Maslov cycle

The stratum $\Sigma_k(L_0)$ has codimension $k(k+1)/2$ in $\Lambda(V)$. This follows from a local chart in which nearby Lagrangians are graphs of symmetric forms: the intersection with $L_0$ becomes the kernel, and the rank-defect-$k$ locus has that codimension. In particular, the first singular stratum has codimension three, so a generic one-parameter path meets only the smooth stratum Robbin–Salamon, §290052-W).

At $L\in\Sigma_1(L_0)$, a tangent vector to $\Lambda(V)$ is represented by a quadratic form on $L$. Restricting it to the line $L\cap L_0$ gives a scalar; its sign distinguishes the two normal directions and defines the coorientation. For a smooth path $L(t)$, the corresponding restriction is the crossing form. A nondegenerate crossing is isolated, and its signature gives the local intersection contribution.

When $V=\mathbb R^2$, every line is Lagrangian and $\Lambda(V)\cong\mathbb RP^1\cong S^1$. For a fixed line $L_0$, the Maslov cycle consists of the single point $L_0$. A rotating line crosses this point once during a half-turn, with sign determined by the direction of rotation. There are no singular strata because two one-dimensional Lagrangians cannot intersect in dimension at least two.

Some authors use $\Sigma_k$ for the closed locus $\dim(L\cap L_0)\geq k$, while others use it for the exact-$k$ stratum. Here $\Sigma_k(L_0)$ means exact intersection dimension, and $\Sigma(L_0)$ is their union. The cycle is not a smooth hypersurface globally; treating it as one discards the higher-incidence strata that matter for nongeneric endpoints.

1. V. I. Arnol'd, “On a characteristic class entering into conditions of quantization,” Functional Analysis and Its Applications 1 (1967), 1–14. DOI record. Relevant: the Maslov cycle and its associated characteristic class.
2. Joel Robbin and Dietmar Salamon, “The Maslov index for paths,” Topology 32 (1993), 827–844. DOI record90052-W). Relevant: §§1–2, strata, crossing forms, and path indices.