Transversality and intersection theory / Sard’s theorem, topological analysis approach Introduction

The concept of transversality deals with the intersection of two objects; in several ways, one may consider it the reverse of tangency. For transversality to occur between two sub- manifolds, their tangential spaces at every intersection point need to extend across the ambient manifold’s tangent space. Transversality, invariably, particularly fails in case of tangency between two sub- manifolds. However, a more notable point is, tangency lacks stability: all situations involving tangency between two objects may be effortlessly and somewhat disturbed into non- tangent situations, which isn’t true when it comes to transversality. Part of the reason why transversality is so sound a tool is its stability.

Rene Thom, a French mathematician, introduced the idea of transversality during the 50s. In his doctoral thesis performed in the year 1954, he included the proof and statement of his Transversality Theorem (Greenblatt, 2015), which proves transversality’s generic nature (i.e., all non- transverse intersections may be deformed (through small arbitrary deformations) into transverse intersections). This property is sounder as compared to stability.

Theory of Transversality

In the domain of mathematics, transversality represents a concept describing intersection between spaces; it may be perceived to be the reverse of tangency, contributing to general position. The theory formalizes the concept of a broad intersection within differential topology and is described through taking into consideration intersecting spaces’ linearizations at intersection points. Surface arcs form a non- trivial and most basic example of the phenomenon. Intersection points between arcs are transverse iff they aren’t tangencies (in other words, in the event of distinct tangent lines within the surface’s tangent plane). Transverse curves fail to intersect in 3D spaces (Thom, 1954). Curves that are transverse to a surface will intersect one another in points, whilst surfaces that are transverse to one another will intersect in the form of curves. Curves tangent to surfaces at any given point (e.g., curves that lie on a given surface) don’t transversally intersect surfaces.

When the value y is regular, f?1 (y) is manifold. The above statement is ‘generalizable’ to complete co- domain subsets Z, so long as the transversality condition is met with.

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

Im (dfx) + Tf(x) (Z) = Tf(x) (Y )

In other words, all components of Tf(x) (Y) may be considered the sum of components in Tf(x) (Z) and Im (dfx).

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

A special case would be: X, Z ? Y being sub- manifolds with f: X ? Y taken as inclusion. According to the transversality condition:

Tp (X) + Tp (Z) = Tp (Y )

In other words, in case of intersection of X and Z at any point, their tangential spaces at the point need to span those of Y. Two manifolds are said to transversally intersect Z ? X in the event the above condition is fulfilled.

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

Stability

A key facet of the analysis of maps’ properties is the stability of the properties in instances of slight deformations.

Definition: Two maps f0, f1: X ? Y may be considered smoothly homotopic in the event some F : X × [0, 1] ? Y such that F (x, 0) = f0 (x) and F (x, 1) = f1 (x).

Definition: P (a property of maps) may be deemed to be stable in the event it remains unchanged when subject to slight deformation. In specific, if f0 : X ? Y is able to fulfill stable property (i.e., P) with F being some homotopy with F(x, 0) = f0, there exists ? > 0 such that F (x, ?) = f? fulfills P for every ? < ?.

Theorem: Stable map properties on manifolds are listed below:

(a) immersion

(b) local diffeomorphism

(c) embedding

(d) submersion

(e) diffeomorphism

(f) transverse to some sub- manifold Z

Intersection Theory

Orientation of Manifolds

Defining manifold orientations requires firstly defining it on a vector space.

Vector Space (V)

Consider {vi} and {ui} to be the ordered bases of any given vector space (i.e., V). Change in basis matrix (i.e., P) maps two bases, inducing equivalence relations on the sets of the vector space’s ordered bases ?, ?’ equivalent when det (P) > 0.

Definition:...

A vector space isomorphism A is said to be orientation preserving in case ? ? ? 0 ? A? ? A?’ (if not, 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 Tx (X). In this context, the term ‘smooth’ implies that around all x ? X, a parameterization, i.e., h : U ? X must exist such that dh : Rk ? Th(u) (X) becomes orientation preserving (Rk has normal orientation).

Remark: Every manifold doesn’t admit orientations (e.g., he M¨obius Strip). In case an orientation exists for manifold X, it will inevitably have another orientation, ?X – the converse basis choice at all points.

X and Y (two sub- manifolds) within the ambient space M may be said to transversally intersect each other if, for every,

Here, the addition is within ;  represents’s tangent map. Two sub- manifolds will automatically be transversal in the event they fail to intersect. For instance, a couple of curves in  will be transversal only in case they entirely fail to intersect. Transversal meeting of  and  means that  denotes a ‘smooth’ sub- manifold of (the anticipated dimension)

In a way, two sub- manifolds should transversally intersect; furthermore, according to Sard's theorem, all intersections may be perturbedly transversal. Homological intersection only seems sensible since intersections may be rendered transversal (Sard, 1942).

Transversality suffices for stability of intersections following perturbation. For instance, the lines and  as well as (perturbed lines) transversally intersect, at a single point only. But,  and don’t transversally intersect. They do so in a single point, whereas  intersects in a couple of points or no points at all (this depends on whether or not  is negative).

If , transversal intersections become isolated points. In case of vector space orientations of the spaces, the transversality condition allows assigning of signs to the intersections. When  form oriented bases for , with   being oriented bases for , the intersection becomes  when  is oriented in , or else it is .

In broader terms, the smooth maps and  will be transversal in the event that, whenever , .

It is said that two sub- manifolds of any finite- dimensioned smooth manifold transversally intersect if, at all intersection points, their individual tangent spaces at the point combined generate ambient manifold tangent spaces at the point (Thom, 1954). Non- intersecting manifolds are vacantly transverse. In case of complementary dimensional manifolds (that is, dimensions adding up to ambient space dimensions), the condition implies tangent spaces to ambient manifolds are directly the sum of smaller tangential spaces. In case of transverse intersections, the intersection is a sub- manifold with co- dimension equivalent to the sum of the two manifolds’ co- dimensions. Without transversality, the intersection might not be a sub- manifold, and have a kind of singular point.

Sard’s Theorem

The pre- images of regular values, y, of smooth mappings f: X ? Y will be smooth manifolds, raising the natural question of:

Q: What quantity of Y constitutes regular values?

A: Sard’s theorem solves this issue. For stating this, the notion of measure on manifolds must be defined.

Definition: Any subset A ? R? is said to have a measure of zero if it is ‘coverable’ by a finite number of arbitrarily small rectangles.

Definition: A subset A of manifold M (i.e., A ? M) is said to have measure zero iff:

1. For all parameterizations ? : U ? M, the pre- images ??1 (A) have a measure of zero as subset of R?.

2. A covering for M exists by charts (??, U?) such that ??1 ? (A) is characterized by measure of zero as subset of R? for every ?.

Theorem (Sard’s): If f: X ? Y represents a smooth manifold map, nearly all points in Y are regular. In other words, if C forms a set of its critical points, f (C) has a measure of zero.

Importance and application of Sard’s theorem

Sard's theorem has chiefly been applied for escaping from a function’s critical values, through slight perturbation. Additionally, it helps extend definitions only valid at typical values, to the critical values. Homotopy…