transversality

Transversality is a fundamental concept in differential topology. We say that two smooth submanifolds $A,B$ of a smooth manifold $M$ intersect transversely, if at any point $x\in A\cap B$, we have

$$T_{x}A+T_{x}B=T_{x}X,$$

where $T_x$ denotes the tangent space at $x$, and we naturally identify $T_{x}A$ and $T_{x}B$ with subspaces of $T_{x}X$.

In this case, $A$ and $B$ intersect properly in the sense that $A\cap B$ is a submanifold of $M$, and

$$\mathrm{codim}(A\cap B)=\mathrm{codim}(A)+\mathrm{codim}(B).$$

A useful generalization is obtained if we replace the inclusion $A\hookrightarrow M$ with a smooth map $f:A\to M$. In this case we say that $f$ is transverse to $B\subset M$, if for each point $a\in f^{-1}(B)$, we have

$$d{f}_a(T_{a}A)+T_{f(a)}B=T_{f(a)}M.$$

In this case it turns out, that $f^{-1}(B)$ is a submanifold of $A$, and

$$\mathrm{codim}(f^{-1}(B))=\mathrm{codim}(B).$$

Note that if $B$ is a single point $b$, then the condition of $f$ being transverse to $B$ is precisely that $b$ is a regular value for $f$. The result is that $f^{-1}(b)$ is a submanifold of $A$. A further generalization can be obtained by replacing the inclusion of $B$ by a smooth function as well. We leave the details to the reader.

The importance of transversality is that it’s a stable and generic condition. This means, in broad terms that if $f:A\to M$ is transverse to $B\subset M$, then small perturbations of $f$ are also transverse to $B$. Also, given any smooth map $A\to M$, it can be perturbed slightly to obtain a smooth map which is transverse to a given submanifold $B\subset M$.