Geometry of Maslov cycles 3 1 Davide Barilari∗and Antonio Lerario† 0 2 n a J Abstract 2 We introduce the notion of induced Maslov cycle, which describes and unifies geometricalandtopologicalinvariantsofmanyapparentlyunrelatedproblems,from ] G Real Algebraic Geometry to sub-Riemannian Geometry. S . h 1 Introduction t a m In this paper, dedicated to Andrei A. Agrachev in the occasion of his 60th birthday, we [ survey and develop some of his ideas on the theory of quadratic forms and its applica- 1 tions, from real algebraic geometry to the study of second order conditions in optimal v controltheory. Theinvestigation oftheseproblemsandtheirgeometric interpretation in 8 thelanguageofsymplecticgeometryisinfactoneofthemaincontributionofAgrachev’s 6 2 research of the 80s-90s (see for instance [1, 5, 6]) and these techniques are still at the 0 core of his more recent research (see the forthcoming preprints [7, 8]). . 1 Also, this survey can be interpreted as an attempt of the authors to give a uni- 0 fied presentation of the two a priori unrelated subjects of their dissertations under 3 1 Agrachev’s supervision, namely sub-Riemannian geometry and the topology of sets de- : fined by quadratic inequalities. The unifying language comes from symplectic geometry v i and uses the notion of Maslov cycle, as we will discuss in a while. X To start with we introduce some notation. The set L(n) of all n-dimensional La- r a grangian subspaces of R2n (with the standard symplectic structure) is called the La- grangian Grassmannian; it is a compact submanifoldof theordinary Grassmannian and once we fix one of its points ∆, we can consider the algebraic set: Σ = {Π ∈ L(n)|∆∩Π 6= 0}, (this is what is usually referred to as the train of ∆, or the universal Maslov cycle). The main idea of this paper is to study generic maps f : X → L(n), for X a smooth ∗CNRS, CMAP, E´cole Polytechnique, Paris and E´quipe INRIA GECO Saclay-ˆIle-de-France, Paris, France - Email: [email protected] †Department of Mathematics, PurdueUniversity - Email: [email protected] 1 manifold, and the geometry of the preimage under f of the cycle Σ. Such a preimage f−1(Σ) is what we will call the induced Maslov cycle. It turns out that many interesting problems can be formulated in this setting and our goal is to describe a kind of duality that allows to get geometric information on the map f by replacing its study with the geometry of f−1(Σ). To give an example, the Maslov cycle already provides information on the topology of L(n) itself. In fact Σ is a cooriented algebraic hypersurface smooth outside a set of codimension three and its intersection number with a generic map γ : S1 → L(n) computes [γ] ∈π (L(n)) ≃ Z. 1 The theory of quadratic forms naturally appears whenwe look at the local geometry of the Lagrangian Grassmannian: in fact L(n) can be seen as a compactification of the space Q(n) of real quadratic forms in n variables and, using this point of view, the Maslov cycle Σ is a compactification of the space of degenerate forms. Given k quadratic forms q ,...,q we can construct the map: 1 k f :Sk−1 → L(n), (x ,...,x ) 7→ x q +···+x q . 1 k 1 1 k k In fact the image of this map is contained in the affine part of L(n) and its homotopy invariants are trivial. Neverthless the induced Maslov cycle f−1(Σ) has a nontrivial geometry and can be used to study the topology of: X = {[x] ∈ RPn−1|q (x) = ··· = q (x) = 0}. 1 k More specifically, it turns outthat as a firstapproximation for thetopology of X we can take the “number of holes” of f−1(Σ). Refining this approximation procedure amounts to exploit how the coorientation of Σ is pulled-back by f. In some sense this is the idea of the study of (locally defined) families of quadratic forms and their degenerate locus, and the set of Lagrange multipliers for a variational problemadmitsthesamedescription. Infactonecanconsidertwosmoothmapsbetween manifolds F : U → M and J : U → R and ask for the study of critical points of J on level sets of F. With this notation the manifold of Lagrange multipliers is defined to be: C = {(u,λ) ∈ F∗(T∗M)|λD F −d J = 0}. F,J u u Attached to every point (u,λ) ∈C there is a quadratic form, namely the Hessian F,J of J|F−1(F(u)) evaluated at u, and using this family of quadratic forms we can still define an induced Maslov cycle Σ (the definition we will give in the sequel is indeed more F,J intrinsic). This abstract setting includes for instance the geodesic problem in Riemannian and sub-Riemannian geometry (and even more general variational problems). In this case the set U parametrizes the space of admissible curves, F is the end-point map (i.e. the 2 map that assigns to each admissible curve its final point), and J is the energy of the curve. The problem of finding critical points of the energy on a fixed level set of F corresponds precisely to the geodesic problem between two fixed points on the manifold M. In this context Σ corresponds to points where the Hessian of the energy is degen- F,J erate and its geometry is related to the structure of conjugate locus in sub-Riemannain geometry. Moreover the way this family of quadratic forms (the above mentioned Hes- sians) degenerates translates into optimality properties of the corresponding geodesics. Rather than a systematic and fully detailed treatment we try to give the main ideas, givingonlysomesketchesoftheproofs(providingreferenceswherepossible)andoffering a maybe different perspective in these well-estabilished research fields. Our presentation is strongly influenced by the deep insight and the ideas of A. A. Agrachev. We are extremely grateful to him for having shown them, both in mathemat- ics and in life, the elegance of simpleness. 2 Lagrangian Grassmannian and universal Maslov cycles 2.1 The Lagrangian Grassmannian Let us consider R2n with its standard symplectic form σ. A vector subspace Λ of R2n is called Lagrangian if it has dimension n and σ| ≡ 0. The Lagrange Grassmannian L(n) Λ in R2n is the set of its n-dimensional Lagrangian subspaces. Proposition 1. L(n) is a compact submanifold of the Grassmannian of n-planes in R2n; its dimension is n(n+1)/2. ⋔ Consider indeed the set ∆ = {Λ ∈ L(n)|Λ∩∆ = 0} of all lagrangian subspaces ⋔ that are transversal to a given one ∆ ∈ L(n). Clearly ∆ ⊂ L(n) is an open subset and ⋔ L(n) = ∆ . (1) ∆∈L(n) [ It is then sufficient to find some coordinates on these open subsets. Let us fix a Lagrangian complement Π of ∆ (which always exists but is not unique). Every n- dimensional subspace Λ ⊂ R2n which is transversal to ∆ is the graph of a linear map fromΠto∆. Choosingabasison∆andΠ, thislinearmapisrepresented incoordinates by a matrix S such that: Λ∩∆ = 0 ⇔ Λ ={(x,Sx), x ∈ Π≃ Rn}. ⋔ Hencetheopenset∆ ofallLagrangiansubspacesthataretransversalto∆isparametrized by the set of symmetric matrices, that gives coordinates on this open set. This also 3 proves that the dimension of L(n) is n(n+1)/2. Notice finally that, being L(n) a closed set in a compact manifold, it is itself compact. Fix now an element Λ ∈ L(n). The tangent space T L(n) to the Lagrange Grass- Λ mannian at the point Λ can be canonically identified with set of quadratic forms on the space Λ itself: T L(n) ≃ Q(Λ). Λ Indeed consider a smooth curve Λ(t) in L(n) such that Λ(0) = Λ and denote by Λ˙ ∈ T L(n) its tangent vector. For any point x ∈ Λ and any smooth extension x(t) ∈ Λ(t) Λ we define the quadratic form: Λ˙ : x 7→ σ(x,x˙), x˙ = x˙(0). An easy computation shows that this is indeed well defined; moreover writing Λ(t) = {(x, S(t)x), x ∈ Rn} then the quadratic form Λ˙ associated to the tangent vector of Λ(t) at zero is represented by the matrix S˙(0), i.e. Λ˙(x) = xTS˙(0)x. We stress that this representation using symmetric matrices works only for coordi- nates induced by a Lagrangian splitting R2n = Π⊕∆,i.e. Π and ∆ are both lagrangian. Example 1 (The Lagrange Grassmannians L(1) and L(2)). Since every line in R2 is Lagrangian (the restriction of a skew-symmetric form to a one-dimensional subspace must be zero), then L(1) ≃ RP1. The case n = 2 is more interesting. We first notice that each 2-plane W in R4 defines a unique (up to a multiple) degenerate 2-form ω in Λ2R4, by W = kerω. Thus there is a map: p :G(2,4) → P(Λ2R4)≃ RP5. This map is called the Plu¨cker embedding; its image is a projective quadric of signature (3,3). The restriction of p to L(2) maps to p(L(2)) = {[ω]| kerω 6= 0 and ω∧σ = 0}, which is the intersection of the image of p with an hyperplanein RP5, i.e. the zero locus of the restriction of the above projective quadric to such hyperplane. In particular L(2) is diffeomorphic to a smooth quadric of signature (2,3) in RP4. 2.2 Topology of Lagrangian Grassmannians It is possible to realize the Lagrange Grassmannian as a homogeneous space, through an action of the unitary group U(n). In fact we have a homomorphism of groups φ :GL(n,C) → GL(2n,R) defined by: A B φ :A+iB 7→ , −B A (cid:18) (cid:19) 4 and the image of the unitary group is contained in the symplectic one. In particular for every lagrangian subspace Λ ⊂ R2n and every M in U(n) the vector space φ(M)Λ is still lagrangian. This defines the action of U(n) on L(n); the stabilizer of a point is readily verified to be the group O(n) and we get: L(n)≃ U(n)/O(n). The cohomology of L(n) can be studied applying standard techniques to the fibration U(n) → L(n) and working with Z coefficients1 we get a ring isomorphism H∗(L(n)) ≃ 2 H∗(S1×···×Sn); we refer the reader to [14] for more details. For our purposes we will need an explict description of the fundamental group of L(n) and this can be obtained as follows. We first consider the map det2 : U(n) → S1 (the square of the determinant). Multiplication by a matrix of O(n) does not change the value of the square of the determinant, thus we get a surjective map det2 :L(n) → S1. This map also is a fibration and with simply connected fibers, each one of them being diffeomorphic to SU(n)/SO(n). In particular it follows that it realizes an isomorphism of fundamental groups and: π (L(n)) ≃ Z. 1 2.3 The universal Maslov cycle Since the fundamental group of L(n) is Z, then the 1-form dθ/2π on S1 (the class of this form generates its first cohomology group with integer coefficients) pulls-back via det2 to a 1-form on L(n) whose cohomology class µ generates H1(L(n),Z): 1 µ = (det2)∗ dθ ∈ H1(L(n),Z). 2π (cid:20) (cid:21) Such a class is usually referred to as the universal Maslov class (see [9, 11]). Once we fix a Lagrangian space ∆ ∈ L(n) it is possible to define a cooriented algebraic cycle in L(n) which is Poincar´e dual to µ; such cycle is called the train of ∆ and is defined as follows: ⋔ Σ = {Λ ∈L(n)|Λ∩∆ 6= 0} = L(n)\∆ . ∆ Here the subscript denotes the dependence on ∆ and when no confusion arises we will omit it: a different choice of ∆ produces an homologous train (in fact just differing by a symplectic transformation). We will discuss the geometry of Σ in greater detail in the next section; what we need for now is that this is an algebraic hypersurface whose 1Unless differently stated, all homology and cohomology groups will be with Z2 coefficients. 5 singularitieshavecodimensionthree andiscooriented. Thefactthatitisanalgebraicset makes it acycle, thefact that itis an hypersurfacewhosesingularities have codimension three allows to define intersection number with it and the fact that is cooriented makes this intersection number an integer. Here coorientation means that Σ is two-sided in L(n), i.e. there is a canonical orientation of its normal bundle along its smooth points. UsingtheabovediffeomorphismL(n)≃ U(n)/O(n)itiseasytochooseapositivenormal at a smooth point Λ ∈ Σ: we represent Λ as [M] for a unitary matrix M and we take the velocity vector in zero of the curve t 7→ [eitM]. Example 2 (The train in L(2)). We have seen that L(2) is diffeomorphic to a quadric of signature (2,3) in RP4; thus it is double covered by S1×S2 (i.e. the set of points in R5 satisfying the equation x2+x2 = x2+x2+x2 and of norm one). 0 1 2 3 4 We fix now a plane ∆ and study the geometry of the train Σ . We let Π be a ∆ ⋔ Lagrangian complement to ∆ and using symmetric matrices chart on Π we have: ⋔ Σ ∩Π ≃ {S| det(S) = 0}. ∆ The set of symmetric matrices with determinant zero is a quadratic cone in R3 with singular point at the origin; to get Σ we have to add its limit points in L(2) and this ∆ results into an identification of the two boundaries components of such a cone. What we get is a Klein bottle with one cycle collapsed to a point. The main idea of this paper is to study generic maps f : X → L(n), for X a smooth manifold, and the geometry of the preimage under f of the cycle Σ (together with its coorientation). Such a preimage f−1(Σ) is what we will call the induced Maslov cycle. Sometimes in the sequel the map f will be defined only locally but it will still produce a Maslov type cycle on X. Our goal is to describe a kind of duality that allows to get geometric information on the map f by replacing its study with the one of the geometry of f−1(Σ). We will discuss these ideas in greater detail in the next section. Example 3 (Generic loops). Consider a smooth map: γ :S1 → L(n) transversal to the smooth points of Σ. Such a property is generic and we might ask for the meaning of the number of points in γ−1(Σ). Since the intersection number with Σ computes the integer [γ] ∈ π (L(n)), in a very rough way we can write: 1 |[γ]| ≤ b(γ−1(Σ)), (2) where the r.h.s. denotes the sum of the Betti numbers, which in this case coincides with the number of connected components (i.e. number of points). This inequality is simply what we obtain by forgetting the coorientation in the sum defining the intersection 6 number. Thecomparisonthroughtheinequality betweenwhatappearsonthel.h.s. and what on the r.h.s. is the first mirror of the mentioned duality between the geometric properties of γ and the topological ones of γ−1(Σ). Remark 4 (Schubert varieties). It is indeed possible to give L(n) a cellular structure using Schubert varieties in a fashion similar to the ordinary Grassmannian: the cells are inonetoonecorrespondencewithsymmetric Youngdiagrams;givenoneofsuchdiagram thecorrespondingSchubertcellis theoneobtained byconsideringaflagthatisisotropic with respect to the symplectic form. More precisely let {0} ⊂ V ⊂ V ··· ⊂ V = R2n 1 2 2n be a complete flag such that σ(V ,V ) = 0 for every j = 1,...,n (this means the j 2n−j flag is isotropic; in particular V is Lagrangian). If now we let a be the partition n a : n ≥ a ≥ a ≥ ··· ≥ a ≥ 0, then the corresponding Schubert variety is: 1 2 n Y = {Λ∈ L(n)| dim(Λ∩V )≥ i for i = 1,...,n}. a n+i−ai The codimension of Y is (|a|+l(a))/2, where l(a) is the number of boxes on the main a diagonal of the associated Young diagram (such a diagram has a boxes in its i-th row). i Since this diagram must be symmetric along its diagonal we see that there are only 2n possible good partitions (see [16] for more details on this approach). Geometrically this shows that the combinatorics of the cell structure of the Grassmannian G(n,2n) descends (by intersection) to the one of L(n). Moreover, since the incidence maps have even degree, cellular homology with Z coefficients gives again the above formula for 2 H∗(L(n)). Notice in particular that Σ is a Schubert variety: letting the n-th element of the isotropic flag to be ∆ itself, then: Σ = {Λ ∈ L(n)| dim(Λ∩∆) ≥ 1} = Y . ∆ (1,0,...,0) Example 5 (Schubert varieties of L(2)). We consider again the case of L(2) and fix an isotropic flag {0} ⊂ V ⊂ ∆ ⊂ V ⊂ R4. The cell structure is given by the four following 1 3 possible partitions (0,0),(1,0),(2,1),(2,2). Let us see how the corresponding Schubert varieties look like. To this end let us write R2n = ∆ ⊕ Π, where Π is a Lagrangian complement to ∆. In this way every Λ in Π⋔ is of the form Λ = {(x,Sx)|S = ST}. We immediately get Y = L(n); moreover we have already seen that Σ = Y . (0,0) ∆ (1,0) The Schubert variety Y equals ∆ itself (in the symmetric matrices coordinates it is (2,2) the zero matrix). Finally we have Y = {Λ|Λ ⊃ V , Λ ⊂ V }. The intersection of this variety with (2,1) 1 3 ⋔ Π equals all the symmetric matrices S whose kernel contains V ⊂ ∆: such matrices 1 are all multiple one of the other and they form a line, thus Y ≃ RP1. (2,1) 7 3 Pencils of real quadrics 3.1 Local geometry and induced Maslov cylces In this section we study in more detail the local geometry of the Lagrangian Grassman- nian. If no data are specified, being a differentiable manifold, each one of its points looks exactly like the others. Once we fix one of them, say ∆, the situation drastically enriches: we have seen, for example, that we can choose a cycle Σ representing the ∆ generator of the first cohomology group. The following proposition gives a more precise structure of the local geometry we obtain on L(n) after we have fixed one of its points ∆. Proposition 2. Let ∆ in L(n) be fixed. Every Λ ∈ L(n) has a neighborood U and a smooth algebraic submersion: φ: U → W, where W is an open set of the space of quadratic forms on ∆∩Λ ≃ Rk, satisfying the following properties: 1. (d φ)Λ˙ = Λ˙| ; Λ ∆∩Λ 2. dim(kerφ(Π)) =k−dim(∆∩Π) for every Π in U. 3. for every Π in W the fiber φ−1(Π) is contractible. Let ∆′ be a lagrangian complement to ∆ transversal to Λ. Then, giving coordinates to the open set {Π ∈ L(n)|Π ⋔ ∆′} using symmetric matrices, the Proposition is just a reformulation of Lemma 2 from [1]. The fact that φ is a submersion allows to reduce the study of properties of L(n) to smaller Grassmannians, via the Implicit Function Theorem. For the first property, recall that we have a natural identification of the vector space T L(n) with the space Λ of quadratic forms on Λ; each one of these quadratic forms can be restricted to the subspace ∆∩Λ and this restriction operation is what d φ does. The second property Λ says that φ transforms the combinatorics of intersections with ∆ with the one of the kernels of the corresponding quadratic forms. Thus locally Σ looks like the space of degenerate quadratic forms and it is interest- ∆ ing to see how all these local charts are glued together. Let us consider a Λ in Σ and ∆ some Π Lagrangian complement to ∆ such that Π ⋔ Λ. Given a symplectyc transfor- 1 1 mation ψ : R2n → R2n preserving ∆, the matrix T representing it in the coordinates given by the Lagrangian splitting ∆⊕Π has the form: 1 A−1 BAT T = with B = BT. 0 AT (cid:18) (cid:19) 8 If Λ is represented by the symmetric matrix S, the change of coordinates ψ changes its representative to (ATSA)(I + BATSA)−1 (indeed this formula works for every Λ transversal to Π). Remark 6 (Localtopologyofthetrain). ThelocaltopologyofΣ canbedescribedusing ∆ Proposition 2. Let B be a small ball centered at Λ ∈ Σ with boundary S = ∂B . Λ ∆ Λ Λ Then the intersection B ∩Σ is contractible: it is a cone over the intersection S ∩Σ ; Λ ∆ Λ ∆ moreover S ∩ Σ is Spanier-Whitehead dual to a union of ordinary Grassmannians Λ ∆ and: k H∗(S ∩Σ )≃ H (G(j,k)), k = dim(Λ∩∆) (3) Λ ∆ ∗ j=0 M Theorem 3 from [17] gives the statement for Λ = ∆ and the general result follows by applying Proposition 2. For every r ≥ 1 we can define the sets: (r) (r) (r+1) Σ = {Λ ∈ L(n)| dim(Λ∩∆) ≥ r} and Z = Σ \Σ . ∆ r ∆ ∆ Using this notation, Proposition 2 implies that Σ is Whitney stratified by Z and ∆ r r the codimension of each Z in L(n) is r+1 (the reader is referred to [13] for properties r 2 S of such stratifications). (cid:0) (cid:1) Remark 7 (Cooorientation revised). Let Λ be a smooth point of Σ and γ : (−ǫ,ǫ) → ∆ L(n) be a curve transversal to all strata of Σ and with γ(0) = Λ (the transversality ∆ (2) condition ensures that γ meets only Σ \Σ , i.e. the set of smooth points of Σ). Since ∆ ∆ T L(n) ≃ Q(Λ), the velocity γ˙(0) can be interpreted (by restriction) as a quadratic Λ form on Λ∩∆. Proposition 2 together with the transversality condition ensures that this restriction is nonzero. We say that the curve γ is positively oriented at zero if γ˙(0)| > 0. Since this definition is intrinsic, it gives a coorientation on Σ and it is not Λ∩∆ difficult to show that it coincides with the above given one. Definition (Induced Maslov cycle). Let X be a smooth manifold and f : X → L(n) be a map transversal to all strata of Σ = Σ . The cooriented preimage f−1(Σ) will be ∆ called the Maslov cycle induced by f. A generic map f : X → L(n) is indeed transversal to all strata of Σ and f−1(Σ) is itself Nash stratified (its strata beingthe preimage of thestrata of Σ); thetransversality condition ensures that the the normal bundle of the smooth points of f−1(Σ) (which is the pull-back of the normal bundle of Σ) has a nonvanishing section, i.e. the induced Maslov cycle also has a coorientation. 9 3.2 Pencils of quadrics We turn now to the above mentioned duality between the geometry of a map f : X → L(n) transversal to all strata of Σ and the cooriented cycle induced by f. We consider ∆ a specific example, namely the case of a map from the sphere, whose image is contained in one coordinate chart. More precisely let ∆ ⊕ Π ≃ R2n be a Lagrangian splitting and W ≃ Rk be a linear ⋔ subspace of Π ≃ Q(∆) (the space of quadratic forms on ∆): W = span{q ,...,q } with q ,...,q ∈ Q(Λ) ≃ Q(n) 1 k 1 k (here Q(n) denotes the space of quadratic forms in n variables). Notice that the above isomorphism is defined once a scalar product on ∆ is given: this allows to identify symmetric matrices with quadratic forms. In this context W is called a pencil of real quadrics; the inclusion Sk−1 ֒→ W defines a map: f : Sk−1 → Q(n) and for a generic choice of W such a map is transversal to all strata of Σ = Σ . Notice ∆ that Σ equals the discriminant of the set of quadratic forms in n variables and equation (3) gives a descritpion of its cohomology. To every linear space W as above we can associate an algebraic subset X of the W real projective space RPn−1 = P(∆) (usually referred to by algebraic geometers as the base locus of W): X = {[x] ∈RPn−1|q (x) = ··· = q (x) = 0}. W 1 k The study of the topology of X was started by Agrachev in [1, 5] and continued by W Agrachev and the second author in [6]. Remark 8 (The spectral sequence approach). The main idea of Agrachev’s approach is to study the Lebesgue sets of the positive inertia index function on W, i.e. the number of positive eigenvalues i+(q) of a symmetric matrix representing q. More specifically we can consider: Wj = {q ∈ W |i+(q) ≥ j}, j ≥ 1, and Theorem A from [6] says that roughly we can take the homology of these sets as the homology of X : W n H∗(W,Wj) “approximates” H∗(X ). W j=1 M The cohomology classes from H∗(W,Wj) are just the canditates for the homology of X . The requirements they have to fulfill in order to represent effective classes in W 10