UDC 513.833

MATHEMATICS

D. B. FUKS

ON THE CHARACTERISTIC CLASSES OF MASLOV—ARNOLD

(Presented by Academician P. S. Aleksandrov, 17 III 1967)

In the present note we discuss the characteristic classes of Lagrangian manifolds introduced by V. P. Maslov (¹) and V. I. Arnold (²)*. We shall give an expression for these classes in terms of the Stiefel—Whitney and A. Borel classes.

§ 1. Notation. Definitions. By \(E=E^{2n}\) we denote the space of real coordinates \((p_1,\ldots,p_n,q_1,\ldots,q_n)\). Define the transformation \(J:E\to E\) by the formula
\(J(p_1,\ldots,p_n,q_1,\ldots,q_n)=(-q_1,\ldots,-q_n,p_1,\ldots,p_n)\). Clearly, \(J^2=-1\). We shall call an \(n\)-dimensional hyperplane \(\Pi\subset E\) Lagrangian if the plane \(J(\Pi)\) is orthogonal to it. The totality of (nonoriented) Lagrangian planes of the space \(E\) passing through the origin of coordinates will be denoted by \(\Lambda_n\). The manifold \(\Lambda_n\) is diffeomorphic to the quotient space \(U(n)/O(n)\) of the group of unitary matrices by the group of orthogonal matrices.

Let \(X\) be a finite \(CW\)-complex. We shall say that an \(n\)-dimensional Lagrangian bundle with base \(X\) is given if the following are given: (1) an \(O(n)\)-bundle \(\xi\) with base \(X\); (2) an equivalence of the complexification \(c\xi\) of the bundle \(\xi\) and of the trivial \(U(n)\)-bundle with base \(X\). We always denote a Lagrangian bundle by the same letter as the \(O(n)\)-bundle defining it. It is easy to see that classes of equivalent \(n\)-dimensional Lagrangian bundles with base \(X\) are in one-to-one correspondence with homotopy classes of maps \(X\) into \(\Lambda_n\) (where the identity map \(\Lambda_n\to\Lambda_n\) corresponds to the bundle induced by the principal fibration \(U(n)\to U(n)/O(n)=\Lambda_n\)).

Let us note the following important example. Let \(M^n\subset E\) be a smooth submanifold of dimension \(n\). The manifold \(M^n\) is called Lagrangian if all its tangent planes are Lagrangian. The complexification of any Lagrangian plane \(\Pi\subset E\) is canonically isomorphic to \(E\); therefore the tangent bundle to a Lagrangian manifold may be regarded as a Lagrangian bundle. The corresponding map \(M^n\to\Lambda_n\) assigns to a point \(x\in M^n\) the plane \(\Pi\subset E\) passing through the origin of the coordinate space \(E\) and parallel to the tangent plane to the manifold \(M^n\) at the point \(x\).

§ 2. Subsets of the manifold \(M^n\). Let \(l\leq k\leq n\); \(P_k\) is the linear span of the vectors \(p_1,p_2,\ldots,p_k\). Denote by \(\Lambda_n^{k,l}\) the subset of the manifold \(\Lambda_n\) consisting of all Lagrangian planes whose intersection with \(P_k\) is at least \(l\)-dimensional. The sets \(\Lambda_n^{k,l}\) are not submanifolds of \(\Lambda_n\), but they can be represented in the following way as images of manifolds. Denote by \(\widetilde{\Lambda}_n^{k,l}\) the space of pairs \((\Pi,\Pi')\), where \(\Pi\in\Lambda_n\), and \(\Pi'\) is an \(l\)-dimensional plane of the space \(E\) such that \(\Pi'\subset \Pi\cap P_k\). The manifold \(\widetilde{\Lambda}^{k,l}\) is the space of a smooth fibration with base \(G_{k,l}\) (the Grassmann manifold) and fiber \(\Lambda_{n-l}\). The projection into

* In these papers only the first (one-dimensional) of the classes considered here was introduced. The definition of the remaining classes was given by V. I. Arnold in his seminar.

in this fibration takes \((\Pi,\Pi')\) to \(\Pi'\). The mapping
\(i=i_n^{k,l}:\widetilde{\Lambda}_n^{k,l}\to\Lambda_n\), where
\(i(\Pi,\Pi')=\Pi\), has as its image \(\Lambda_n^{k,l}\). On
\(i^{-1}(\Lambda_n^{k,l}\setminus \Lambda_n^{k,l+1})\) it is a diffeomorphism onto
\(\Lambda_n^{k,l}\setminus \Lambda_n^{k,l+1}\); the singularities of this mapping are concentrated on the set \(i^{-1}(\Lambda_n^{k,l+1})\), which, as is easily seen, has codimension \(n-k+1\). In what follows we shall mainly have to consider the sets
\(A_k=\Lambda_n^{\,n-k+1,1}\), \(B_k=\Lambda_n^{\,n-k+1,2}\), and
\(\Lambda_n^k=\Lambda_n^{\,n,k}\). Let us note that the dimension of the manifold \(\Lambda_n\) is \(n(n+1)/2\), and the codimensions of \(A_k\), \(B_k\), and \(\Lambda_n^k\) in \(\Lambda_n\) are respectively \(k\), \(2k+1\), and \(k(k+1)/2\).

§ 3. Orientability. Cohomology classes. The manifold \(\Lambda_n\) is orientable if and only if \(n\) is odd. In order that the manifold \(\widetilde{\Lambda}_n^{\,k,l}\) be orientable, it is necessary and sufficient that the fiber \(\Lambda_{n-l}\) be orientable and that the base \(G_{k,l}\) and the fibration itself be orientable or nonorientable simultaneously. The fiber \(\Lambda_{n-l}\) is orientable if and only if \(n-l\) is odd, and the base \(G_{k,l}\) if and only if \(k\) is even (excluding the cases \(l=k\) and \(l=0\)); the restriction of the fibration over the circle representing a generator of \(\pi_1(G_{k,l})\) is always trivial. Hence, in particular, it follows that if \(n\) is odd, then the manifold \(\Lambda_n^{k,l}\) is orientable if and only if \(k\) and \(l\) are both even, i.e. when the fiber and the base of the fibration are orientable.

The mapping \(i_n^{k,l}\) determines a homology class of the manifold \(\Lambda_n\) (the image of the fundamental class \([\widetilde{\Lambda}_n^{k,l}]\))—with integer coefficients if \(n\) is odd and \(k\) and \(l\) are even, and with coefficients in the group \(Z_2\) for arbitrary \(n,k,l\). If \(k\) is even and \(n\) and \(l\) are odd, then the manifold \(\widetilde{\Lambda}_n^{k,l}\) is nonorientable, and its one-dimensional Stiefel class is equal to the image of the generator of the group \(H^1(\Lambda_n;Z_2)\) under the homomorphism \((i_n^{k,l})^*\). Therefore in this case the mapping \((i_n^{k,l})_*\) determines a homology class with coefficients in \(Z_T\), the only nontrivial local system of groups isomorphic to \(Z\) with base \(\Lambda_n\). Finally, if \(n=k\) is odd and \(l\) is also odd, then although the manifold \(\widetilde{\Lambda}_n^{k,l}\) is nonorientable, the open manifold
\(\widetilde{\Lambda}_n^{k,l}\setminus i^{-1}(\Lambda_n^{k,l+1})\) is orientable (the codimension of the singularities of the mapping \(i\) is equal to \(1\)). This permits one to regard \(\Lambda_n^{\,n,l}\), for odd \(l\), as an integer cycle. If \(l\) is even, then \(\Lambda_n^{\,n,l}\) is a cycle with coefficients in \(Z_T\). Applying Poincaré duality to the homology classes defined by the cycles \(A_k,B_k,\Lambda_n^k\), we obtain, for every odd \(n\), the cohomology classes respectively

\[ \begin{aligned} a_k &\in H^k(\Lambda_n;Z_2), && k=1,2,\ldots,n;\\ b_k &\in \begin{cases} H^{2k+1}(\Lambda_n;Z),\\ H^{2k+1}(\Lambda_n;Z_T), \end{cases} && \begin{aligned} &k=2,4,\ldots,n-1;\\ &k=1,3,\ldots,n-2; \end{aligned}\\ \lambda_n^k &\in \begin{cases} H^{(k(k+1))/2}(\Lambda_n;Z),\\ H^{(k(k+1))/2}(\Lambda_n;Z_T), \end{cases} && \begin{aligned} &k=1,3,\ldots,n;\\ &k=2,4,\ldots,n-1, \end{aligned} \end{aligned} \]

with
\(\rho_2\lambda_n^1=a_1\) \((\rho_2\) is reduction modulo \(2)\).

The same classes can also be defined in the cohomology of \(\Lambda_n\) for even \(n\). This can be done either geometrically (only then the nonorientability, rather than the “coorientability,” of the set \(\Lambda_n^{k,l}\) will be relevant), or by considering the natural embeddings
\(\Lambda_{n-1}\subset \Lambda_n\subset \Lambda_{n+1}\). We shall not dwell on this in detail; in the subsequent formulations of all the theorems we regard \(n\) as arbitrary, and in the proofs as odd.

Since the space \(\Lambda_n\) is classifying for Lagrangian bundles, in the cohomology of the base \(X\) of any Lagrangian bundle \(\xi\) there are defined characteristic classes
\(a_k(\xi)\in H^k(X;Z_2)\), \(b_k(\xi)\in H^{2k+1}(X;Z\)

or \(Z_T\)) and \(\lambda_n^k(\xi)\in H^{(k(k+1))/2}(X; Z\ \text{or}\ Z_T)\). Here \(Z_T\) is the local system of groups isomorphic to \(Z\) with base \(X\), determined by the bundle \(\xi\).

The classes \(a_k(\xi)\) coincide with the Stiefel classes of the bundle \(\xi\) (and thus do not depend on the choice of trivialization of the complexification of the bundle \(\xi\)). The classes \(\lambda_n^k(\xi)\), by definition, are the Maslov–Arnol'd classes of the Lagrangian bundle \(\xi\). Information on the classes \(b_k(\xi)\) is contained in Theorem 4 below.

§ 4. Formulation of the results.

Theorem 1. \(\rho_2\lambda_n^k=a_1\cdots a_k=W^1\cdots W^n\) \((k=1,2,\ldots,n)\).

Corollary. The Maslov–Arnol'd classes, reduced modulo 2, do not depend on the choice of trivialization of the complexification.

Theorem 2. \(\lambda_n^{2s+1}= \lambda_n^1 b_2 b_4\cdots b_{2s}\) \((s=1,2,\ldots,(n-1)/2)\).

Theorem 3. \(\lambda_n^{2s}= b_1 b_3\cdots b_{2s-1}\) \((s=1,2,\ldots,(n-1)/2)\).

Corollary. If any one of the Maslov–Arnol'd classes is equal to zero, then all subsequent ones are equal to zero.

Theorem 4. The ring of weak integral cohomology of the manifold \(\Lambda_n\) for odd \(n\) is an exterior algebra with generators \(\lambda_n^1, b_2, b_4,\ldots,b_{n-1}\).

The last theorem means that the classes \(\lambda_n^1, b_k\) coincide with the generators of the cohomology ring of the homogeneous space \(U(n)/O(n)\), found by A. Borel \((^3)\).

Since \(\Lambda_n\) is the classifying space of Lagrangian bundles, the relations constituting Theorems 1–3 also hold for the characteristic classes \(a_k(\xi), b_k(\xi), \lambda_n^k(\xi)\) of any Lagrangian bundle \(\xi\). The proofs of Theorems 1–3 are analogous. In § 5 we shall prove only Theorem 1. § 6 is devoted to the study of the integral cohomology of the space \(\Lambda_n\). In it, in particular, Theorem 4 is proved.

§ 5. Intersections. We shall prove the equality \(\rho_2\lambda_n^k=a_k(\rho_2\lambda_n^{k-1})\). By definition, the class \(D(\rho_2\lambda_n^{k-1})\) is the image of the fundamental class mod 2 of the manifold \(\widetilde{\Lambda}_n^{\,n,k-1}\) under the mapping \(i_n^{\,n,k-1}\). Fix a small number \(\varepsilon>0\). Denote by \(\widetilde{A}_k^{(\varepsilon)}\) the set of pairs \((\Pi,l)\), where \(\Pi\subset E\) is a Lagrangian plane, and \(l\) is a line lying in the intersection of \(\Pi\) with the linear span of the vectors \(p_1\cos\varepsilon+q_1\sin\varepsilon,\ldots,p_{n-k+1}\cos\varepsilon+q_{n-k+1}\sin\varepsilon\). Obviously,
\[ \widetilde{A}_k^{(\varepsilon)}\approx \widetilde{\Lambda}_n^{\,n-k+1,1}, \]
and the mapping \(i':\widetilde{A}_k^{(\varepsilon)}\to\Lambda_n\), where \(i'(\Pi,l)=\Pi\), is homotopic to \(i_n^{\,n-k+1,1}\). Therefore the image under the mapping \(i'\) of the fundamental class mod 2 of the manifold \(\widetilde{A}_k^{(\varepsilon)}\) is \(D(a_k)\). It turns out that the mapping
\[ j=i_n^{\,n,k-1}\times i' : \widetilde{\Lambda}_n^{\,n,k-1}\times \widetilde{A}_k^{(\varepsilon)}\to \Lambda_n\times\Lambda_n \]
is transversal-regular with respect to the diagonal \(\Delta(\Lambda_n)\subset\Lambda_n\times\Lambda_n\). The complete preimage \(j^{-1}(\Delta(\Lambda_n))\) is precisely the manifold whose image of the fundamental class gives, in the sense of Poincaré duality, the product \(a_k(\rho_2(\lambda_n^{k-1}))\). Denote by \(\widetilde{\Lambda}_n^k\) the manifold whose points are triples \((\Pi,\Pi',l)\), where \(\Pi\subset E\) is a Lagrangian plane, \(\Pi'\subset\Pi\cap P_n\) is a \(k\)-dimensional plane, and \(l\subset \Pi'\cap P_{n-k+1}\) is a line.

Consider the mapping
\[ \eta:\widetilde{\Lambda}_n^k\to \widetilde{\Lambda}_n^{\,n,k-1}\times \widetilde{A}_k^{(\varepsilon)}, \]
where
\[ \eta(\Pi,\Pi',l)=((\varphi_\varepsilon\Pi,\Pi'/l),(\varphi_\varepsilon\Pi,\varphi_\varepsilon l)). \]
Here \(\Pi'/l\) is the orthogonal complement of the line \(l\) in the plane \(\Pi'\), \(\varphi_\varepsilon:E\to E\) is the transformation rotating the plane \((l,J(l))\) through the angle \(\varepsilon\) (so that \((\varphi_\varepsilon\Pi,\varphi_\varepsilon l)\in\widetilde{A}_k^{(\varepsilon)}\)) and equal to the identity on the orthogonal complement of this plane. It is easy to see that this mapping is a diffeomorphism from \(\widetilde{\Lambda}_n^k\) onto \(j^{-1}(\Delta(\Lambda_n))\). The composition
\[ \widetilde{\Lambda}_n^k \xrightarrow{\eta} \widetilde{\Lambda}_n^{\,n,k-1}\times A^{(\varepsilon)} \to \Delta(\Lambda_n)\subset\Lambda_n\times\Lambda_n \]
is homotopic to the mapping taking \((\Pi,\Pi',l)\) to \(\Pi\). The latter is the composition of the mapping
\[ \widetilde{\Lambda}_n^k\to \Lambda_n^{n,k}\quad ((\Pi,\Pi',l)\to(\Pi,\Pi')) \]
of degree 1 and \(i_n^{\,n,k}\). Hence the assertion being proved follows.

§ 6. The rational cohomology of the space \(\Lambda_n\) for odd \(n\) can easily be found from the Cartan–Serre theorem. The ring \(H^*(\Lambda_n;Q)\) for odd \(n\) is the exterior algebra on generators \(\beta_k\in H^{4k-3}(\Lambda_n;Q)\), \(k=1,\ldots,(n-1)/2\).

Consider in the space \(\Lambda_n\) the filtration
\[ *=\Lambda_n^n\subset\cdots\subset\Lambda_n^1\subset\Lambda_n^0=\Lambda_n. \]
Construct, with respect to this filtration, the (integral) spectral sequence of cohomology groups. The term \(E_1^{p,q}\) of this spectral sequence is
\[ H^{p+q}(\Lambda_n^{\,n-p+1},\Lambda_n^{\,n-p}); \]
the term \(E_\infty\) is associated with \(H^*(\Lambda_n)\). It can be shown that this spectral sequence is multiplicative. The difference \(\Lambda_n^p\setminus\Lambda_n^{p-1}\) is the space of a vector bundle with base \(G_{n,p}\) and fiber of dimension \(((n-p)(n-p+1))/2\), and this bundle is orientable if \(p\) is odd and nonorientable if \(p\) is even. Therefore
\[ E_1^p=\sum_q E_1^{p,q} \]
is nothing other than the full cohomology group of the Thom space of this bundle, i.e.
\[ E_1^{p,q}=H^{p+q-(p(p+1))/2}(G_{n,n-p};Z) \]
for even \(p\), and
\[ E_1^{p,q}=H^{p+q-(p(p+1))/2}(G_{n,n-p};Z_T) \]
for odd \(p\), where \(Z_T\) is the only possible nontrivial local system over \(G_{n,n-p}\) of groups isomorphic to \(Z\). In particular, the groups \(E_1^{p,(p(p-1))/2}\) for odd \(p\leq n\) are isomorphic to \(Z\); all the groups \(E_1^{1,q}\) are finite, except for \(E_1^{1,0}=Z\); among the groups \(E_1^{2,q}\), only the groups \(E_1^{2,3+4s}\), \(s=0,\ldots,(n-3)/2\), are infinite, each of them being isomorphic to the direct sum of the group \(Z\) and a finite group.

A simple count of ranks shows that the analogous spectral sequence with coefficients in the rational numbers is trivial. It follows that the elements of the group \(E_1^{p,(p(p-1))/2}\) are not images of any differentials (by dimensional considerations, the cycles of all differentials are elements of this group). Since
\[ E_1^{r,(p(p+1))/2-r}=0 \]
for \(r>p\), the group \(E_\infty^{p,(p(p-1))/2}\) is a subgroup of the group
\[ H^{(p(p-1))/2}(\Lambda_n). \]
This subgroup is characterized by the fact that its elements go to zero under the homomorphism induced by the inclusion
\[ \Lambda_n^{\,n-p-1}\subset\Lambda_n. \]
In particular, the class \(\lambda_n^p\) has this property. Moreover, the value of this class on the cycle \(\Lambda_{n-p}\) is equal to 1; consequently, it is not divisible by any integer and hence is a generator of this subgroup.

Thus the classes
\[ \lambda_n^p=\lambda_n^1 b_2\ldots b_{p-1}\in H^*(\Lambda_n) \]
are indivisible and have infinite order. Hence, together with the computation above of the rational cohomology of the space \(\Lambda_n\), Theorem 4 follows.

Let us note that the generators \(\lambda_n^1,b_2,\ldots,b_{n-1}\) of the ring of weak cohomology of \(\Lambda_n\) correspond in \(E_1\) to the generators of the free parts of the groups
\[ E_1^{1,0},\ E_1^{2,3},\ \ldots,\ E_1^{2,2n-3}. \]

In conclusion we note that all the results of this note can be obtained solely from the spectral sequence of § 6, without invoking any geometry.

Moscow State University
named after M. V. Lomonosov

Received
17 III 1967

References

1. V. P. Maslov, Theory of perturbations and asymptotic methods, Moscow, 1965.
2. V. I. Arnol'd, Functional Analysis, 1, No. 1 (1967).
3. A. Borel, in the collection Fiber Spaces, IL, 1958, p. 163.