nLab transversal maps

Idea

Two differentiable functions $f \colon X \to Z$ and $g \colon Y \to Z$ between differentiable manifolds are called transversal if, roughly, the images of $X$ and $Y$ in $Z$ do not “touch tangentially”.

If $f$ or $g$ are inclusions of (possibly immersed) submanifolds one speaks of these submanifolds being transversal to each other.

Definition

Two differentiable functions $f \colon X \to Z$ and $g \colon Y \to Z$ between differentiable manifold (for instance smooth functions between smooth manifolds) are transversal if for all pairs of points $x \in X$ and $y \in Y$ with $f(x) = z = g(y)$ the differentials of $f$ and $g$ in these points span the entire tangent space at $z$ in the sense that

$$im(d f) + im(d g) \simeq T_z Z \,.$$

(Notice that this is not required to be a direct sum.)

Examples

• A submersion is transversal to all differentiable functions into its codomain.

Properties

Pullbacks along transversal maps

Various constructions involving pullbacks of differentiable manifolds work as expected only for pullbacks involving transversal maps.

For example, two differentiable functions with a common codomain are transversal only if their pullback exists and is preserved by the tangent bundle functor; that is, $T(X \times_Z Y) = T X \times_{T Z} T Y$. (However, the pullback may exist and be preserved without transversality; for example if $X$ and $Y$ are both abstract points, $Z$ is not a point, and the maps $X, Y \to Z$ are equal as concrete points of $Z$.)

This is to be regarded as the dual of the possibly more familiar statement that various constructions involving quotients only work as expected for free actions.

Both of these “problems” are solved by passing from the ordinary 1-category Diff of manifolds to a suitable higher category of generalized smooth spaces.

More precisely:

• the problem with the pushouts (quotients) is resolved by passing to stacks and smooth infinity-stacks.

• the problem with the pullbacks is resolved by passing to derived stacks. Concretely for the case of manifolds this is discussed at derived smooth manifold.

Thom’s transversality theorem

See at Thom's transversality theorem.

Related concepts

• regular value

References

• Theodor Bröcker, Klaus Jänich, Introduction to differentiable topology (1982) [ISBN:9780521284707]

  (translated from the German 1973 edition)

• Morris Hirsch, Differential topology, GTM 33, Springer (1976) [doi:10.1007/978-1-4684-9449-5, gBooks]

• Antoni Kosinski, chapter IV (pp. 59) of: Differential manifolds, Academic Press (1993) [pdf, ISBN:978-0-12-421850-5]

• Ieke Moerdijk, Gonzalo E. Reyes, p. 27 in: Models for Smooth Infinitesimal Analysis, Springer (1991) [doi:10.1007/978-1-4757-4143-8]