The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: , and – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension).
In particular, the fundamental group of is infinite cyclic. Its first homology group is therefore also infinite cyclic, as is its first cohomology group, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov.
For a Lagrangian submanifold M of V, in fact, there is a mapping
which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in
of the distinguished generator of
- .
- V. I. Arnold, Characteristic class entering in quantization conditions, Funktsional'nyi Analiz i Ego Prilozheniya, 1967, 1,1, 1-14, doi:10.1007/BF01075861.
- V. P. Maslov, Théorie des perturbations et méthodes asymptotiques. 1972
- Ranicki, Andrew, The Maslov index home page, archived from the original on 2015-12-01, retrieved 2009-10-23 Assorted source material relating to the Maslov index.