Transversality, Sard's Theorem, and Topological Data Analysis

Introduction

The concept of transversality deals with the intersection of two objects; in several ways, it may be considered the reverse of tangency. For transversality to occur between two sub-manifolds, their tangent spaces at every intersection point must span the ambient manifold's tangent space. Transversality notably fails in cases of tangency between two sub-manifolds. A more significant point, however, is that tangency lacks stability: all situations involving tangency between two objects can be effortlessly disturbed into non-tangent situations, which is not true of transversality. Part of the reason transversality is such a powerful tool is precisely this stability.

René Thom, a French mathematician, introduced the idea of transversality in the 1950s. In his doctoral thesis of 1954, he included the proof and statement of his Transversality Theorem (Greenblatt, 2015), which establishes transversality's generic nature — that is, all non-transverse intersections may be deformed, through small arbitrary deformations, into transverse intersections. This property is stronger than stability alone.

Theory of Transversality

In mathematics, transversality represents a concept describing the intersection between spaces; it may be perceived as the reverse of tangency, and it contributes to the notion of general position. The theory formalizes the concept of a broad intersection within differential topology and is described by considering the linearizations of intersecting spaces at their intersection points. Surface arcs form the most basic non-trivial example of the phenomenon. Intersection points between arcs are transverse if and only if they are not tangencies — in other words, when the arcs have distinct tangent lines within the surface's tangent plane. Transverse curves do not intersect in 3-dimensional spaces (Thom, 1954). Curves that are transverse to a surface will intersect one another at points, while surfaces that are transverse to one another will intersect in the form of curves. Curves tangent to surfaces at any given point — for example, curves lying on a given surface — do not transversally intersect those surfaces.

When the value y is regular, f⁻¹(y) is a manifold. This statement is generalizable to complete co-domain subsets Z, so long as the transversality condition is satisfied.

Definition: All smooth functions f : X → Y are transverse to sub-manifolds Z ⊂ Y at x if:

Im(df_x) + T_f(x)(Z) = T_f(x)(Y)

In other words, every component of T_f(x)(Y) may be expressed as the sum of components in T_f(x)(Z) and Im(df_x).

Example: From X = ℝ, f(t) = (0, t), Y = ℝ², and Z = ⟨1, 0⟩, a transverse mapping is obtained. The transversality holds because Im(df) = ⟨0, 1⟩, and Z + Im(df) = span(e₁, e₂) = ℝ².

A special case arises when X, Z ⊂ Y are sub-manifolds with f : X → Y taken as the inclusion map. According to the transversality condition:

T_p(X) + T_p(Z) = T_p(Y)

In other words, if X and Z intersect at any point, their tangent spaces at that point must span the tangent space of Y. Two manifolds are said to transversally intersect Z ⊂ X when the above condition is fulfilled.

Theorem: If f : X → Y is transverse to Z ⊂ Y (a sub-manifold), then f⁻¹(Z) is also a sub-manifold.

Stability

A key facet of the analysis of maps' properties is the stability of those properties under slight deformations.

Definition: Two maps f₀, f₁ : X → Y are said to be smoothly homotopic if there exists some F : X × [0, 1] → Y such that F(x, 0) = f₀(x) and F(x, 1) = f₁(x).

Definition: A property P of maps is deemed stable if it remains unchanged under slight deformation. Specifically, if f₀ : X → Y satisfies stable property P, and F is some homotopy with F(x, 0) = f₀, then there exists ε > 0 such that F(x, δ) = f_δ satisfies P for every δ < ε.

Theorem: The following map properties on compact manifolds are stable:

(a) immersion; (b) local diffeomorphism; (c) embedding; (d) submersion; (e) diffeomorphism; (f) transversality to some sub-manifold Z.

Intersection Theory and Manifold Orientations

Vector Spaces

Defining manifold orientations requires first defining orientation on a vector space. Let {v_i} and {u_i} be ordered bases of a given vector space V. The change-of-basis matrix P mapping two bases induces an equivalence relation on the set of ordered bases of V: two bases β and β′ are equivalent when det(P) > 0.

Definition: A vector space orientation may be defined as the map B → {±1}, where B is the set of ordered bases and equivalent bases share an identical sign. A vector space isomorphism A is said to be orientation-preserving if β ~ β′ implies Aβ ~ Aβ′; otherwise, it is termed orientation-reversing.

Manifolds

Manifold orientations are grounded in vector space orientations.

Definition: Orientations of bounded smooth manifolds represent a smooth orientation choice on every T_x(X). Here, "smooth" implies that around all x ∈ X, a parameterization h : U → X must exist such that dh : ℝ^k → T_h(u)(X) is orientation-preserving (where ℝ^k carries the standard orientation).

Remark: Not every manifold admits an orientation — the Möbius strip is a classic example. If an orientation exists for manifold X, it will inevitably have another orientation, −X, representing the opposite basis choice at all points.

Two sub-manifolds X and Z within ambient space M are said to transversally intersect if, at every intersection point, their tangent spaces together span the tangent space of M. Two sub-manifolds are automatically transversal if they fail to intersect. For example, two curves in a 3-dimensional space are transversal if they do not intersect at all. The transversal meeting of X and Z means that their intersection is a smooth sub-manifold of the expected dimension.

In a certain sense, two sub-manifolds should generically transversally intersect; furthermore, according to Sard's theorem, all intersections may be perturbed to be transversal. Homological intersection is only meaningful because intersections can be rendered transversal (Sard, 1942).

Transversality suffices for the stability of intersections following perturbation. For instance, two lines that transversally intersect do so at a single point only. Lines that do not transversally intersect may coincide at a single point, while a slight perturbation may cause them to intersect at two points or not at all, depending on the sign of the perturbation parameter.

When the dimensions are complementary, transversal intersections become isolated points. Given vector space orientations on the relevant tangent spaces, the transversality condition allows one to assign signs to intersection points. When the tangent vectors form oriented bases for the respective sub-manifolds, the intersection is assigned +1 when the combined basis is positively oriented in the ambient space, and −1 otherwise.

More generally, two sub-manifolds of any finite-dimensional smooth manifold transversally intersect if, at all intersection points, their individual tangent spaces together generate the ambient manifold's tangent space (Thom, 1954). Non-intersecting manifolds are vacuously transverse. In the case of complementary-dimensional manifolds — that is, manifolds whose dimensions sum to the ambient space's dimension — the condition implies that the tangent spaces to the ambient manifold are directly the sum of the smaller tangent spaces. When intersections are transverse, the intersection is a sub-manifold with co-dimension equal to the sum of the two manifolds' co-dimensions. Without transversality, the intersection may fail to be a sub-manifold and may exhibit singular points.