Transversality

Transversality in mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

 is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency

Tangent

In geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...

, and plays a role in general position

General position

In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...

. It formalizes the idea of a generic intersection in differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.


Two submanifold

Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...

s of a given finite dimensional smooth manifold are said to intersect transversally if at every point of intersection

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

, their separate tangent spaces at that point together generate the tangent space

Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.-Informal description:In...

 of the ambient manifold at that point.

Transversality in mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

 is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency

Tangent

In geometry, the tangent line to a curve at a given point is the straight line that "just touches" the curve at that point...

, and plays a role in general position

General position

In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...

. It formalizes the idea of a generic intersection in differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.

Definition

Two submanifold

Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...

s of a given finite dimensional smooth manifold are said to intersect transversally if at every point of intersection

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

, their separate tangent spaces at that point together generate the tangent space

Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.-Informal description:In...

 of the ambient manifold at that point. Manifolds that do not intersect are vacuously transverse. If the manifolds are of complementary dimension (i.e., their dimensions add up to the dimension of the ambient space

Ambient space

An ambient space, ambient configuration space, or electroambient space, is the space surrounding an object.-Mathematics:In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object...

), the condition means that the tangent space to the ambient manifold is the direct sum of the two smaller tangent spaces. If an intersection is transverse, then the intersection will be a submanifold whose codimension

Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

 is equal to the sums of the codimensions of the two manifolds. In the absence of the transversality

Transversality

Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology...

 condition the intersection may fail to be a submanifold, having some sort of singular point

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

.

In particular, this means that transverse submanifolds of complementary dimension intersect in isolated points (i.e., a 0-manifold). If both submanifolds and the ambient manifold are oriented, their intersection is oriented. When the intersection is zero-dimensional, the orientation is simply a plus or minus for each point.

One notation for the transverse intersection of two submanifolds L₁ and L₂ of a given manifold M is . This notation can be read in two ways: either as “L₁ and L₂ intersect transversally” or as an alternative notation for the set-theoretic intersection L₁ ∩ L₂ of L₁ and L₂ when that intersection is transverse. In this notation, the definition of transversality reads

Transversality of maps

The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. If the maps are embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

s, this is equivalent to transversality of submanifolds.
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Meaning of transversality for different dimensions

Suppose we have transverse maps and where are manifolds with dimensions respectively.

The meaning of transversality differs a lot depending on the relative dimensions of and . The relationship between transversality and tangency is clearest when .

We can consider three separate cases:

1. When , it is impossible for the image of and 's tangent spaces to span 's tangent space at any point. Thus and cannot intersect.
2. When , the image of and 's tangent spaces must sum directly to 's tangent space at any point of intersection. Their intersection thus consists of isolated signed points, i.e. a zero-dimensional manifold.

1. When this sum needn't be direct. In fact it cannot be direct if and are immersion
   Immersion (mathematics)
   In mathematics, an immersion is a differentiable map between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if...
   
   s at their point of intersection, as happens in the case of embedded submanifolds. If the maps are immersions, the intersection of their images will be a manifold of dimension .

Intersection product

Given any two smooth submanifolds, it is possible to perturb either of them by an arbitrarily small amount such that the resulting submanifold intersects transversally with the fixed submanifold. Such perturbations do not affect the homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

 class of the manifolds or of their intersections. For example, if manifolds of complementary dimension intersect transversally, the signed sum of the number of their intersection points does not change even if we isotope the manifolds to another transverse intersection. (The intersection points can be counted modulo 2, ignoring the signs, to obtain a coarser invariant.) This descends to a bilinear intersection product on homology classes of any dimension, which is Poincaré dual to the cup product

Cup product

In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q...

 on cohomology

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

. Like the cup product, the intersection product is graded-commutative.

Examples of transverse intersections

The simplest non-trivial example of transversality is of arcs in a surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R³ — for example, the surface of a ball...

. An intersection point between two arcs is transverse if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...

 it is not a tangency, i.e., their tangent lines inside the tangent plane to the surface are distinct.

In a three-dimensional space, transverse curves do not intersect. Curves transverse to surfaces intersect in points, and surfaces transverse to each other intersect in curves. Curves that are tangent to a surface at a point (for instance, curves lying on a surface) do not intersect the surface transversally.

Optimal control

In fields utilizing the calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. Such functionals can for example be formed as integrals involving an unknown function and its derivatives...

 or the related Pontryagin maximum principle, the transversality condition is frequently used to control the types of solutions found in optimization problems. For example, it is a necessary condition for solution curves to problems of the form:

Minimize where one or both of the endpoints of the curve are not fixed. In many of these problems, the solution satisfies the condition that the solution curve should cross transversally the nullcline

Nullcline

Nullclines, sometimes called zero-growth isoclines , are encountered in two-dimensional systems of differential equations...

 or some other curve describing terminal conditions.

Smoothness of solution spaces

Using Sard's Theorem--whose hypothesis is a special case of the transversality of maps--it can be shown that transverse intersections between submanifolds of a space of complementary dimensions or between submanifolds and maps to a space are themselves smooth submanifolds. For instance, if a smooth section

Section

Section may refer to:* Section , papers folded during bookbinding* Section * Section , also in homological algebra, and including:** Section , in topology** Part of a sheaf...

 of an oriented manifold's tangent bundle

Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x₁ and x₂ of manifold M the tangent spaces T₁ and T₂ have no common vector...

--i.e. a vector field

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...

--is viewed as a map from the base to the total space, and intersects the zero-section (viewed either as a map or as a submanifold) transversely, then the zero set of the section--i.e. the singularities of the vector field--forms a smooth 0-dimensional submanifold of the base, i.e. a set of signed points. The signs agree with the indices of the vector field, and thus the sum of the signs--i.e. the fundamental class of the zero set--is equal to the Euler characteristic of the manifold. More generally, for a vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

 over an oriented smooth closed finite-dimensional manifold, the zero set of a section transverse to the zero section will be a submanifold of the base of codimension equal to the rank of the vector bundle, and its homology class will be Poincare dual

Poincaré duality

In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...

 to the Euler class

Euler class

In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...

 of the bundle.

An extremely special case of this is the following: if a differentiable function from reals to the reals has nonzero derivative at a zero of the function, then the zero is simple, i.e. it the graph is transverse to the x-axis at that zero; a zero derivative would mean a horizontal tangent to the curve, which would agree with the tangent space to the x-axis.

For an infinite-dimensional example, the d-bar operator is a section of a certain Banach space

Banach space

In mathematics, Banach spaces are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions , spaces of Lebesgue integrable functions known as L^p spaces,...

 bundle over the space of maps from a Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

 into an almost-complex manifold. The zero set of this section consists of holomorphic maps. If the d-bar operator can be shown to be transverse to the zero-section, this moduli space

Moduli space

In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...

 will be a smooth manifold. These considerations play a fundamental role in the theory of pseudoholomorphic curves and Gromov-Witten theory.