synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given two $k$-times differentiable manifolds (or smooth manifolds), then a diffeomorphism
is a differentiable function such that there exists an inverse differentiabe function $f^{-1}$ (a function which is an inverse function on the underlying sets and is itself differentiable to the given degree).
Diffeomorphisms are the isomorphisms in the corrresponding category Diff of differentiable manifolds/smooth manifolds.
It is clear that
Every diffeomorphism is in particular a homeomorphism between the underlying topological spaces.
The converse in general fails. There exist differentiable maps with only continuous inverse. There are also differentiable bijections whose inverse is not even continuous.
The function $f : \mathbb{R} \to \mathbb{R}$ given by $x \mapsto x^3$ is a homeomorphism but not a diffeomorphism. The diffeomorphism property fails at the origin, where the differential $d f : T_0 \mathbb{R} \to T_0 \mathbb{R}$ is not onto.
But there is a rich collection of theorems about cases when the converse is true after all.
For $n \in \mathbb{N}$, the open n-ball $\mathbb{B}^n$ is the open subset
of the Cartesian space $\mathbb{R}^n$ of all points of distance lower than 1 from the origin. This inherits the structure of a smooth manifold from the embedding into $\mathbb{R}^n$.
In dimension $d \in \mathbb{N}$ for $d \neq 4$ we have:
every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.
See the first page of (Ozols) for a list of references.
What’s a good/canonical textbook reference for this?
In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$.
For $X$ and $Y$ smooth manifolds of dimension $d = 1$, $d = 2$ or $d = 3$ we have:
if there is a homeomorphism from $X$ to $Y$, then there is also a diffeomorphism.
See the corollary on p. 2 of (Munkres).
For the following kinds of manifolds $\Sigma$ it is true that every homotopy equivalence
(hence every equivalence of their fundamental infinity-groupoids) is homotopic to a diffeomorphism
i.e. that given $\alpha$ there is $a$ with
for $\Sigma$ any surface (Zieschang-Vogt-Coldeway)
for $\Sigma$ a Haken 3-manifold (Waldhausen)
for $\Sigma$ any hyperbolic manifold of finite volume and of dimension $\geq 3$ (by Mostow rigidity theorem) (check)
A review of results and relevant literature is also on the first page of (Hass-Scott 92)-
V. Ozols, Largest normal neighbourhoods Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor) (AMS: pdf)
James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms Bull. Amer. Math. Soc. Volume 65, Number 5 (1959), 332-334. (Euclid)(AMS: pdf)
Zieschang, Vogt and Coldeway, Surfaces and planar discontinuous groups
Friedhelm Waldhausen, On Irreducible 3-Manifolds Which are Sufficiently Large, Annals of Mathematics Second Series, Vol. 87, No. 1 (Jan., 1968), pp. 56-88 (JSTOR)
Joel Hass, Peter Scott, Homotopy equivalence and homeomoprhism of 3-manifolds, Topology, Vol. 31, No. 3 (1992) pp. 493-517 (pdf)