On An Extension of the Mean Index to the Lagrangian Grassmannian

For a symplectic vector space, recall the identification of the symplectic group Sp(V) with an open and dense subset of the Lagrangian Grassmannian via the map sending each linear symplectomorphism to its graph. Our central result is in extending the mean index, using this embedding and a formal construction of the mean index in terms of a `circle map' on Sp(V), from the set of continuous paths in Sp(V) to those contained in a subset L with a complement of codimension two in the Lagrangian Grassmannian. Namely, we continuously extend the square of the circle map to a circle-valued map on L so that by applying the aforementioned construction to this new map, we reduce the existence and continuity of our extended index to the simpler problem of producing this continuous extension. Our secondary results concern the algebraic properties of the extended index with respect to a set-theoretic composition operation on linear relations which extends the usual group structure of Sp(V) to that of a monoid on the Lagrangian Grassmannian. To derive these we define an open and dense subset of differentiable paths in L which we call `stratum-regular', equip them with an equivalence relation and show that the point-wise composition of any two equivalent stratum-regular paths is piece-wise differentiable. We then show that over the stratum-regular paths, the extended index is homogeneous (for non-negative integers) and satisfies a quasimorphism-type bound for any equivalent pair of paths