nLab smooth homotopy

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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)

Homotopy theory

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

Contents

Definition

In differential topology, given a pair XX, YY of smooth manifolds, and a pair of parallel smooth functions Xf 1f 0YX \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 η:X×Y \eta \;\colon\; X \times \mathbb{R} \longrightarrow Y on the product manifold of XX with the real line, which restricts to f if_i at X×{i}X \times \{i\}, for i{0,1}i \in \{0,1\} \subset \mathbb{R}.

More generally, for XX, YY differentiable manifolds and f 0f_0 and f 1f_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 XX and YY are closed manifolds, then a smooth homotopy exists between any pair of parallel smooth functions (f 0,f 1)(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)(f_0, f_1) are n+2n+2-fold differentiable, then an ordinary continuous homotopy between them implies an nn-fold differentiable homotopy.

(Pontrjagin 55, Thm. 8, p. 41))

More in detail:

Proposition

Let XX and YY by smooth manifolds where XX may be a manifold with boundary. If continuous functions f,g:XYf,g \;\colon\; X \xrightarrow{\;} Y are continuously homotopic relative to a closed subset AXA \subset X, then they are also smoothly homotopic relative to AA

(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]X \,=\, [0,1] is the closed interval and YY any smooth manifold, then Prop. say that continuous homotopy classes of smooth paths in YY, 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 XX and YY by smooth manifolds, both possibly with boundary. If continuous functions f,g:XYf,g \;\colon\; X \xrightarrow{\;} Y are continuously homotopic, then they are also smoothly homotopic

(Lee 2012, Thm. 9.28 (2nd ed.))

References

Last revised on August 14, 2022 at 09:23:19. See the history of this page for a list of all contributions to it.