Transversality theorem

The transversality theorem is a theorem of differential topology , which goes back to René Thom and forms the basis for numerous topological constructions such as the Pontryagin-Thom construction , the cobordism theory , the theory of surgery and the definition of cutting numbers and entanglement numbers .

sentence

Let be a differentiable mapping between differentiable manifolds and a submanifold of . Then for every strictly positive function (and every metric on ) there is an approximation of which is transversal to . ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle U}$ ${\displaystyle N}$ ${\displaystyle \delta \colon M\rightarrow \mathbb {R} }$ ${\displaystyle N}$ ${\displaystyle \delta }$ ${\displaystyle f}$ ${\displaystyle U}$

Explanations: A differentiable map is transversal to the submanifold if ${\displaystyle g\colon M\rightarrow N}$ ${\displaystyle U}$

${\displaystyle T_{g(x)}N=T_{g(x)}U+d_{x}g(T_{x}M)\quad \forall \,x\in g^{-1}(U)}$

applies. (Especially if .) A mapping is a δ-approximation of if ${\displaystyle g^{-1}(U)=\emptyset }$ ${\displaystyle g\colon M\rightarrow N}$ ${\displaystyle f\colon M\rightarrow N}$

${\displaystyle d(f(x),g(x))<\delta (x)\quad \forall \,x\in M,}$

applies. For sufficiently small , every δ-approximation is homotopic to . In particular, the transversality principle implies the existence of a mapping that is too homotopic and that is transversal to . For each there is a , so that for each δ-approximation of there is a homotopy between and , for which the mapping for each is an ε-approximation of . ${\displaystyle \delta >0}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle \epsilon \colon M\rightarrow \mathbb {R} }$ ${\displaystyle \delta \colon M\rightarrow \mathbb {R} }$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle H\colon M\times \left[0,1\right]\rightarrow N}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle t\in \left[0,1\right]}$ ${\displaystyle H(.,t)}$ ${\displaystyle f}$

Examples

• ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} ^{2},\;t\mapsto (t,t^{2})}$is not transverse to the x-axis, but for each the mapping is transverse to the x-axis. ${\displaystyle \epsilon >0}$ ${\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} ^{2},\;t\mapsto (t,t^{2}+\epsilon )}$
• If so , it follows from the transversality theorem that there is a δ-approximation for every mapping , the image of which is disjoint . ${\displaystyle \dim(M)+\dim(U)<\dim(N)}$ ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle U}$

Relative version and homotopy transversality theorem

Let be a differentiable mapping between differentiable manifolds and a submanifold of . Let be a submanifold of and let the constraint be transverse to . Then for every strictly positive function (and every metric on ) there is an approximation of which is transversal to and which corresponds to. ${\displaystyle f\colon M\rightarrow N}$ ${\displaystyle U}$ ${\displaystyle N}$ ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle f\mid _{A}}$ ${\displaystyle U}$ ${\displaystyle \delta \colon M\rightarrow \mathbb {R} }$ ${\displaystyle N}$ ${\displaystyle \delta }$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle A}$ ${\displaystyle f}$

The homotopy transversality theorem is obtained as a special case :

Let be differentiable manifolds and a submanifold of . Let be a differentiable map for which and are transversal to . Then there is a mapping that is transversely to and corresponds to or with or . ${\displaystyle M,N}$ ${\displaystyle U}$ ${\displaystyle N}$ ${\displaystyle F\colon M\times \left[0,1\right]\rightarrow N}$ ${\displaystyle f_{0}:=F(.,0)\colon M\rightarrow N}$ ${\displaystyle f_{1}:=F(.,1)\colon M\rightarrow N}$ ${\displaystyle U}$ ${\displaystyle G\colon M\times \left[0,1\right]\rightarrow N}$ ${\displaystyle U}$ ${\displaystyle M\times \left\{0\right\}}$ ${\displaystyle M\times \left\{1\right\}}$ ${\displaystyle f_{0}}$ ${\displaystyle f_{1}}$

In words: if two transverse mappings are homotopic, then there is also a transverse homotopy.

Individual evidence

1. ↑ René Thom: Un lemme sur les applications différentiables . In: Boletin de la Sociedad Matemática Mexicana / 2. Series , Vol. 1 (1956), pp. 59-71, ISSN  0037-8615 .
2. ^ Theodor Bröcker , Tammo tom Dieck : Kobordismentheorie (Lecture Notes in Mathematics; Vol. 178). Springer Verlag, Berlin 1970, ISBN 3-540-05341-7 .