The essential Lagrangian–Grassmannian and the homotopy type of the Fredholm Lagrangian–Grassmannian

Let H be a separable infinite dimensional Hilbert space endowed with a symplectic structure and let L 0 ⊂ H be a Lagrangian subspace. Using the results of [A. Abbondandolo, P. Majer, Infinite dimensional Grassmannians, math.AT/0307192], we show that the Fredholm Lagrangian–Grassmannian F L 0 ( Λ ) has the homotopy type of G c ( L 0 ) , the Grassmannian of all Lagrangian subspaces of H that are compact perturbations of L 0 . It is well known that the latter has the homotopy type of the quotient U ( ∞ ) / O ( ∞ ) . As a corollary, we recover a result by B. Booss-Bavnbek and K. Furutani (see [B. Booss-Bavnbek, K. Furutani, Symplectic functional analysis and spectral invariants, Contemp. Math. 242 (1999) 53–83; K. Furutani, Fredholm–Lagrangian–Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004) 269–331]) that the L 0 -Maslov index is an isomorphism between the fundamental group of F L 0 ( Λ ) and the integers.