# nLab smooth isotopy

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

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A smooth isotopy is an isotopy that varies smoothly.

In knot theory, one typically does not want to identify knots $S^1 \to S^3$ by plain isotopy, as that makes all tame knots? be equivalent. A common fix is to use ambient isotopy instead. But one may also use smooth isotopy. (see e.g. Greene 13 or MO discussion here).

## Definition

Let $\Sigma$ and $X$ be smooth manifolds, and let

$\gamma_0, \gamma_1 \;\colon\; \Sigma \hookrightarrow X$

be two embeddings of smooth manifolds. Then a smooth isotopy between them is a smooth homotopy between them via embeddings: a smooth function

$\eta \;\colon\; [0,1] \times \Sigma \longrightarrow X$

such that

$\eta(0,-) \;=\; \gamma_0 \phantom{AAA} \eta(1,-) \;=\; \gamma_1$

and such that for each $t \in [0,1]$,

$\gamma(t,-) \;\colon\; \Sigma \longrightarrow X$

(e.g. Greene 13, Def. 1.7)

• Josh Greene, Combinatorial methods in knot theory, 2013 (pdf)