nLab regular value

Idea

A regular value of a map $f$ between smooth manifolds is an element of the codomain of $f$ that is not a critical value, i.e., not the image of a critical point of $f$.

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

Related concepts

• immersion, submersion

• formally smooth morphism

• transverse map

• Thom's transversality theorem

• Pontryagin-Thom isomorphism

References

• Lev Pontrjagin, p. 5 of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)

• René Thom, p. 18 of: Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, digiz:GDZPPN002056259, pdf)

• Antoni Kosinski, Differential manifolds, Academic Press 1993 (pdf)