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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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)
manifolds and cobordisms
cobordism theory, Introduction
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).
(Smale conjecture)
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 , 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
Zieschang, Vogt and Coldeway, Surfaces and planar discontinuous groups
J. S. Dowker, Note on the structure constants for the diffeomorphisms of the two-sphere [arXiv:2301.09487]
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)
For 4-manifolds the analogue of the Smale conjecture fails:
Tadayuki Watanabe, Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (arXiv:1812.02448)
Tadayuki Watanabe, Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (homological interpretation) (arXiv:2109.01609)
(via graph complexes)
Last revised on January 24, 2023 at 07:22:40. See the history of this page for a list of all contributions to it.