Transversality of a map to a submanifold ($f⋔S$)

A smooth map $f:M\to N$ is transverse to an embedded submanifold $S\subset N$ whenThis is a sum of vector subspaces, not necessarily a direct sum. Equivalently the composition of $d{f}_p$ with the quotient to $T_{f(p)}N/T_{f(p)}S$ is surjective. In a slice chart for an embedded submanifold of codimension $k$, the $k$ normal coordinate functions composed with $f$ have surjective derivative on their common zero set. If the preimage is empty the condition holds vacuously.

Smooth maps of finitely many circles into a manifold of dimension at least three can be perturbed within their homotopy classes to disjoint embeddings. The transversality dimension count for two curve branches is $1+1-m<0$, so their images avoid one another after perturbation; one also removes self-intersections and obtains immersions by the jet version of transversality. This allows disjoint representative loops for relators in high-dimensional surgery constructions of prescribed fundamental groups.

If $f:M\to N$ satisfies transversality of a map to a submanifold $S$, its nonempty preimage is an embedded submanifold of the displayed codimension. Choose a slice chart for an embedded submanifold around $f(p)$ in which $S$ is the zero set of the first $k$ coordinates. Their composition with $f$ has surjective derivative on its zero set, so the regular level set theorem gives a local embedded slice of dimension $\dim M-k$. These slices give the subspace topology and a compatible smooth atlas on the preimage. Its tangent space is