Transverse complementary-dimensional intersection sets

Let $M$ be a smooth $n$-manifold without boundary. Let $f:X\to M$ and $g:Z\to M$ be smooth maps from smooth manifolds of dimensions $x=\dim X$ and $z=\dim Z$, with $x+z=n$, and suppose $f$ and $g$ are transverse in the sense of Transverse smooth maps. The transverse intersection set, or fibre product, of $f$ and $g$ is

$X{\times }_{M}Z:=\{(a,b)\in X\times Z:f(a)=g(b)\},$

with the subspace topology. In the submanifold case, if $A^a$ and $B^b$ are transverse embedded submanifolds of $M$ in the sense of Transverse embedded submanifolds with $a+b=n$, the transverse intersection set is $A\cap B$.

Both constructions are $0$-dimensional embedded submanifolds. For the fibre product this is Transverse fibre products are embedded submanifolds applied through the diagonal, whose codimension in $M\times M$ is $n$, so that

$\dim (X{\times }_{M}Z)=x+z-n=0.$

For the submanifold case it is Transverse embedded submanifolds intersect in the expected codimension, which gives codimension additivity, hence $\dim (A\cap B)=a+b-n=0$. The empty set is an allowed value and is a $0$-dimensional embedded submanifold (Embedded submanifolds and slice charts).

At a point $(a,b)\in X{\times }_{M}Z$ the tangent space is

$T_{(a,b)}(X{\times }_{M}Z)=\{(v,w)\in T_{a}X\oplus T_{b}Z:d{f}_a(v)=d{g}_b(w)\},$

the kernel of $(d{f}_a,-d{g}_b):T_{a}X\oplus T_{b}Z\to T_{g(b)}M$; transversality makes this map surjective, so the kernel has dimension $x+z-n=0$ by Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$. At $p\in A\cap B$ the tangent space is $T_p(A\cap B)=T_{p}A\cap T_{p}B$ by Transverse embedded submanifolds intersect in the expected codimension.

The intersection set is not an intersection number: a number requires finiteness of the transverse intersection and a parity or orientation convention.