# nLab smooth homotopy

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

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.

## Properties

### Continuous homotopy implies smooth homotopy

###### Proposition

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.

More in detail:

###### Proposition

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 $A$

(Lee 2012, Thm. 10.22 (1st ed.) / Thm. 6.29 (2nd ed.))

###### Example

(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.

###### Proposition

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

## References

Last revised on August 14, 2022 at 13:23:19. See the history of this page for a list of all contributions to it.