  We give a formula for the parity of the Maslov index of a triple of
Lagrangian subspaces of a skew symmetric bilinear form over the real numbers.
We define an index two subcategory (the even subcategory) of a 3-dimensional
cobordism category. The objects of the category are surfaces are equipped with
Lagrangian subspaces of their real first homology. This generalizes a result of
the first author where surfaces are equipped with Lagrangian subspaces of their
rational first homology.
