Regularity via Links and Stein Factorization

TL;DR

This work extends this type of construction to the codomain of higher Morse functions, using the singular locus to induce a stratification of which sub-posets are equivalent to multi-parameter filtrations.

Abstract

Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.

Figures (7)

• Figure 2.1: The four critical values of $-h$ are shown on the upright $T^2$ along with their corresponding images in $\mathbb R$. The direction of $-h$ is indicated by the leftmost arrow. Two strata defined by stable manifolds, $X_{s_1}$ and $X_{s_2}$, are shown in orange (upper circle) and teal (inner circle), respectively.
• Figure 3.1: The leftmost images illustrates why $p \in X$, an interior point, is regular and the rightmost image illustrates why $m \in X$ is a critical simplex, as described in Example \ref{['ex:icos']}
• Figure 3.2: A generic projection $\pi : \partial \Delta^3 \to \mathbb R^2$ along with a regular point, $r$, and a critical point, $p$. The Jacobi set of this projection forms a subcomplex and is highlighted.
• Figure 3.3: This figure is a piece of a larger triangulated torus. Although the PL differential of the height function at the saddle point $p$ is surjective, the link of $p$ does not satisfy the condition of L-regularity.
• Figure 4.1: The above is an example of an intersections in $f(J_f)$ for some map $f\colon X \to \mathbb R^3$. By adding vertices and and edges on intersection, we refine $f(J_f)$ into eight vertices, seven edges, and two faces. This can then be stratified by containment, as in Proposition \ref{['prop:Jsimp']}. Note that, although more refinement may be necessary for maps into higher dimensional Euclidean space, no further refinement is necessary for $k \leq 3$ since intersections are one- or zero-dimensional, and are thus already guarenteed to be simplices.

Theorems & Definitions (60)

• Theorem
• Definition 2.1.1
• Definition 2.1.2
• Proposition 2.1.3
• Remark 2.1.4
• Definition 2.1.5
• Proposition 2.1.6
• Example 2.1.7
• Definition 2.1.8
• Example 2.1.9