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