n (p i) 0 = j B 0

10 10 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON Let cat E 0 i F = m. By Theorem 2.13, there exists a function : F! T m E 0 such that the following diagram is homotopy commutative F y i ;;;! T m C 0 y j E 0 E 0 m E ;;;! 0 m E 0 : Thus there exists a continuous function K : F I! m E 0 such that K( 0) = m E 0 i K( 1) = j E 0. Since (B0 ) is a closed cobred pair, by the Strm's theorem [11], (E 0 F) is a cobred pair. That is, F i ;! E 0 is cobration. Thus for a space m E 0, a map g = m E 0 i and a continuous function K : F I! m E 0 such that K( 0) = g, there is a continuous function K : E0 I! m E 0 such that K( 0) = m E 0 K (i 1) = K: Let = K( 1) : E0! m E 0. Then i = K jf f1g = K( 1) = j E 0 : That is, (2) i = j E 0 Since the following diagram is commutative ;;;! E 0 E 0 E 0 n E 0 1 y E 0 y n ( m E 0) mn E 0 ;;;! (E 0 E 0 ) (E 0 E 0 ) n () 0 K 0 n ( m E 0) n ( m 0 E 0)G n ( m E 0) n E =mn. That is, 0 Let x 2 E 0. We know, by (1), that n () 0 mn E 0 : n (p i) 0 (x) =j B 0 (x) =(x) 2 T n B 0 : E 0

11 THE RELATIVE LUSTERNIK-SCHNIRELMANN CATEGORY 11 Thus there is an element x f 2 F suchthat 0 (x) =(x 1 x 2 x f x n ), where F is subset of E 0. From (2), we know that i = j E 0. Thus n () 0 (x) = n ()(x 1 x 2 x f x n ) = ((x 1 ) (x 2 ) (x f ) (x n )) = ((x 1 ) (x 2 ) (x f ) (x n )) = ((x 1 ) (x 2 ) ( ) (x n )) where : F! T m E 0 and ( ) 2 T m E 0. Therefore at least one coordinates of n () 0 (x) is the base point ofe 0. That is, there is a map n () 0 : E 0! T mn E 0 such that j n () 0 mn E 0 : Hence by Theorem 2.13, cat E E 0 mn = cat E 0 i F cat B 0 p E 0. In particular, we have, from Corollary 2.9, that cat E 0 i F cat E 0 i(f )=cat E 0 F cat B 0 p E 0 cat B 0 p(e 0 )=cat B 0 B 0 = cat B 0 : Thus cat E E 0 cat E 0F cat B 0. In Theorem 2.15, taking B 0 = B, we have the following corollary. Corollary [4] Let F ;! i E ;! p B be a bration, and B a path connected categorically well based space and E a normal categorically well based path connected space. Then cat E cat i cat p. In particular, cat E cat F cat B. References 1. I. Berstein and T. Ganea, The category of a map and of acohomology class, Fund. Math. 50 (1961/2), Y. Felix and J. M. Lemaire, On the mapping theorem for Lusternik-Schnirelmann category, Topology Vol. 24 (1985), R. H. Fox, On the Lusternik-Schnirelmann category, Ann. Math. 42(1941),

12 12 EUN JU MOON, CHANG KYU HUR AND YEON SOO YOON 4. I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology. Vol. 17(1978), I. M. James, Lusternik-Schnirelmann Category : Handbook of Algebraic Topology, Elsevier Science B. V. (1995), L. F. Liu, On some mapping properties of the Lusternik-Schnirelmann category, Topology and its Applications 55(1994), L. Lusternik and L. Schnirelmann, Method es Topologiques dan les Problemes Variationnels, Herman, Paris (1934). 8. C. McCord and J. Oprea, Rational Lusternik-Schnirelmann Category and the Arnol'd conjecture for nilmanifolds, Topology 32(4)(1993), C. R. F. Mounder, Algebraic Topology, Van Nostrand-Reinhold, London, A. J. Sieradski, An Introduction to Topology and Homotopy, PWS-KENT Publishing Company Boston, E. H. Spanier, Algebraic Topology, McGraw-Hill Book Company, A. Strom, Note on cobrations II, Math. scand. 22 (1968), R. M. Switzer, Algebraic Topology : Homotopy and Homology, Springer-Verlag, G. W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, Department of Mathematics Hannam University Taejon, , Korea