# transversality

Transversality is a fundamental concept in differential topology. We say that two smooth submanifolds $A,B$ of a smooth manifold^{} $M$ intersect *transversely*, if at any point $x\in A\cap B$, we have

$${T}_{x}A+{T}_{x}B={T}_{x}X,$$ |

where ${T}_{x}$ denotes the tangent space^{} at $x$, and we naturally identify ${T}_{x}A$ and ${T}_{x}B$ with subspaces^{} of ${T}_{x}X$.

In this case, $A$ and $B$ intersect properly in the sense that $A\cap B$ is a submanifold of $M$, and

$$\mathrm{codim}(A\cap B)=\mathrm{codim}(A)+\mathrm{codim}(B).$$ |

A useful generalization is obtained if we replace the inclusion $A\hookrightarrow M$ with a smooth map $f:A\to M$. In this case we say that $f$ is transverse to $B\subset M$, if for each point $a\in {f}^{-1}(B)$, we have

$$d{f}_{a}({T}_{a}A)+{T}_{f(a)}B={T}_{f(a)}M.$$ |

In this case it turns out, that ${f}^{-1}(B)$ is a submanifold of $A$, and

$$\mathrm{codim}({f}^{-1}(B))=\mathrm{codim}(B).$$ |

Note that if $B$ is a single point $b$, then the condition of $f$ being transverse to $B$ is precisely that $b$ is a regular value for $f$. The result is that ${f}^{-1}(b)$ is a submanifold of $A$. A further generalization can be obtained by replacing the inclusion of $B$ by a smooth function as well. We leave the details to the reader.

The importance of transversality is that it’s a stable and generic^{} condition. This means, in broad terms that if $f:A\to M$ is transverse to $B\subset M$, then small perturbations of $f$ are also transverse to $B$. Also, given any smooth map $A\to M$, it can be perturbed slightly to obtain a smooth map which is transverse to a given submanifold $B\subset M$.

Title | transversality |
---|---|

Canonical name | Transversality |

Date of creation | 2013-03-22 13:29:46 |

Last modified on | 2013-03-22 13:29:46 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 57R99 |

Defines | transversal |

Defines | transverse |

Defines | transversally |

Defines | transversely |