---
tag: R2TQ82PCNB
kind: concept
name: Smooth structure
msc: [57R55, 58A05]
sources:
  - url: https://en.wikipedia.org/wiki/Smooth_structure
same_as:
  wikidata: Q7546440
  wikipedia: Smooth structure
---

## Summary

A smooth structure on a [[topological manifold|QQQ9B1DK4P]] is a maximal atlas of charts whose transition maps are smooth.
It is the extra data that turns a topological manifold into a [[smooth manifold|F45VQ47GPV]], so that one can differentiate functions and maps on it.
A topological manifold may have no smooth structure, exactly one up to diffeomorphism, or many.
The study of these possibilities is a central theme of differential topology.

## Explanations

### L1 human

A topological manifold only tells you which points are near which.
A smooth structure also tells you which functions on it count as smooth, by fixing a family of coordinate systems that fit together without kinks.
For example, the coordinate $t$ on the line $\R$ gives the usual smooth structure.
The coordinate $t^3$ gives a different atlas, because $t \mapsto t^{1/3}$ is not smooth at $0$, but the map $t \mapsto t^{1/3}$ is a diffeomorphism between the two structures.
Some spaces are more surprising: the $7$-sphere carries smooth structures that are not diffeomorphic to the standard one.