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)
The diffeomorphism group $Diff(X)$ of a smooth manifold $X$ is the group of its diffeomorphisms: the automorphism group of $X$ as an object of the category SmoothMfd.
Beware that when $X$ is assumed orientable then sometimes, but not always, $Diff(X)$ is implicitly taken to be the group of orientation-preserving diffeomorphisms.
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)
The homotopy type $\Pi(Diff(\Sigma))$ of the diffeomorphism group $Diff(\Sigma)$ is of interest (e.g. Hatcher 12).
For instance this is the automorphism ∞-group of a manifold, regarded as a k-morphism in an (∞,n)-category of cobordisms.
Specifically, the group of connected components is the mapping class group
For $\Sigma$ a closed orientable surface, then the bare homotopy type of its diffeomorphism group is
if $\Sigma$ is the sphere then
if $\Sigma$ is the torus then
in all other cases all higher homotopy groups vanish:
The first statement is due to (Smale 58), see also at sphere eversion. The second and third are due to (Earle-Eells 67, Gramain 73).
The bare homotopy type of the diffeomorphism group of the 3-sphere is that of the orthogonal group $O(4)$
the equivalence being exhibited by the canonical inclusion
Also
After being conjectured by Smale, this was proven in (Hatcher 1983).
Generally:
For every smooth 3-manifold the canonical map
sending diffeomorphisms to their underlying homeomorphisms of topological spaces is a weak homotopy equivalence.
That this follows from the Smale cojecture, theorem 1, was shown in (Cerf). For discussion see (Hatcher, 1978).
If a 3-manifold $X$ is not a Seifert 3-manifold via an $S^1$-action then
If $X$ is Seifert via an $S^1$-action, then the component of $Diff(X)$ are typically $\Pi(S^1)$-s.
The observation that infinite-dimensional smooth groups such as diffeomorphism groups (and quantomorphism groups etc.) are naturally regarded as internal groups in diffeological spaces – diffeological groups – is due to
C.J. Earle, J. Eells, The diffeomorphism group of a compact Riemann surface, Bulletin of the American Mathematical Society 73(4) 557–559, 1967
J. Cerf, Sur les difféomorphismes de la sphère de dimension trois ($\Gamma_4 = 0$), Lecture Notes in Math., vol. 53, Springer-Verlag, Berlin and New York, 1968_
Allen Hatcher, Linearization in 3-dimensional topology, Proceedings of the international congress of Mathematicians, Helsinki (1978)
Allen Hatcher, On the diffeomorphism group of $S^1\times S^2$, Proceedings of the AMS 83 (1981), 427-43 (pdf)
Allen Hatcher, A proof of the Smale conjecture, $Diff(S^3) \simeq O(4)$, Annals of Mathematics 117 (1983) (jstor)
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)