---
tag: A2PDZ8GA9C
kind: concept
name: Transversality
msc: [57R35, 57N75]
sources:
  - url: https://en.wikipedia.org/wiki/Transversality
same_as:
  wikidata: Q7835414
  wikipedia: Transversality
---

## Summary

Transversality is the condition that two [[submanifolds|VKFW27WXEZ]], or a smooth map and a submanifold, meet in general position.
At each meeting point their tangent spaces together span the tangent space of the ambient manifold.
Transverse intersections are again submanifolds, of the expected dimension, and they persist under small perturbations.
Transversality is the basic notion of general position in differential topology and underlies intersection numbers, degree theory and cobordism.

## Explanations

### L1 human

Two lines in the plane that cross at an angle meet transversally, while two lines that touch along a common direction do not.
A small wiggle cannot destroy a transverse crossing, but it can easily separate two curves that only touch.
In space, a plane and a line meet transversally when the line pierces the plane rather than lying in it.
Two curves in $\R^3$ meet transversally only if they do not meet at all, because two lines cannot span three dimensions.