Stratified space

A space partitioned into manifold strata, with a locally finite frontier-compatible partition. In the conical setting, a neighbourhood of a point in an $s$-dimensional stratum has the form ${\mathbb{R}}^s\times cL$, respecting the strata, for a compact stratified link $L$.

A stratum is one manifold piece of a stratification, with locally constant dimension and prescribed conical normal neighbourhoods. Its codimension in a pure $N$-dimensional space is $N$ minus its dimension.

An $N$-dimensional conically stratified space has a closed filtration $\varnothing =X_{-1}\subseteq X_0\subseteq \cdots \subseteq X_N=X$, with each nonempty $X_j\setminus X_{j-1}$ a $j$-manifold, $X_{N-1}=X_{N-2}$, and dense top stratum. At an $s$-stratum the conical link is a compact $(N-s-1)$-dimensional stratified pseudomanifold. In dimension zero a compact link is a finite set. Connected links are an additional normality condition, not part of this definition.

The link of an isolated stratum of a closed two-dimensional pseudomanifold is a finite disjoint union of circles. Separate these branches, replacing the cone on their union by separate disks. This produces a closed surface and a finite surjective map to the original space, a homeomorphism on the regular stratum. Orientations on the regular stratum extend over the disks.

A rational Witt space is a stratified pseudomanifold such that the link $L$ of every stratum of odd codimension $2r+1$ satisfies $I{H}_r^{\bar{m}}(L;\mathbb{Q})=0$. This removes the only difference between lower and upper middle extensions and gives rational middle-perversity Poincare duality on closed oriented Witt spaces.

A compact stratified space with boundary is a Witt space with collared boundary when its interior and boundary are Witt spaces and a stratified neighbourhood of the boundary is $\partial X\times [0,\epsilon )$, with product stratification. Orientations are carried by the regular strata and induce the usual boundary orientation.

The compact space $L$ in a stratified neighbourhood ${\mathbb{R}}^s\times cL$ is the link of the stratum. Its dimension is one less than the codimension of that stratum. A disconnected link records distinct local branches.