
Differential topology is essentially the study of differentiable manifolds.




Manifold

Chart
Atlas
Differentiable manifold

- diffeomorphism
\begin{definition}
A \emph{diffeomorphism} between two differential manifolds $N$ and $M$ is a bijective function $f : N \to M$ such that both $f$ and $f^{-1}$ are continuously differentiable.
\end{definition}
- autodiffeomorphism
- inverse function theorem (manifolds)

- compatible pair of charts
- compatible chart to atlas
- stereographic projection
- manifold
- n-sphere
- real projective space
- complex projective space
- Grassmanian


- tangent space




Diffeomorphism



De Rham cohomology

##Lie theory
Lie group
Lie algebra
