Morse-Bott functions and the Lusternik-Schnirelmann category, I Hiroyuki Kadzisa and Mamoru Mimura Abstract The Lusternik-Schnirelmann category of a space is a homotopy in- variant. Cone-decompositions are used to give an upper bound for Lusternik-Schnirelmann categories of topological spaces. The pur- pose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C1 and their gradient flows, and to apply the result to some homogeneous spaces to determine their Lusternik-Schnirelmann categories. In particular, the Morse- Bott functions on the Stiefel manifolds considered by Frankel are effectively used for constructing all the cone-decompositions in this paper. 1 Introduction In this paper, every space is assumed to have the homotopy type of a finite dimensional CW-complex. The Lusternik-Schnirelmann category of a space is a homotopy invariant defined as follows: Definition 1.1. Let X be a space. The non-negative integer S n min { n | X = k=0Uk, and each Uk is open and contractible in X } is denoted by cat(X) and called the Lusternik-Schnirelmann category (ab- breviated L-S category) of X. To determine the L-S category of a space, we often use a cone-decomposition of the space, which is defined as follows: Definition 1.2. Let X be a space with base point *. A cone-decomposition ik of X with length m is a sequence of m cofibration sequences Ak ! Xk ! Xk+1, 0 k < m, satisfying X0 ' * and Xm ' X. 1 The cone-decomposition gives a homotopy invariant of a space, which is called the cone-length defined as follows: Definition 1.3. Let X be a space. The non-negative integer min { m | X has a cone-decomposition with length m } is called the cone-length of X and is denoted by cl(X). The cone-length gives an upper bound for the L-S category. Our aim in this paper is to construct cone-decompositions of manifolds by using functions of class C1 and their gradient flows on them so as to apply the result to complex Stiefel manifolds Vm (Cn) = U(n)=U(n - m) and symmetric Riemann spaces U(n)=O(n), U(2n)=Sp (n), and to determine the L-S categories and the cone-lengths of these manifolds. We remark that the cone-length and the L-S category of Vm (Cn) are already determined by the first author and Singhof in [7] and [14] respectively. This paper is organized as follows: Section 2. We will discuss various notions related to L-S category. Section 3. We will state a theorem which is the main result of this paper. Section 4. We will study ANR's and NDR-pairs constructed in the previous and present sections which are needed to prove the main theorem. Section 5. We will prove the main theorem. Section 6. We will study the Morse-Bott functions considered by Frankel [3] and the filtrations defined by Miller [9]. They are used to construct cone-decompositions of the complex Stiefel manifolds. Section 7. We will discuss a relation between the cellular decomposition of the Stiefel manifolds in [15, Ch. IV] and Miller's filtration in [9]. Section 8. We will construct cone-decompositions of the complex Stiefel man- ifolds by using the main result and results in Sections 6 and 7. Section 9. We will prepare the necessary propositions and lemmas to ex- plain our method of constructing cone-decompositions of U(n)=O(n) and U(2n)=Sp (n) which are entirely similar to each other. Section 10. We will state a concluding remark in which we discuss a topolog- ical characteristic of Vm (Cn) by using the cone-decomposition of this paper. 2 The present work started with the observation by the second author that the Morse-Bott functions considered by Frankel are closely related to the L-S category of Stiefel manifolds, especially Sp(n); based on this the first author gave a talk [7] at a seminar held at Okayama University in Fall, 2005. In fact, the present work resulted as a by-product from our efforts to understand the works of Frankel [3] and Miller [9] in order to estimate the L-S category of the symplectic group Sp(n). Throughout the paper the notation ' means homotopy equivalence and does homeomorphism. The authors wish to thank J.Korba~s for giving us useful comments and K.Morisugi for pointing out that U(n)=O(n) and U(2n)=Sp (n) are related to real and quaternionic projective spaces respectively. 2 Lusternik-Schnirelmann category In this section we will discuss the relation of the L-S category to other ho- motopy invariants. We often use a cone-decomposition of a space to determine the L-S cat- egory of a space, since the cone-length gives an upper bound for the L-S category. Some other invariants are used to determine the L-S category; for example, the cup-length is used for a lower bound of the L-S category and the strong L-S category for an upper bound. Their definitions are stated as follows: Definition 2.1 (see Iwase-Mimura [6]). Let X be a space. For each multi- plicative cohomology theory h, the non-negative integer max { m | 9x1, . .,.xm 2 eh*(X) such that x1 . .x.m6= 0} is denoted by cup(X; h). The non-negative integer max { cup(X; h) | h is a multiplicative cohomology theory} is denoted by cup(X) and called the cup-length of X. Definition 2.2. Let X be a space. The non-negative integer S m min{ m | X = k=0Uk, and each Uk is open and contractible in itself} is denoted by gcat(X) and called the geometric category of X. The non- negative integer min { gcat(Y ) | Y ' X } is denoted by Cat(X) and called the strong Lusternik-Schnirelmann category of X. 3 For each space X, it is easy to see from the definitions and the result of Schweitzer [13, Prop. 1.6] that cup(X) cat(X) Cat(X) gcat(X). We recall a formula due to Ganea [4, Prop. 2.1]: Cat (X) = cl(X), which holds for each pathwise connected space X. We will mainly use in this paper the following inequalities and equation: cup(X) cat(X) Cat(X) = cl(X). 3 The main result Let X be a compact manifold with a base point *, f : X ! R be a function of class C1, {y0, . .,.ym } be the ordered set of all critical values of f such that y0 < . . .< ym , and { k}mk=0be the family of all critical subsets of f satisfying that f( k) = {yk} for each k = 1, . .,.m and 0 = {*}. The flow of the vector field -grad f on X will be denoted by : R x X ! X. We consider the unstable subset Uk associated with k, which is defined by ae oe Uk = x 2 X | lim (t, x) 2 k t!-1 for each k = 0, . .,.m. When a closed subset Fk of X is defined by [k Fk = Uk i=0 for each k = 0, . .,.m, the family {Fk}mk=0gives rise to a filtration of X. Under these notations we consider an inclusion e'kof the unreduced cone C k over k into Fk as follows: Definition 3.1. An inclusion e'k: C k ! Fk is along the gradient flow if for each [t, x], [s, y] 2 C k, there hold e'k[0, x] = *, e'k[1, x] = x, f(e'k[t, x]) = f(e'k[s, y]) when t = s, f(e'k[t, x]) < f(e'k[s, y]) when t < s, and (R x e'k(C k)) e'k(C k). 4 An inclusion of the cone along the gradient flow means a deformation of the critical subset k to the base point along the gradient flow. The main result of this paper is the following theorem, which gives rise to a cone-decomposition of X: Theorem 3.2. Suppose that (1) { k}mk=0is a family of ANR's ; (2) {Fk}mk=0is an NDR-filtration of X; (3) the unreduced cone C k is embedded in Fk along the gradient flow for each k = 1, . .,.m. Then there exist spaces Xk (k = 0, . .,.m - 1) and subspaces Ak Xk (k = 0, . .,.m - 1) satisfying that Xk ' Fk, Xk [ eCAk ' Fk+1, where eCAk denotes the reduced cone over Ak. The reader is referred to [5] and [16] for the definitions and properties of ANR and NDR. One can expect that the conditions (1) and (2) of Theorem 3.2 are satisfied for many manifolds and many functions of class C1 on them. In particular, the condition (1) is satisfied when f is a Morse-Bott function. When applying the above theorem, it is the most important to see whether there exist inclusions of the cones over the critical subsets into the filtrati* *on- sets along the gradient flow. 4 ANR's and NDR-pairs We will construct spaces Lk, Lk-, Lk+, Bk, Bk- , Bk+ and prove that all the spaces given in Section 3 as well as in this section are ANR's. We fix a number k, 0 k m - 1. We restrict the function f and the flow to the filtration-set Fk+1 as follows: Notation 4.1. The restriction of f to the domain Fk+1 and the codomain [y0, yk+1] is denoted by fk+1 : Fk+1 ! [y0, yk+1]. Notation 4.2. The restriction of to Fk+1 is denoted by k+1 : R x Fk+1 ! Fk+1. 5 We define three subspaces Lk, Lk-, Lk+ of Fk+1 by using the function fk+1. Definition 4.3. Subspaces Lk, Lk-, Lk+ of Fk+1 are defined by ae oe yk + yk+1 Lk = fk+1-1 _________ , 2 ~ ~ ~ ~ yk + yk+1 + -1 yk + yk+1 Lk - = fk+1-1 y0, _________ , Lk = fk+1 _________, yk+1 . 2 2 When we regard the function f as the height function, we can describe the shape of Fk+1 in the following figure: | |6height | critical subset | ......................................................_|||||||* *||||||||||y| + ..........................................................|| * * k+1 Lk ...............................................|| ..........................................._||||||||||||yk+yk+1* *_| Lk ..........................................|2 ...........................................||gradient flow ' -$ ............................................|||||||||||||||||||* *||||||||y| Lk .............................................| * * k ..........................................................|| k+1 ..............................................................* *.....F|| .............................................................* *.......k|| ...........................................................* *...........................|| &%.........................................................* *..........................s||||||||||||||||||||||||||||? |y0 base point | | We introduce three subspaces Bk, Bk- , Bk+ of Fk+1. Definition 4.4. Subspaces Bk, Bk- , Bk+ of Fk+1 are defined by Bk = Lk \ e'k+1(C k+1), Bk - = Lk- \ e'k+1(C k+1), Bk + = Lk+ \ e'k+1(C k+1). One can describe the shape of e'k+1(C k+1) in the following figure: | |6height Fk+1 e'k+1(C k+1) | critical subset critical subset | .....................................................................* *.................._...................................................._|||||* *||||||||||||y + ......................................................................* *.............................................................................* *.............................................................................* *...+|k+1 Lk ......................................................................* *.............................................................................* *........................................................................Bk| ......................................................................* *.................................._..........................................* *........................................._|||||||||||||||||||||||||||||||||||* *yk+yk+1_ Lk ......................................................................* *.......................................................Bk|2 ......................................................................* *..........................................| ' -$ ......................................................................* *........................................-| Lk ......................................................................* *................................................Bk .....................................................................* *.................................................................| .....................................................................* *.........................F...................................................* *.........| ....................................................................* *.............................k...............................................* *................| ..................................................................* *.............................................................................* *...........................................| &%................................................................* *.....................................................................s.......* *.............................................................................* *.......................s||||||||||||||||||||||||||| |y0 base point base point | | 6 We will show that the spaces Fk+1, Fk, Lk, Lk -, Lk +, e'k+1(C k), Bk, Bk -, Bk + thus constructed are ANR's. Proposition 4.5. The filtration-sets Fk+1 and Fk are ANR's. Proof. The space X is an ANR, since it is a compact manifold. The family __ {Fk}mk=0is an NDR-filtration of X. Therefore Fk+1 and Fk are ANR's. |__| We recall a well-known theorem from the Morse theory to show that the spaces Lk, Lk- and Lk+ are ANR's: Theorem 4.6. Let M be a smooth manifold and g a function of class C1 from M to R. If a subset g-1[a, b] is compact and contains no critical points, then (1) g-1(-1, a] is a deformation retract of g-1(-1, b], (2) g-1[b, 1) is a deformation retract of g-1[a, 1), (3) g-1{a+b_2} is a deformation retract of g-1[a, b]. (See Milnor [10, Thm. 3.1] for a proof of Theorem 4.6.) In the proof of Theorem 4.6, the gradient flow of g is used as the retracting deformation. Consequently we obtain the following lemma by using the same deformation: Lemma 4.7. Let M be a smooth manifold and g a function of class C1 from M to R. Let be a critical subset of g and U the unstable subset associated with . If a subset g-1[a, b] is compact and contains no critical points, then (1) U \ g-1(-1, a] is a deformation retract of U \ g-1(-1, b), (2) U \ g-1[b, 1) is a deformation retract of U \ g-1(a, 1), (3) U \ g-1{a+b_2} is a deformation retract of U \ g-1(a, b). Now we show Proposition 4.8. The spaces Lk, Lk- and Lk + are ANR's. Proof. The space Fk+1 is an ANR by Proposition 4.5. The sets defined by ` ' 3yk + yk+1 yk + 3yk+1 fk+1-1 ___________, ___________, 4 4 ~ ' ` ~ yk + 3yk+1 -1 3yk + yk+1 fk+1-1 y0, ___________, fk+1 ___________, yk+1 4 4 7 are ANR's, since they are open subsets of Fk+1. Obviously the set ~ ~ 3yk + yk+1 yk + 3yk+1 f-1 ___________, ___________ 4 4 is compact and contains no critical points. Hence Lk, Lk- and Lk + are de- formation retracts of ` ' 3yk + yk+1 yk + 3yk+1 fk+1-1 ___________, ___________, 4 4 ~ ' ` ~ yk + 3yk+1 -1 3yk + yk+1 fk+1-1 y0, ___________, fk+1 ___________, yk+1 4 4 respectively by Lemma 4.7. The spaces Lk, Lk- and Lk + are closed subsets __ of Fk+1. Therefore Lk, Lk- and Lk+ are ANR's. |__| We use the following theorem to show that the spaces e'k+1(C k+1), Bk, Bk- and Bk + are ANR's: Theorem 4.9. If h : (X, A) ! (Y, B) is a relative homeomorphism, where X, A, B are compact ANR's and Y is a Hausdorff space, then Y is also an ANR. (See Hu [5, Ch. VI, Thm. 1.4] for a proof of Theorem 4.9.) We will show that the spaces e'k+1(C k+1), Bk, Bk- and Bk + are ANR's. Proposition 4.10. The spaces e'k+1(C k+1), Bk, Bk- and Bk + are ANR's. Proof. It is clear that e'k+1(C k+1) [0, 1]x k+1 = {0} x k+1, Bk k+1, Bk - [0, t]x k+1 /{0} x k+1, Bk + [t, 1]x k+1 for some t 2 [0, 1]. The spaces Bk and Bk + are ANR's, since k+1 is an ANR. The canonical quotient map from ([0, 1] x k+1, {0} x k+1) to (C k+1, *) is a relative homeomorphism, where * is a vertex of the cone. The space k+1 is compact, since it is a closed subset of a compact manifold. Consequently [0, 1] x k+1 and {0} x k+1 are compact ANR's. It is clear that one point set {*} is a compact ANR. The space C k+1 is a Hausdorff space, since it is the space shrinking a closed subspace {0} x k+1 of a compact Hausdorff space [0, 1] x k+1 to a point. Therefore the spaces e'k+1(C k+1) and Bk - are __ ANR's by Theorem 4.9. |__| 8 The following proposition relates ANR's to NDR-pairs: Proposition 4.11. Let (X, A) be a pair of a metrizable space and its closed subspace. If X and A are ANR's, then (X, A) is an NDR-pair. The reader is referred to [5, Ch. IV, Thm. 3.2] and [16, Ch. I, (5.1)] for a proof. Thus we obtain the following lemma: Lemma 4.12. A topological pair formed by any two of the spaces in { Fk+1, Fk, Lk, Lk -, Lk +, e'k+1(C k+1), Bk, Bk -, Bk + } is an NDR-pair. __ Proof. It is clear from Propositions 4.5, 4.8, 4.10, and 4.11. |_* *_| 5 The proof of Theorem 3.2 We will prove Theorem 3.2 in this section. We define spaces Xk and Ak by Xk = Lk- =Bk -, Ak = Lk=Bk. Our goal is to show that Fk ' Lk- ' Lk- =Bk - = Xk, Fk+1 ' Fk+1=e'k+1(C k+1) = (Lk -=Bk -) [ (Lk +=Bk +) Xk [ eCAk and that Xk and Ak have the homotopy type of CW-complexes. The following lemma implies that Lk - ' Fk. Lemma 5.1. The space Fk is a deformation retract of Lk -. Proof. The pair (Lk -, Fk) is an NDR-pair by Lemma 4.12. There exist an open neighborhood U of Fk in Lk- and a homotopy { _t : U ! Lk- | t 2 [0, 1] } such that _0 is equal to the inclusion map, _1 is a retraction, and _t(x) = x for each (t, x) 2 [0, 1] x Fk. The subspace Lk - \ U of Fk+1 is compact and the family of the spaces { k+1({t} x (Fk+1 \ Lk- ) | t 2 [0, 1) } 9 is an open covering of Lk -\ U. Hence there exist real numbers s1, . .,.sl 2 [0, 1) such that { k+1({t} x (Fk+1 \ Lk- ) | t = s1, . .,.sl } is an open covering of Lk- \ U. We put s = max {s1, . .,.sl}. Then we have Lk- \ U k+1({s} x (Fk+1 \ Lk- )) = Fk+1 \ k+1({s} x Lk- ), Lk - \ (Lk - \ U) Lk- \ (Fk+1 \ k+1({s} x Lk- )), Lk- \ U Lk- \ k+1({s} x Lk- ). Consequently we have U k+1({s} x Lk- ), since the spaces U and k+1({s} x Lk- ) are subsets of Lk -. The spaces Fk and k+1([s, 0] x Lk) are disjoint closed subsets of the metric space Lk- . There exists a continuous function u : Lk- ! [0, 1] such that Fk = u-1{0}, k+1([s, 0] x Lk) = u-1{1}. Define a homotopy { 't : Lk- ! Lk- | t 2 [0, 1] } by 't(x) = k+1(u(x)st, x) for each (t, x) 2 [0, 1] x Lk- . Then '0 is equal to the identity map on Lk -, '1(Lk -) U, and 't(x) = (0, x) = x for each (t, x) 2 [0, 1] x Fk. Define a homotopy { ht : Lk- ! Lk- | t 2 [0, 1] } by ( '2t if 0 t 1_ ht = 1 2 _2t-1O '1 if _2 t 1 for each t 2 [0, 1]. Then h0 is equal to the identity map on Lk- , h1(Lk -) F* *k, and ht(x) = x for each (t, x) 2 [0, 1] x Fk. __ Thus Fk is a deformation retract of Lk- . |__| The following lemma implies that CeAk = eC(Lk=Bk) Lk+ =Bk +. Lemma 5.2. There exists a homeomorphism eg: eC(Lk=Bk) ! Lk+ =Bk + such that eg[1, x] = x for each x 2 Lk=Bk. 10 Proof. Denote by ss the natural projection from Lk + to Lk +=Bk +. Define a map g : [0, 1] x Lk ! Lk+ =Bk + by ( ss k+1 1 - 1_, x if t 6= 0 g(t, x) = + t [Bk ] if t = 0 for each (t, x) 2 [0, 1] x Lk, where [Bk +] denotes the base point obtained by collapsing Bk +. It is clear that g is continuous at (t, x) 2 (0, 1] x Lk. We will show that g is continuous at (0, x) for each x 2 Lk. Take an open neighborhood U of g(0, x), which is [Bk +]. The subspace Lk + \ ss-1(U) of Fk+1 is compact and the family of the spaces { k+1({t} x (Fk+1 \ Lk+ ) | t 2 (-1, 0] } is an open covering of Lk+ \ss-1(U). Hence there exist real numbers s1, . .,.sl2 (-1, 0] such that { k+1({t} x (Fk+1 \ Lk+ ) | t = s1, . .,.sl } is an open covering of Lk +\ ss-1(U). We put s = min {s1, . .,.sl}. Then we have Lk+ \ ss-1(U) k+1({s} x (Fk+1 \ Lk+ )) = Fk+1 \ k+1({s} x Lk+ ), Lk+ \ (Lk + \ ss-1(U)) Lk + \ (Fk+1 \ k+1({s} x Lk+ )), Lk+ \ ss-1(U) Lk + \ k+1({s} x Lk+ ). Consequently we have ss-1(U) k+1({s} x Lk+ ), since the spaces ss-1(U) and k+1({s} x Lk+ ) are subsets of Lk+ . Thus `~ ~ ' `~ ' ' 1 1 U ss( k+1({s} x Lk+ )) = g 0, _____ x Lk g 0, _____ x Lk 1 - s 1 - s and ~ ' 1 g-1(U) 0, _____ x Lk 3 (0, x). 1 + s Clearly the map g is continuous at (0, x). Therefore the map g is continuous. The set g([0, 1] x Bk) is equal to [Bk +], since e'k+1(C k+1) is embedded in Fk+1 along the gradient flow . It is clear that the set g({0} x Lk) is equal to [Bk +]. Hence g naturally induces a continuous map eg: eC(Lk=Bk) ! Lk+ =Bk +. 11 The canonical base point of eC(Lk=Bk) is denoted by *. The restriction eg|(Ce(Lk=Bk) \ {*}) : eC(Lk=Bk) \ {*} ! (Lk +=Bk +) \ [Bk +] is bijective. Consequently egis bijective. The space Ce(Lk=Bk) is compact, since Lk is compact. The space Lk+ =Bk + is a Hausdorff space, since Lk+ is a compact Hausdorff space and since Bk + is a closed subset of Lk+ . Therefore the map egis a homeomorphism. It is clear that eg[1, x] = x for each x 2 __ Lk=Bk. |__| Now we are ready to prove Theorem 3.2. Proof of Theorem 3.2. We obtain a homotopy equivalence Fk ' Lk- by Lemma 5.1. Hence we have Fk ' Lk- ' Lk- =Bk - = Xk, since (Lk -, Bk- ) is an NDR-pair and since Bk - is contractible in itself. The space Xk has the homotopy type of a CW-complex, since Fk is an ANR. We deduce that Xk [ eCAk = Xk [ eC(Lk=Bk) (Lk -=Bk -) [ (Lk +=Bk +) from Lemma 5.2. Hence we have that Fk+1 ' Fk+1=e'k+1(C k+1) = (Lk -=Bk -) [ (Lk +=Bk +) Xk [ eCAk, since (Fk+1, e'k+1(C k+1)) is an NDR-pair and since e'k+1(C k+1) is contractible in itself. Since (Lk, Bk) is an NDR-pair, we obtain a homotopy equivalence Ak = Lk=Bk ' Lk [ CBk. The space Bk is a compact ANR. The space CBk is a Hausdorff space, since Bk is a Hausdorff space. Consequently CBk is an ANR by Theorem 4.9. The space Lk is an ANR. Hence Lk [ CBk is an ANR. Therefore Ak has the __ homotopy type of a CW-complex. |__| Remark 1. Suppose given a manifold X and a function f : X ! R of class C1, with the properties required at the beginning of Section 3. In order to construct a cone-decomposition, we have deformed a critical subset to a point along the gradient flow. However, if critical subsets are contractible in a given manifold X, then one could prove, even without deforming along the gradient flow, that the function f gives an upper bound of the L-S category, possibly under the assumptions that the critical subsets are ANR and that the filtration constructed from them is an NDR-filtration. 12 6 Some results due to Frankel and Miller Frankel [3] and Miller [9] provide us with some information about Morse- Bott functions on the real, complex, and quaternionic Stiefel manifolds. We will recall their results and prove in this section a new lemma, which will be used to construct a cone-decomposition of Vm (Cn). Frankel and Miller studied Morse-Bott functions and related topics on the real, complex, and quaternionic Stiefel manifolds simultaneously. Similarly we will proceed by using a field F which denotes the field of real numbers R, the field of complex numbers C, or the quaternionic skew-field H according as d = 1, 2, or 4 respectively. The Stiefel manifold Vm (Fn) consisting of all m-frames in Fn is defined by Vm (Fn) = (u1, . .,.um ) | u1, . .,.um 2 Fn, ui*uj = ffiij, where ffiijis the Kronecker delta and u* is a conjugate transpose of a vector u 2 Fn. The space Vm (Fn) is identified with a homogeneous space U(n, F) = U(n - m, F) x {Im }, where U(n, F) is defined by U(n, R) = O(n), U(n, C) = U(n), U(n, H) = Sp(n) and Im is the m x m unit matrix. The canonical quotient map is denoted by pnm: U(n, F) ! Vm (Fn). First of all, we recall some results of Frankel from [3], in which he con- structed a function on Vm (Fn) and proved that it is a Morse-Bott function: Notation 6.1. A function f : Vm (Fn) ! R is defined by _ m ! X f(U) = -< ui+n-m i i=1 for U = (uij) 2 Vm (Fn), where < indicates the real part. Remark 2. Frankel [3] considered the Stiefel manifold U(n, F) = {Im }xU(n- m, F) and used a Morse-Bott function f : Vm (Fn) ! R defined by _ m ! X f(U) = < uii i=1 for U = (uij) 2 Vm (Fn). We use, however, the previous definition, since it is suitable for Theorem 3.2 as well as the cellular decomposition constructed in Steenrod [15, Ch. IV]. 13 The function f gives rise to a gradient flow on Vm (Fn): Notation 6.2. The flow of the vector field -grad f on Vm (Fn) is denoted by : R x Vm (Fn) ! Vm (Fn). Frankel [3] proved that the critical subset of the function f is a disjoint union of Grassmann manifolds. For any natural numbers m and k such that k m, the Grassmann manifold Gk(Fm ) over F is defined by Gk(Fm ) = { P is an m x m matrix in F | P *= P, P 2= P, rank P = k }. A matrix P 2 Gk(Fm ) in this definition represents the orthogonal projection to the k-plane which is the image of P . Following Frankel [3] we embed the space Gk(Fm ) in Vm (Fn) as follows: Notation 6.3. An embedding 'k : Gk(Fm ) ! Vm (Fn) is defined by ` ' O 'k(P ) = Im - 2P for each P 2 Gk(Fm ). For each P 2 Gk(Fm ), the matrix Im - 2P transforms a vector v in the image of P to -v and a vector u in the kernel of P to u. The following theorem is a result on the critical subset of f stated in his paper [3, Thm. 2]: Theorem 6.4 (Frankel). The critical subset of f : Vm (Fn) ! R is equal to ma 'k(Gk(Fm )). k=0 He used the following lemma ([3, Lem. 1]) to prove Theorem 6.4: Lemma 6.5 (Frankel). Let T be a maximal torus of U(m, F). Then grad f is tangent to T at each point h 2 T . Second of all, we recall a result of Miller from [9], in which he gives a filtration {FkVm (Fn)}mk=0defined by ae ` ` ' ' oe O FkVm (Fn) = V 2 Vm (Fn) | dim ker V - m - k Im for all k = 0, . .,.m. 14 Remark 3. Miller [9] mainly used a filtration {FkVm (Fn)}mk=0defined by ae ` ` ' ' oe Im FkVm (Fn) = V 2 Vm (Fn) | dim ker V + m - k O for his specific calculation. We use, however, the previous filtration, since it is suitable for Theorem 3.2 as well as the cellular decomposition constructed in Steenrod [15, Ch. IV]. There is no essential difference between them. We consider the unstable subset associated with 'k(Gk(Fm )). Miller re- lated in his paper [9, Prop. 4.1] the filtration to the unstable subsets associ- ated with 'k(Gk(Fm )) as follows: Proposition 6.6 (Miller). The unstable subset associated with 'k(Gk(Fm )) is equal to FkVm (Fn) \ Fk-1Vm (Fn). Finally, we generalize Lemma 6.5 to give a proof of a proposition which will be used later in this paper. For simplicity the n x m matrix ` ' O Im is denoted by Inm, which is equal to pnm(In). The matrix Inmis identified with the embedding of U(m, F) to Vm (Fn). For each subset A of U(m, F), we denote by InmA the set defined by { InmU 2 Vm (Fn) | U 2 A }. We denote by {e1, . .,.en} the canonical basis of Fn satisfying that (e1, . .,.en) = In. The tangent space of a manifold M at a point p is denoted by TpM. The sub- space of RN spanned by vectors w1, . .,.wl2 RN is denoted by . We now generalize Lemma 6.5. Lemma 6.7. Let T be a maximal torus of U(m, F). Then grad f is tangent to InmT at each point V 2 InmT . Proof. The manifold Vm (Fn) is a subset of the Euclidean space Rdnm and has the metric induced from the Euclidean metric of Rdnm . Take an n x m matrix V 2 InmT and define unit vectors v1, . .,.vm by (v1, . .,.vm ) = V. 15 It is clear that the matrix V belongs to InmU(m, F). The vectors v1, . .,.vm are perpendicular to the unit vectors e1, . .,.en-m . For each index (i, j) 2 {1, . .,.n - m} x {1, . .,.m}, an n x m matrix Eij denotes the matrix whose (i, j)-entry is 1 and 0 otherwise. We use a parameter t 2 R and define smooth curves V ij(t), Wi ij(t), Wj ij(t), Wk ij(t) in Vm (Hn) by V ij(t) = (v1, . .,.vj-1, vjcos t + eisint, vj+1, . .,.vm ), Wi ij(t) = (v1, . .,.vj-1, vjcos t + eii sint, vj+1, . .,.vm ), Wj ij(t) = (v1, . .,.vj-1, vjcos t + eij sint, vj+1, . .,.vm ), Wk ij(t) = (v1, . .,.vj-1, vjcos t + eik sint, vj+1, . .,.vm ) for all (i, j) 2 {1, . .,.n - m} x {1, . .,.m}, where {1, i, j, k} is the usual* * basis of H over R. Then the curve V ijlies in Vm (Rn) and the curves V ij, Wiij lie in Vm (Cn). These curves go through the point V when t = 0. Velocities of the curves V ij(t), Wiij(t), Wjij(t), Wk ij(t) at V are given by i i dV_ij(0)_ i dWi ij(0) i dWj j(0) i dWk j(0) i = E j, _________= iE j, _________= jE j, _________ = kE j dt dt dt dt for all (i, j) 2 {1, . .,.n - m} x {1, . .,.m} respectively. Observe that the velocities Eij, iEi j, jEi j, kEi jare tangent vectors of Vm (Hn), that Eij, iE* *i j are tangent vectors of Vm (Cn), and that Eij is a tangent vector of Vm (Rn) at V . They are perpendicular to the tangent spaces of InmU(m, F) at V . It is clear that dim (TV Vm (Hn)) = 4mn - 2m2 + m, dim (TV (InmSp(m))) = 2m2 + m, dim (TV Vm (Cn)) = 2mn - m2, dim (TV (InmU(m))) = m2, 2mn - m2 - m n m2 - m dim (TV Vm (Rn)) = _______________, dim (TV (Im O(m))) = ________ 2 2 and that 4mn - 4m2 = dim , 2mn - 2m2 = dim , mn - m2 = dim . Hence the spaces , , 16 are orthogonal complements of TV (InmSp(m)), TV (InmU(m)), TV (InmO(m)) re- spectively. The gradient of f at V is perpendicular to all the velocities Eij, iEi j, jEi j, kEi j for (i, j) 2 {1, . .,.n - m} x {1, . .,.m}, since we have i i d(f_O_V_ij)_ d(f O Wiij) d(f O Wj j) d(f O Wk j) (0) = ___________(0) = ___________(0) = ____________(0) = 0. dt dt dt dt Hence the gradient of f at V belongs to TV (InmU(m, F)) and is equal to the gradient of f|(InmU(m, F)) at V . __ Therefore the gradient of f is tangent to InmT at V by Lemma 6.5. |__| 7 The cellular decomposition of Miller's fil- tration To show that Miller's filtration {FkVm (Fn)}mk=0is an NDR-filtration, we observe that the filtration is compatible with the cellular decomposition of Vm (Fn) stated in the following theorem (see Steenrod [15, Ch. IV]): Theorem 7.1. The Stiefel manifold Vm (Fn) has a cellular decomposition 0 1 m[ [ pnm(e0) [ @ pnm ednj-1ednj-1-1. .e.dn1-1A . j=1 n nj>nj-1>...>n1>n-m The following theorem describes the relationship between Miller's filtra- tion and the cellular decomposition: Theorem 7.2. The 0-th filtration-set F0Vm (Fn) is equal to pnm(e0), and for each k = 1, . .,.m, the k-th filtration-set FkVm (Fn) has a cellular decompo- sition: 0 1 [k [ pnm(e0) [ @ pnm ednj-1ednj-1-1. .e.dn1-1A . j=1 n nj>nj-1>...>n1>n-m Before proving Theorem 7.2, we put S0F= { ~ 2 F | k~k = 1 } and define a map ~ : (S0F)k x Vk(Fn) ! U(n, F) by Xk ~ ((~1, . .,.~k), (v1, . .,.vk))= In + vi(~i- 1)vi* i=1 17 for each ((~1, . .,.~k), (v1, . .,.vk))2 (S0F)k x Vk(Fn). In the case k = 1, the map ~ : S0Fx V1(Fn) ! U(n, F) is used in Steenrod [15, Ch. IV] to construct a cellular decomposition of the Lie group U(n, F). One can easily show that F1U(n, F) = ~(S0Fx V1(Fn)). Thus a cellular decomposition of F1U(n, F) is given by _ ! [ e0 [ edn1-1 . n n1>0 The following lemma will imply Theorem 7.2 in the case Vm (Fn) = U(n), Sp (n). Lemma 7.3. Let F be C or H. Then for each k = 1, . .,.n, there holds that ~((S0F)k x Vk(Fn)) = ~(S0Fx V1(Fn))k = FkU(n, F). Proof. It is clear that ~((S0F)k x Vk(Fn)) ~(S0Fx V1(Fn))k, since we have Xk In + vi(~i- 1)vi* = (In + v1(~1 - 1)v1 *) . .(.In + vk(~k - 1)vk *) i=1 for each ((~1, . .,.~k), (v1, . .,.vk))2 (S0F)k x Vk(Fn). We will show that ~(S0Fx V1(Fn))k FkU(n, F). Take U 2 ~(S0Fx V1(Fn))k and suppose that U is represented as (In + v1(~1 - 1)v1 *) . .(.In + vk(~k - 1)vk *), where ~1, . .,.~k 2 S0Fand v1, . .,.vk 2 V1(Fn). There exists an orthonor- mal (n - k)-frame (u1, . .,.un-k) of the orthogonal complement of the space spanned by v1, . .,.vk. The matrix U belongs to the filtration- set FkU(n, F), since Uui = ui for all i = 1, . .,.n - k. Consequently we have ~(S0Fx V1(Fn))k FkU(n, F). We will show that FkU(n, F) ~((S0F)k x Vk(Fn)). 18 Take U 2 FkU(n, F). There exists an orthonormal basis {v1, . .,.vn} whose elements are eigenvectors of U. We may suppose that the eigenvalues of vk+1, . .,.vn are 1, since the dimension of the eigenspace with eigenvalue 1 is greater than or equal to n - k. For each i = 1, . .,.k, let us denote by a scalar ~i the eigenvalue of vi. Hence we have Xk X n Xk _ Xk ! U = vi~ivi*+ vivi*= vi~ivi*+ In - vivi* i=1 i=k+1 i=1 i=1 Xk = In + vi(~i- 1)vi*, i=1 and U 2 ~((S0F)k x Vk(Fn)). Consequently we have FkU(n, F) ~((S0F)k x Vk(Fn)). __ Therefore we have ~((S0F)k x Vk(Fn)) = ~(S0Fx V1(Fn))k = FkU(n, F). |__| We prove the following lemma which will give Theorem 7.2 in the case Vm (Fn) = O(n): Lemma 7.4. For each k = 1, . .,.n, we have (F1O(n))k = FkO(n). Proof. It follows from Lemma 7.3 that (F1O(n))k (F1U(n))k \ O(n) = FkU(n) \ O(n) = FkO(n). It remains to show that FkO(n) (F1O(n))k. Take a matrix U 2 FkO(n). There exists l k such that U 2 FlO(n)\Fl-1O(n), where we regard F-1O(n) as empty set. Since U is an orthogonal matrix, there exist an orthogonal matrix P and real numbers `1, . .,.`L for some L 2 N such that 0 1 cos`1 - sin`1 B sin`1 cos`1 O C B C B .. C B . C B C B cos`L - sin`L C U = P BB sin`L cos `L CCP -1. B C B 1 C B C B O 1 C B C @ ... A 1 19 An orthonormal basis {v1, . .,.vn} of Rn is defined by (v1, . .,.vn) = P. The dimension of the eigenspace with eigenvalue 1 is n - l. We may suppose that the eigenspace with eigenvalue 1 is the subspace spanned by vl+1, . .,.vn. If l is even, then l = 2L. Consequently we have XL ` ' ` *' Xn cos `i - sin`i v2i-1 * U = v2i-1 v2i * + vivi i=1 sin`i cos `i v2i i=l+1 XL ` ' ` *' Xl cos `i - sin`i v2i-1 * = v2i-1 v2i * + In - vivi i=1 sin`i cos `i v2i i=1 XL ` ' ` *' -1 + cos`i - sin`i v2i-1 = In + v2i-1 v2i * ` i=1 ` sin`i -1 + cos`i'` v2i'' -1 + cos`L - sin`L vl-1* = In + vl-1 vl * sin`L -1 + cos`L vl ` ` ' ` ' ' -1 + cos`1 - sin`1 v1* . . .In + v1 v2 * . sin`1 -1 + cos`1 v2 Define unit vectors u1, . .,.ul by `i `i u2i-1= v2i-1, u2i= v2i-1cos __+ v2isin__ 2 2 for all i = 1, . .,.L. Then we have (In - 2u2iu2i*)(In - 2u2i-1u2i-1*)v2i-1 = (In - 2u2iu2i*)(-v2i-1) `i = -v2i-1+ 2u2icos __ ` 2 ' ` ' `i `i `i = v2i-1 -1 + 2 cos2__ + v2i 2 cos__ sin__ 2 2 2 = v2i-1cos `i+ v2isin`i ` ` ' ` ' ' -1 + cos`i - sin`i v2i-1* = In + v2i-1 v2i * v2i-1, sin `i -1 + cos`i v2i and (In - 2u2iu2i*)(In - 2u2i-1u2i-1*)v2i = (In - 2u2iu2i*)v2i `i = v2i- 2u2isin__ 2 20 ` ' ` ' `i `i 2 `i = v2i-1 -2 cos__ sin__ + v2i 1 - 2 sin __ 2 2 2 = v2i-1(- sin`i) + v2icos`i ` ` ' ` ' ' -1 + cos`i - sin`i v2i-1* = In + v2i-1 v2i * v2i. sin `i -1 + cos`i v2i For all vectors v which are perpendicular to v2i-1 and v2i, we have (In - 2u2iu2i*)(In - 2u2i-1u2i-1*)v = v` ` ' ` ' ' -1 + cos`i - sin`i v2i-1* = In + v2i-1 v2i * v. sin `i -1 + cos`i v2i Thus we have (In - 2u2iu2i*)(In - 2u2i-1u2i-1*) ` ' ` ' -1 + cos`i - sin`i v2i-1* = In + v2i-1 v2i * , sin `i -1 + cos`i v2i and hence we obtain that ` ` ' ` '' -1 + cos`L - sin`L vl-1* U = In + vl-1 vl * sin`L -1 + cos`L vl ` ` ' ` ' ' -1 + cos`1 - sin`1 v1* . . .In + v1 v2 * sin`1 -1 + cos`1 v2 = (In - 2ulul*) . .(.In - 2u1u1*), and so U 2 (F1O(n))l (F1O(n))k. If l is odd, then l - 1 = 2L and the eigenvalue of vl is equal to -1. Consequently we obtain that X L ` ' ` *' Xn cos`i - sin`i v2i-1 * * U = v2i-1 v2i * - vlvl + vivi i=1 sin`i cos`i v2i i=l+1 XL ` ' ` *' -1 + cos`i - sin`i v2i-1 * = In + v2i-1 v2i * - 2vlvl i=1 ` sin`i ` -1 + cos`i v2i ' ` ' ' -1 + cos`L - sin`L vl-2* = (In - 2vlvl*) In + vl-2 vl-1 * sin`L -1 + cos`L vl-1 ` ` ' ` ' ' -1 + cos`1 - sin`1 v1* . . .In + v1 v2 * . sin`1 -1 + cos`1 v2 21 Define unit vectors u1, . .,.ul by `i `i u2i-1= v2i-1, u2i= v2i-1cos __+ v2isin__, ul= vl 2 2 for all i = 1, . .,.L. Then we have U = (In - 2ulul*) . .(.In - 2u1u1*), and so U 2 (F1O(n))l (F1O(n))k. __ Therefore we have FkO(n) = (F1O(n))k. |__| Now we can prove Theorem 7.2 in the case m = n. Proof of Theorem 7.2 in the case m = n. It is already shown in Steenrod [15, Ch. IV] that (F1U(n, F))k has a cellular decomposition 0 1 [k [ e0 [ @ ednj-1ednj-1-1. .e.dn1-1A. j=1 n nj>nj-1>...>n1>n-m __ So we obtain by Lemmas 7.3 and 7.4 that (F1U(n, F))k = FkU(n, F). |__| The following corollary will be needed to prove the remaining cases of Theorem 7.2: Corollary 7.5. To every m-frame V 2 Vm (Fn), there exists a matrix U 2 Fm U(n, F) satisfying that pnm(U) = V . Proof. Take an m-frame V 2 Vm (Fn). There exists a matrix U0 2 U(n, F) satisfying pnm(U0) = V . By Theorem 7.1, there exist scalars ~1, . .,.~n 2 S0F and vectors ui 2 V1(Fi) for i = 1, . .,.n such that U0 = (In + vn(~n - 1)vn *) . .(.In + v1(~1 - 1)v1 *). Define a matrix U by ___ * ______ * U = U0(In + v1(~1 - 1)v1 ) . .(.In-m + vn-m (~n-m - 1)vn-m ). Since we have __ * (In + vi(~i- 1)vi*)(In + vi(~i - 1)vi ) = In for all i = 1, . .,.n, we obtain U = (In + vn(~n - 1)vn *) . .(.In + vn-m+1 (~n-m+1 - 1)vn-m+1 *). The matrix U belongs to Fm U(n, F) by Lemmas 7.3 and 7.4. We obtain that ___ * ______ * (In + v1(~1 - 1)v1 ) . .(.In + vn-m (~n-m - 1)vn-m ) 2 U(n - m, F) x {Im }, __ which implies that pnm(U) = pnm(U0) = V . |__| 22 In order to show that Theorem 7.2 holds for all the remaining cases, it is sufficient to prove the following lemma: Lemma 7.6. For each k = 0, . .,.m, there holds pnm(FkU(n, F)) = FkVm (Fn). Proof. We will show that pnm(FkU(n, F)) FkVm (Fn). Take a matrix U 2 FkU(n, F). We denote by W the eigenspace of U with eigenvalue 1. Then dim W n - k. Thus we have dim (W \ )= dim W + m - dim(W + ) (n - k) + m - n = m - k. Hence there exists an orthonormal (m - k)-frame (v1, . .,.vm-k ) in the space W \ . We denote by Imn the transposed matrix of Inm. Then the matrix (Imnv1, . .,.Imnvm-k ) is an orthonormal (m - k)-frame in the space Fm , since v1, . .,.vm-k 2 . Then there holds that pnm(U)Imnvi = UInmImnvi = Uvi = vi = InmImnvi for all i = 1, . .,.m - k. Thus we have dim ker(pnm(U) - Inm) m - k, that is, pnm(U) 2 FkVm (Fn). Therefore pnm(FkU(n, F)) FkVm (Fn). It remains to show that FkVm (Fn) pnm(FkU(n, F)). Take a matrix V 2 FkVm (Fn). There exists an (m - k)-frame (uk+1, . .,.um ) 2 Vm-k (Fm ) such that V ui = Inmui for all i = k + 1, . .,.m. Adding unit vectors u1, . .,.uk 2 Fm appropriately to them, we obtain an orthonormal basis {u1, . .,.um } of Fm . Define U1 and V1 respectively by ` ' In-m O U1 = (u1, . .,.um ), V1 = -1 V U1. O U1 23 Then we have ` ' In-m O V1(ek+1, . .,.em )= -1 V U1(ek+1, . .,.em ) O U1 ` ' In-m O = -1 V (uk+1, . .,.um ) O U1 ` ' In-m O n = -1 Im (uk+1, . .,.um ) O U1 = InmU1-1(uk+1, . .,.um ) = Inm(ek+1, . .,.em ). Hence there exists a matrix V2 2 Vk(Fn-m+k ) satisfying that ` ' V2 O V1 = . O Im-k It follows from Corollary 7.5 that there exists a matrix U2 2 FkU(n-m+k, F) such that V2 = pn-m+kk(U2). Then the dimension of the eigenspace of U2 with eigenvalue 1 is greater than or equal to n - m. Define a matrix U by ` ' ` ' ` ' In-m O U2 O In-m O U = -1 . O U1 O Im-k O U1 ` * * ' U2 O The matrix U belongs to the filtration-set FkU(n, F), since the matrix ` ' O Im* *-k In-m O belongs to FkU(n, F) and since is a unitary matrix which has ` ' O U1 In-m O the inverse matrix -1 . Thus we have O U1 ` ' ` ' ` ' In-m O U2 O In-m O n pnm(U) = -1 Im O U1 O Im-k O U1 ` ' ` ' In-m O U2 O n -1 = Im U1 O U1 O Im-k ` ' ` ' In-m O U2In-m+kk O -1 = U1 O U1 O Im-k ` ' ` ' In-m O V2 O -1 = U1 O U1 O Im-k ` ' In-m O -1 = V1U1 O U1 = V, 24 and this fact implies V 2 pnm(FkU(n, F)). Therefore we have pnm(FkU(n, F)) = __ FkVm (Fn). |__| Finally, we can finish the proof of Theorem 7.2. Proof of Theorem 7.2. It is already shown in Steenrod [15, Ch. IV] and the proof of Theorem 7.2 for the case m = n, that pnm(FkU(n, F)) has a cellular decomposition 0 1 [k [ pnm(e0) [ @ pnm(ednj-1ednj-1-1. .e.dn1-1)A. j=1 n nj>nj-1>...>n1>n-m __ It follows from Lemma 7.6 that pnm(FkU(n, F)) = FkVm (Fn). |__| 8 Cone-decompositions of the complex Stiefel manifolds In this section we will give a cone-decomposition of the complex Stiefel mani- fold Vm (Cn) with length m by using Theorem 3.2. The base point of Vm (Cn) is the matrix Inm. Recall here that Frankel considered the Morse-Bott function f on Vm (Cn) in [3], in which, for each k = 0, . .,.m, the critical subset k is identified with the complex Grassmann manifold Gk(Cm ) by the inclusion 'k : Gk(Cm ) ! Fk. Then Miller's filtration {Fk}mk=0of Vk(Cn) is an NDR- filtration by Theorem 7.2. We define an inclusion e'k+1: CGk+1(Cm ) ! Fk+1 as follows: Definition 8.1. An inclusion e'k+1: CGk+1(Cm ) ! Fk+1 is defined by ` ' O e'k+1[t, P ] = isst Im - P + e P for all [t, P ] 2 CGk+1(Cm ). The m x m matrix Im - P + eisstP in Definition 8.1 transforms a vector v in the image of P to -eisstv and a vector u in the kernel of P to u. We need a lemma: Lemma 8.2. The inclusion e'k+1is along the gradient flow. Proof. It is easy to see that e'k+1[0, P ] = Inm, e'k+1[1, P ] = 'k+1(P ) 25 for each P 2 Gk+1(Cm ). For each [t, P ] 2 CGk+1(Cm ), we have f(e'k+1[t, P ]) = -m + (k + 1) - (k + 1) cossst, which implies that f(e'k+1[t, x]) = f(e'k+1[s, y]) when t = s, f(e'k+1[t, x]) < f(e'k+1[s, y]) when t < s for each [t, x], [s, y] 2 e'k+1(C k). The set (R x e'k(C k)) is a subset of e'k(C k) by Lemma 6.7. __ Therefore the inclusion e'k+1is along the gradient flow. |__| We will use Theorem 3.2 to construct a cone-decomposition of Vm (Cn) with length m. We have already seen that {e'k(Gk(Cm ))}mk=0is a family of ANR's by Theorem 6.4, that Miller's filtration {FkVm (Cn)}mk=0is an NDR- filtration by Theorem 7.2, and that all the inclusions are along the gradient flow by Lemma 8.2. Thus we have constructed a cone-decomposition of Vm (Cn) with length m. Theorem 8.3. The complex Stiefel manifold Vm (Cn) has a cone-decomposition with length m. Remark 4. The construction of a cone-decomposition of SU (n) Vn-1(Cn) given above considerably simplifies the argument in the proof given in [8]. It is easy to see from the structure of cohomology of Vm (Cn) that cup(Vm (Cn)) m. We obtain the following corollary: Corollary 8.4. cup(Vm (Cn)) = cat(Vm (Cn)) = Cat(Vm (Cn)) = cl(Vm (Cn)) = m. Remark 5. Singhof proved in [14] that cup(Vm (Cn)) = cat(Vm (Cn)) = Cat(Vm (Cn)) = gcat(Vm (Cn)) = m for all 0 < m n. The estimation problem of the L-S category of real and quaternionic Stiefel manifolds seems to us more difficult than that of complex Stiefel manifolds, since we could not give inclusions of the unreduced cones to the filtration-sets along the gradient flows of the Morse-Bott functions considered by Frankel on real and quaternionic Stiefel manifolds. 26 9 Cone-decompositions of submanifolds of U(n) We fix a matrix A 2 U(n) and define a space SA by SA = { B is a complex n x n matrix | tB = A*BA }. where tB denotes the transposed matrix of B. We use the following propo- sition: Proposition 9.1. Let U be a unitary matrix. Suppose that U has a unique spectral resolution: Xl U = ~kPk, k=1 where ~1, . .,.~l are the distinct eigenvalues of U and P1, . .,.Pl are idem- potent Hermitian matrices so that Pk is the eigenspace of ~k for each k = 1, . .,.l. Then U 2 SA if and only if Pk 2 SA for all k = 1, . .,.l. Proof. It is clear that if Pk 2 SA for all k = 1, . .,.l then U 2 SA . So we will show that if U 2 SA then Pk 2 SA for all k = 1, . .,.l. Suppose that U 2 SA . Then X l Xl ~ktPk = tU = A*UA = ~kA*PkA. k=1 k=1 From the uniqueness of the spectral resolution, there holds tPk = A*PkA for __ all k = 1, . .,.l. |__| Supposing that U(n)\SA is an ANR, we define a function bfon U(n)\SA by the restriction of the Morse-Bott function f on U(n) defined in Notation 6.1. We will calculate the critical subset of bfas follows: Lemma 9.2. The critical subset of bfis equal to an Gk(Cn) \ SA . k=0 Proof. Let C denote the critical subset of bf. It is clear that an C Gk(Cn) \ SA . k=0 27 We will show that a critical point of fb is a critical point of f. Take U 2 U(n) \ SA and suppose that U is not a critical point of f. The matrix U has a spectral resolution, which is represented as Xl ~kPk. k=1 Then Pk 2 SA for all k = 1, . .,.l by Proposition 9.1. Lemma 6.5 implies that the tangent vector (grad f)U is tangent to the maximal torus of U(n). Hence (grad f)U is tangent to TU (U(n) \ SA ) and is equal to gradfb U by Proposition 9.1. Consequently U is not a critical`point of bf. __ Therefore the critical subset C is equal to nk=0Gk(Cn) \ SA . |__| It follows from Lemma 6.5 and Proposition 9.1 that the unstable subset associated with Gk(Cn) \ SA is equal to (FkU(n) \ Fk-1U(n)) \ SA for each k = 0, . .,.n. Hence the family {FkU(n) \ SA }nk=0is a filtration of U(n) \ SA . Now we will show that the filtration {FkU(n) \ SA }nk=0is an NDR- filtration. Lemma 9.3. The filtration {FkU(n) \ SA }nk=0is an NDR-filtration. Proof. It suffices to show that (FkU(n) \ SA , Fk-1U(n) \ SA ) is an NDR-pair. We call (m1, . .,.ml) 2 Nla partition of k with length l if m1+. .+.ml= n. For each partition (m1, . .,.ml), we define the complex flag manifold of type (m1, . .,.ml), which is denoted F(m1, . .,.ml; Cn), by F(m1, . .,.ml; Cn) = { (P1, . .,.Pl) 2 Gm1 (Cn) x . .x.Gml(Cn) | PiPj = O if i 6= j}. Let F(m1, . .,.ml; Cn)|SA denote the intersection F(m1, . .,.ml; Cn) \ (SA x . .x.SA ). As usual, l denotes an l-simplex defined by l= { (`1, . .,.`l) 2 [0, 2ss]l | `1 `2 . . .`l }. We define a map ~ : lx F(m1, . .,.ml; Cn)|SA ! FkU(n) \ SA by Xl ~((`1, . .,.`l), (P1, . .,.Pl)) = In + (ei`i- 1)Pi i=1 28 for each ((`1, . .,.`l), (P1, . .,.Pl)) 2 lx F(m1, . .,.ml; Cn)|SA . We define* * a family of closed subsets {Fk,l}kl=0of FkU(n) \ SA by _ ! a Fk,l= (Fk-1U(n) \ SA ) [ ~ lx F(m1, . .,.ml; Cn)|SA m1+...+ml=k for each l = 1, . .,.k and Fk,0= Fk-1U(n) \ SA . The family {Fk,l}kl=0gives rise to a filtration of FkU(n) \ SA by Lemma 7.3 and Proposition 9.1. It is easy to see that _ ! a Fk,l-1= ~ @ lx F(m1, . .,.ml; Cn)|SA m1+...+ml=n and that ~ is bijective on a ( l- @ l) x F(m1, . .,.ml; Cn)|SA m1+...+ml=n for each l = 1, . .,.k. Hence the map l l a n ~ : , @ x F(m1, . .,.ml; C )|SA ! (Fk,l, Fk,l-1) m1+...+ml=n is a relative homeomorphism. It is clear that the pair l l a n , @ x F(m1, . .,.ml; C )|SA m1+...+ml=n is an NDR-pair. Consequently (Fk,l, Fk,l-1) is an NDR-pair. Therefore (FkU(n)\SA , Fk-1U(n)\SA ) are NDR-pairs for all k = 1, . .,.n, __ that is, the filtration {FkU(n) \ SA }k=0 is an NDR-filtration. |__| We will use the notations in the proof of Lemma 9.3 to prove the following lemma: Lemma 9.4. The critical subset Gk(Cn) \ SA of bffor each k = 0, 1, . .,.n is an ANR. Proof. When k = 0, the lemma is obvious. For each k = 1, . .,.n, the pair (U(n) \ SA , Fk,1) is an NDR-pair by the proof of Lemma 9.3. Hence Fk,1is an ANR, since U(n) \ SA is an ANR. The space Fk,1\ Fk,0is an open subset of Fk,1and is homeomorphic to (0, 2ss) x F(k, Cn)|SA . It is easy to see that Gk(Cn) \ SA is a deformation retract of Fk,1\ Fk,0. __ Therefore Gk(Cn) \ SA is an ANR. |__| 29 We define an inclusion b'k: C(Gk(Cn) \ SA ) ! FkU(n) \ SA as the restriction of the inclusion e'k: CGk(Cn) ! FkU(n). It is immediate to obtain the following lemma: Lemma 9.5. The inclusion b'k: C(Gk(Cn) \ SA ) ! FkU(n) \ SA is along the gradient flow of bf. Proof. The inclusion e'k: CGk(Cn) ! FkU(n) is along the gradient flow of f. Therefore by Lemma 6.7 and Proposition 9.1 the inclusion b'k: C(Gk(Cn) \ __ SA ) ! FkU(n) \ SA is along the gradient flow of bf. |__| As applications, we will construct cone-decompositions of U(n)=O(n) and U(2n)=Sp (n) with length n. To that end we need embeddings of symmetric spaces into Lie groups, called Cartan models of symmetric spaces, as follows (for a proof, the reader is referred to [2, Ch. 4, Thm. 15.1]): Theorem 9.6. Let G be a Lie group and oe an involution of G. Then we have G=Goe { goe(g-1) 2 G | g 2 G } { g 2 G | g-1 = oe(g) }. Moreover if the space { g 2 G | g-1 = oe(g) } is connected, then G=Goe { goe(g-1) 2 G | g 2 G } = { g 2 G | g-1 = oe(g) }. (1) U(n)=O(n). We will construct a cone-decomposition of U(n)=O(n) with length n. We obtain the Cartan model of U(n)=O(n) from Theorem 9.6 as follows: Lemma 9.7. The symmetric space U(n)=O(n) is homeomorphic to { U 2 U(n) | tU = U } = U(n) \ SIn. __ Proof. Set G = U(n) and define an involution oe of U(n) by oe(U) = U for each U 2 U(n). Hence Goe= O(n). It is clear that __ t { U 2 U(n) | U-1 = U } = { U 2 U(n) | U = U } = U(n) \ SIn. The space { U 2 U(n) | tU = U } is pathwise connected from Proposition 9.1. We obtain that U(n)=O(n) { U 2 U(n) | tU = U } from Theorem __ 9.6. |__| We can construct a cone-decomposition of U(n)=O(n). 30 Theorem 9.8. The symmetric space U(n)=O(n) has a cone-decomposition with length n. Proof. We consider a homeomorphism U(n)=O(n) U(n) \ SIn given in Lemma 9.7. All the critical subsets {Gk(Cn) \ SIn}nk=0are ANR's by Lemmas 9.2 and 9.4, the filtration {FkU(n) \ SIn}nk=0is an NDR-filtration by Lemma 9.3, and all the inclusions of the unreduced cones on the critical subsets are along the gradient flow by Lemma 9.5. Thus we have constructed a cone-decomposition __ of U(n)=O(n) with length n by Theorem 3.2. |__| It is immediate from the structure of Z2-cohomology of U(n)=O(n) (see for example [12, Ch. 3, Thm. 6.7(3)]) that cup(U(n)=O(n)) n. Consequently, we obtain the following corollary: Corollary 9.9. cup (U(n)=O(n)) = cat(U(n)=O(n)) = Cat(U(n)=O(n)) = cl(U(n)=O(n)) = n. (2) U(2n)=Sp (n). We will construct a cone-decomposition of U(2n)=Sp (n) with length n. We consider that Sp(n) is embedded in U(2n) by the map ` ___' B1 -B2 B1 + jB2 7! ___ , B2 B1 where B1 + jB2 2 Sp (n) and B1, B2 are complex matrices. We obtain the Cartan model of U(2n)=Sp (n) from Theorem 9.6 as follows: Lemma 9.10. The symmetric space U(2n)=Sp (n) is homeomorphic to { U 2 U(2n) | tU = J*UJ } = U(2n) \ SJ, where J denotes the matrix ` ' O -In . In O __ Proof. Set G = U(2n) and define the involution oe of U(2n) by oe(U) = J*U J for each U 2 U(2n). Hence Goe= Sp(n). It is clear that __ t * { U 2 U(2n) | U-1 = J*U J } = { U 2 U(2n) | U = J UJ } = U(2n) \ SJ. The space { U 2 U(2n) | tU = J*UJ } is pathwise connected by Proposition 9.1. We obtain that U(2n)=Sp (n) { U 2 U(2n) | tU = J*UJ } by Theorem __ 9.6. |__| 31 For the sake of completeness, we give a proof of the following lemma, along the idea in [11]. Lemma 9.11. If B 2 U(2n) satisfies tB = J*BJ, then the dimension of the eigenspace of B with eigenvalue 1 is even. Proof. The lemma is obvious, if B does not have eigenvalue 1. Suppose that B has eigenvalue 1. Take an eigenvector v with eigenvalue 1. Then we have ________ ____________ ______ ____ __ B*(-J__v) = tB(J*v) = J*BJ(J*v) = J*Bv = J*v = -Jv . Consequently we have -J__v= BB*(-J__v) = B(-J__v). The vector J__vis an eigenvector with eigenvalue 1. The vector J__vis perpen- dicular to v, since v*(J__v) = 0. If the dimension of the eigenspace of 1 is not equal to 2, one can apply inductively the same method to the orthogonal complement of the subspace spanned by v, J__v. Therefore we obtain that the dimension of the eigenspace __ of B with eigenvalue 1 is even. |__| Now we consider critical subsets and filtration-set as follows. Corollary 9.12. For each k = 0, . .,.n - 1, there holds F2k+1U(2n) \ SJ = F2kU(2n) \ SJ. We can construct a cone-decomposition of U(2n)=Sp (n). Theorem 9.13. The symmetric space U(2n)=Sp (n) has a cone-decomposition with length n. Proof. We consider that U(2n)=Sp (n) U(2n) \ SJ by Lemma 9.10. It is already shown that all critical subsets {Gk(C2n) \ SJ}2nk=0are ANR's by Lemmas 9.2 and 9.4, that the filtration {FkU(n) \ SJ}2nk=0is an NDR- filtration by Lemma 9.3, and that all the inclusions of the unreduced cones on the critical subsets are along the gradient flow by Lemma 9.5. Thus we have constructed a cone-decomposition of U(2n)=Sp (n). Its __ length is equal to n from Corollary 9.12. |__| It is immediate from the structure of cohomology of U(2n)=Sp (n) (see for example [12, Ch. 3, Thm. 6.7(1)]) that cup(U(2n)=Sp (n)) n. Consequently, we obtain the following corollary: 32 Corollary 9.14. cup (U(2n)=Sp (n)) = cat(U(2n)=Sp (n)) = Cat(U(2n)=Sp (n)) = cl(U(2n)=Sp (n)) = n. 10 Concluding remark In this paper, we have obtained not only algebraic invariants but also geo- metric and topological characteristics of complex Stiefel manifolds. The L-S category of a space may be considered as a measure of contractibility of it; for example, the L-S category of a space X is equal to 0 if and only if X is contractible. In fact, if m 6= 0, then the complex Stiefel manifold Vm (Cn) is not contractible, since we have cat Vm (Cn) = m. Let us examine which part of Vm (Cn) can be an obstruction of the contractibility of it. For each k = 1, . .,.m, the cone-decomposition of Vm (Cn) given in Section 8 gives an inequality cl FkVm (Cn) k. The cellular decomposition of FkVm (Cn) given in Section 7 gives an inequal- ity k cup FkVm (Cn). It follows from them that we have equations cup FkVm (Cn) = cat FkVm (Cn) = Cat FkVm (Cn) = clFkVm (Cn) = k. Hence we see that the subspace FkVm (Cn) \ Fk-1Vm (Cn) gives rise to an increment in the L-S category of Vm (Cn). Recall that the space FkVm (Cn) is divided into two parts, FkVm (Cn) \ Fk-1Vm (Cn) and Fk-1Vm (Cn) whose L-S category is k - 1. However, Fk-1Vm (Cn) is not a maximal subspace of FkVm (Cn) in the subsets of FkVm (Cn) with L-S category k - 1. Let us obtain a maximal subspace of FkVm (Cn) among the subsets of FkVm (Cn) with L-S category k - 1, because the complement of it seems to give an essential increment in the L-S category. The gradient flow of the Morse-Bott function on Vm (Cn) defined in Notation 6.1 gives a retracting deformation of FkVm (Cn) \ Fk-1Vm (Cn) to Gk(Cm ) and a retracting defor- mation of FkVm (Cn)\Gk(Cm ) to Fk-1Vm (Cn). Hence, we see that the space FkVm (Cn) is divided into two parts, Gk(Cm ) and FkVm (Cn) \ Gk(Cm ) with L-S category k - 1. If we take a point p 2 Gk(Cm ) and construct the union {p} [ (FkVm (Cn) \ Gk(Cm )), then we obtain an inequality cup ({p} [ (FkVm (Cn) \ Gk(Cm ))) k. 33 This follows from the fact that {p} [ (FkVm (Cn) \ Gk(Cm )) is homotopy equivalent to Fk-1Vm (Cn)[pnm e2k-1. . .e1, where pnm e2k-1. . .e1is the cell considered in Section 7. It follows from this fact that the space FkVm (Cn) \ Gk(Cm ) is the maximal subspace in FkVm (Cn) with L-S category k - 1. In general, a manifold is a union of contractible subsets, but it is never contractible if it is a closed manifold of positive dimension. Thus the con- tractibility is one of global properties of a manifold. The result of this paper indicates that the Morse-Bott function on Vm (Cn) considered by Frankel reflects most effectively the obstruction to the contractibility of Stiefel man- ifolds Vm (Cn). In the case of U(n)=O(n) and U(2n)=Sp (n), we have obtained cl(FkU(n) \ SIn) k, and cl(F2kU(2n) \ SJ) k. So, we could discuss similarly the reasons of the increment of the L-S category of U(n)=O(n) and U(2n)=Sp (n), if we know the cup-length of FkU(n) \ SIn and F2kU(2n) \ SJ; we do not know, however, at present the cup-length of FkU(n) \ SIn and F2kU(2n) \ SJ. We expect to solve this problem by con- structing particular cellular decompositions of U(n)=O(n) and U(2n)=Sp (n), similarly to that of Vm (Cn) given in Section 7. Remark 6. Cellular decompositions of U(n)=O(n) and U(2n)=Sp (n). In order to construct a cellular decomposition of U(n) (see, Steenrod [15, Ch. IV]) we have identified Miller's first filtration-set F1U(n) with a quasi-suspension, where a quasi-suspension of a topological space X is a suspension of X [ {*}. Here we have used complex numbers of absolute value equal to 1 for the parameters of the quasi-suspension and complex idempotent Hermite matrices for the complex projective space. It follows from Proposition 9.1 and [1] that, for the symmetric space U(n)=O(n) (resp. U(2n)=Sp (n)), we can identify the first filtration-set F1U(n)\SIn (resp. F2U(2n)\SJ) with the quasi-suspension of the real (resp. quaternionic) projective space. Then one can construct a cellular decomposition of the first filtration-set F1U(n) \ SIn (resp. F2U(2n) \ SJ) by using complex numbers of absolute value equal to 1 for the parameters of the quasi-suspension and real (resp. quaternionic) idempotent Hermite matrices for the real (resp. quaternionic) projective space. Then the cellular decomposition of the first filtration-set F1U(n) \ SIn (resp. F2U(2n) \ SJ) should be useful in defining concretely characteristic maps of cells of U(n)=O(n) (resp. U(2n)=Sp (n)). This is a rough idea to construct cellular decompositions of U(n)=O(n) and U(2n)=Sp (n). P. Landweber informed us that the L-S category of U(n)=O(n) should be useful in symplectic geometry. 34 In our forthcoming paper we will construct cone-decompositions of the irreducible symmetric Riemann spaces SU (n)=SO (n) and SU (2n)=Sp (n) as well as real and quoternionic Stiefel manifolds so that they give concrete NDR-filtrations, by making use of Morse-Bott functions and the main result of the present paper. Remark 7. The L-S categories of SU (n)=SO (n) and SU (2n)=Sp (n) are inde- pendently determined in [11] as cat(SU (n)=SO (n)) = n - 1 and cat(SU (2n)=Sp (n)) = n - 1. References [1]M. C. Crabb, S. A. Mitchell, The loops on U(n)=O(n) and U(2n)=Sp(n), Math. * *Proc. Cambridge Philos. Soc. 104 (1988), 95-103. [2]A. T. Fomenko, Differential Geometry and Topology, Contemporary Soviet Math* *e- matics, Consultants Bureau, New York, 1987. [3]T. 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Milnor, Morse Theory, Annals of Mathematics Studies, 51, Princeton Unive* *rsity Press, Princeton, N.J., 1963. [11]M. Mimura, K. Sugata, On the Lusternik-Schnirelmann category of SU(n)=SO (n* *) and SU (2n)=Sp(n), to appear in the proceedings of the conference "Groups, Homo* *topy and Configuration Spaces - in honor of Fred Cohen". [12]M. Mimura, H. Toda, Topology of Lie Groups, I, II, Translations of Mathemat* *ical Monographs, 91, American Mathematical Society, Providence, RI, 1991. [13]P. A. Schweitzer, Secondary cohomology operations induced by the diagonal m* *apping, Topology 3 (1965), 337-355. [14]W. Singhof, On the Lusternik-Schnirelmann category of Lie groups. II, Math.* * Z. 151 (1976), 143-148. 35 [15]N. E. Steenrod, Cohomology Operations, Annals of Mathematics Studies, 50, P* *rince- ton University Press, Princeton, N.J., 1962. [16]G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics, 61, Springer-Verlag, New York, 1978. (Hiroyuki Kadzisa) Department of Mathematics, Tokyo Institute of Tech- nology, Meguro, Tokyo 152-8551, Japan E-mail address : kadzisa@math.titech.ac.jp (Mamoru Mimura) Department of Mathematics, Faculty of Sciences, Okayama University, Okayama 700-8530, Japan E-mail address : mimura@math.okayama-u.ac.jp 36