nLab
transversal maps

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

$$\mathrm{im}(df)+\mathrm{im}(dg)\simeq T_{z}Z.$$im(d f) + im(d g) \simeq T_z Z \,.

Note that this is not required to be a direct sum. Also, if $f$ (say) is a submersion, then it is transversal to all $g$.

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

Here is a slick category-theoretic way to phrase the definition: Two maps with a common target are transversal iff their pullback exists and is preserved by the tangent bundle functor; that is, $T(X{\times }_{Z}Y)=TX{\times }_{TZ}TY$.

Properties

• Thom's transversality theorem

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

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:

• 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.

References

• 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