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 $$ df_a(T_a A) + T_{f(a)}B = T_{f(a)}M. $$ 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 $B\subset M$