Transversality

In differential topology , transversality describes a term that describes the mutual position of two submanifolds . In a certain sense, transversality describes the opposite of tangency and represents the "normal case" (see stability and transversality theorem ).

definition

Let be and differentiable manifolds , with denotes the tangent space at the point and with the total differential . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T_{x}Y}$ ${\displaystyle x\in Y}$ ${\displaystyle d}$

• Furthermore, let be a differentiable map and a submanifold . The mapping is called transversal to if: ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle Z\subseteq Y}$ ${\displaystyle f}$ ${\displaystyle Z}$

${\displaystyle T_{f(x)}Y=T_{f(x)}Z+d_{x}f(T_{x}X)\quad \forall \,x\in f^{-1}(Z)}$

• Be submanifolds. The submanifold is called transversal to if: ${\displaystyle W,Z\subseteq Y}$ ${\displaystyle W}$ ${\displaystyle Z}$

${\displaystyle T_{x}Y=T_{x}Z+T_{x}W\quad \forall \,x\in W\cap Z}$.

This is equivalent to the natural inclusion map being transverse to . ${\displaystyle i\colon W\hookrightarrow Y}$ ${\displaystyle Z}$

Remarks

• The sum of the vector spaces is generally not a direct sum.
• The transversality of submanifolds is a symmetrical relation :, therefore one also says “ and intersect transversely”. ${\displaystyle X{\mbox{ transversal zu }}Z\Longleftrightarrow Z{\mbox{ transversal zu }}X}$ ${\displaystyle X}$ ${\displaystyle Z}$
• Two disjoint submanifolds always intersect transversely.

Examples

• ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ^{2},\;t\mapsto (t,t^{2}+\varepsilon )}$is transverse to if and only if : ${\displaystyle \mathbb {R} \times \left\{0\right\}\subset \mathbb {R} ^{2}}$ ${\displaystyle \varepsilon \neq 0}$
  • ${\displaystyle \varepsilon =0}$: The tangent spaces coincide at the single point of intersection , their sum does not give the entire tangent space of . ${\displaystyle (0,0)}$ ${\displaystyle \mathbb {R} ^{2}}$
  • ${\displaystyle \varepsilon >0}$: No intersection, i.e. transversal.
  • ${\displaystyle \varepsilon <0}$: At the (two) points of intersection, the sum of the tangent spaces of the submanifolds gives the whole tangent space.
• Two straight lines in intersect transversely if and only if they are not identical. ${\displaystyle \mathbb {R} ^{2}}$
• Two straight lines in intersect each other transversely if and only if they do not intersect. ${\displaystyle \mathbb {R} ^{3}}$
• ${\displaystyle \mathbb {R} ^{k}\times \left\{0\right\}}$and in exactly intersect then transversely when . ${\displaystyle \left\{0\right\}\times \mathbb {R} ^{l}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle k+l\geq n}$

motivation

The original motivation for the definition of transversality lies in the question of when the archetype of a submanifold under a differentiable map is again a submanifold (of ). This is the case when transversal is too . ${\displaystyle Z\subseteq Y}$ ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle Z}$

To show this, one writes locally as a level set of a differentiable map , that is . The condition to be fulfilled reads: is the regular value of , that is, the tangential mapping is surjective for all . Elementary transformations show that this condition is equivalent to what corresponds to the definition of transversality. ${\displaystyle Z}$ ${\displaystyle g\colon V\rightarrow \mathbb {R} ^{l},\;V\subseteq Y{\mbox{ offen}}}$ ${\displaystyle Z\cap V=g^{-1}(0)}$ ${\displaystyle 0}$ ${\displaystyle g\circ f\colon X\rightarrow \mathbb {R} ^{l}}$ ${\displaystyle d_{x}(g\circ f)\colon T_{x}X\rightarrow \mathbb {R} ^{l}}$ ${\displaystyle x\in (g\circ f)^{-1}(0)}$ ${\displaystyle T_{f(x)}Y=T_{f(x)}Z+d_{x}f(T_{x}X)\quad \forall \,x\in f^{-1}(Z)}$

The reverse of the above statement is not true. This can be seen as follows: Let , and be embeddings of the curves in the form of closed curves that meet tangentially at one point (figure-eight). Think of it as an embedded submanifold. Then , as embedding. ${\displaystyle Y=S^{2}}$ ${\displaystyle X=S^{1}}$ ${\displaystyle f_{i}\colon X\to Y}$ ${\displaystyle S^{1}}$ ${\displaystyle \operatorname {Im} f_{1}=:Z}$ ${\displaystyle f_{2}^{-1}(Z)=f_{2}^{-1}(\{*\})=\{*\}}$ ${\displaystyle f_{2}}$

stability

A property of a differentiable map is called stable if the following applies to every differentiable homotopy : If this property exists, then one exists in such a way that it also has this property for all . ${\displaystyle X\rightarrow Y}$ ${\displaystyle F\colon X\times [0,1]\rightarrow Y}$ ${\displaystyle x\mapsto F(x,0)}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle x\mapsto F(x,t)}$ ${\displaystyle t\in [0,\varepsilon )}$

The stability theorem states that for differentiable mappings the transversality to a closed submanifold is a stable property if is compact . ${\displaystyle X\rightarrow Y}$ ${\displaystyle Z\subseteq Y}$ ${\displaystyle X}$

More sentences

Other important sentences in this context are the transversality theorem and the homotopy transversality theorem . They essentially mean that for every differentiable map there is a homotopic map which is transversal to a given submanifold and that transverse homotopias consist of families of maps which are transversal for almost all parameter values. These sentences enable the general definition of numbers of cuts with the help of homotopy, since these can only be defined directly for transversal cuts.

Genericity

A property of functions is called generic if the set of functions with this property is open and dense in the space of all functions.

Transversality (for a given submanifold ) is a generic property of differentiable mappings : the stability results in openness and the transversality principle gives rise to the density of the transversal mappings in the space of all differentiable mappings. ${\displaystyle Z\subset Y}$ ${\displaystyle X\rightarrow Y}$

philosophy

In philosophy, the concept of transversality is taken up by Wolfgang Welsch .

Individual evidence

1. ^ W. Welsch: reason. The contemporary critique of reason and the concept of transversal reason. Frankfurt a. M. 1995.

literature

• Victor Guillemin , Alan Pollack: Differential topology. Prentice-Hall, Englewood Cliffs NJ 1974, ISBN 0-13-212605-2 .
• Theodor Bröcker, Tammo tom Dieck: Kobordismentheorie, Lecture Notes in Mathematics 178, Springer Verlag (1970).