# nLab regular value

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

## Definition

###### Definition

(regular value)

For

$f \;\colon\; X \longrightarrow Y$

a differentiable function between differentiable manifolds (e.g. a smooth function between smooth manifolds) a point $q \in f(X) \subset Y$ in the image of $f$ is called a regular value of $f$ if at all points $p \in f^{-1}(\{q\})$ in its preimage, the differential

$d f_p \;\colon\; T_p X \longrightarrow T_{f(x)} Y = T_{q}Y$

is a surjective function between the corresponding tangent spaces.

A function all whose values are regular values is called a submersion.

(e.g. Kosinski 93, II (2.4))

###### Remark

(relation to transversality)

That $q \in Y$ is a regular value (Def. ) of $f \colon X \to Y$ means equivalently that $f$ is a transverse map to the submanifold-inclusion $\ast \overset{q}{\hookrightarrow} Y$.

In this sense transversality generalizes the concept of regular values.

## Properties

### Inverse image

The inverse function theorem implies that:

###### Proposition

The inverse image $f^{-1}(q) \subset X$ of a smooth function $f \colon X \to Y$ at a regular value $q \in Y$ is a smooth manifold of $X$.

Together with the Thom's transversality theorem, this is the key to the proof of the Pontryagin-Thom isomorphism.