Contents

# Contents

## Idea

Two maps $f : X \to Z$ and $g : Y \to Z$ of manifolds are transversal roughly if the images of $X$ and $Y$ in $Z$ do not “touch tangentially”.

## Definition

Two maps $f : X \to Z$ and $g : Y \to Z$ of smooth manifolds are transversal if for all point $x \in X$ 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.)

In particular, a submersion is transversal to all functions.

In particular, $f$ or $g$ may be inclusions of (possibly immersed) submanifolds in which case we talk about the transversality of submanifolds.

## Properties

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

For example, two maps with a common target 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 of manifolds to a suitable higher category of generalized smooth spaces.

More precisely:

• Antoni Kosinski, chapter IV.1 of Differential manifolds, Academic Press 1993 (pdf)

• T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; ∃ also 1990 German 2nd edition)

• Morris W. Hirsch, Differential topology, Springer GTM 33, gBooks