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)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In differential topology, given a pair $X$, $Y$ of smooth manifolds, and a pair of parallel smooth functions $X \underoverset{f_1}{f_0}{\rightrightarrows} Y$ between them, a smooth homotopy between these is (typically defined to be) a left homotopy by a smooth function, hence a smooth function $\eta \;\colon\; X \times \mathbb{R} \longrightarrow Y$ on the product manifold of $X$ with the real line, which restricts to $f_i$ at $X \times \{i\}$, for $i \in \{0,1\} \subset \mathbb{R}$.
More generally, for $X$, $Y$ differentiable manifolds and $f_0$ and $f_1$ differentiable functions, one may ask for differentiable homotopies, all to any given order(s) of differentiability.
At least if $X$ and $Y$ are closed manifolds, then a smooth homotopy exists between any pair of parallel smooth functions $(f_0, f_1)$ between them as soon as there exists an ordinary (i.e. continuous) left homotopy between their underlying continuous functions.
More generally, if $(f_0, f_1)$ are $n+2$-fold differentiable, then an ordinary continuous homotopy between them implies an $n$-fold differentiable homotopy.
(Pontrjagin 55, Thm. 8, p. 41))
More in detail:
Let $X$ and $Y$ by smooth manifolds where $X$ may be a manifold with boundary. If continuous functions $f,g \;\colon\; X \xrightarrow{\;} Y$ are continuously homotopic relative to a closed subset $A \subset X$, then they are also smoothly homotopic relative to $A$
(homotopy of smooth paths relative to their endpoints)
In the case that $X \,=\, [0,1]$ is the closed interval and $Y$ any smooth manifold, then Prop. say that continuous homotopy classes of smooth paths in $Y$, relative to their endpoints are equivalent to smooth such homotopy classes. This is relevant for instance for the characterization of the fundamental groupoid and the universal cover of a smooth manifold.
Let $X$ and $Y$ by smooth manifolds, both possibly with boundary. If continuous functions $f,g \;\colon\; X \xrightarrow{\;} Y$ are continuously homotopic, then they are also smoothly homotopic
(Lee 2012, Thm. 9.28 (2nd ed.))
Lev Pontrjagin, Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)
John Lee, Introduction to Smooth Manifolds, Springer (2012) [doi:10.1007/978-1-4419-9982-5, book webpage, pdf]
Last revised on August 14, 2022 at 13:23:19. See the history of this page for a list of all contributions to it.