nLab
smooth isotopy

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

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

Idea

A smooth isotopy is an isotopy that varies smoothly.

In knot theory, one typically does not want to identify knots S 1S 3S^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 Σ\Sigma and XX be smooth manifolds, and let

γ 0,γ 1:ΣX \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

η:[0,1]×ΣX \eta \;\colon\; [0,1] \times \Sigma \longrightarrow X

such that

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

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

γ(t,):ΣX \gamma(t,-) \;\colon\; \Sigma \longrightarrow X

is an embedding of smooth manifolds.

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

References

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

Created on July 20, 2018 at 05:46:12. See the history of this page for a list of all contributions to it.