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}{}& ʃ &\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)

References

Created on February 3, 2021 at 10:43:11. See the history of this page for a list of all contributions to it.