Smooth fiber product

Let $f:M\to P$ and $g:N\to P$ be smooth maps between finite-dimensional smooth manifolds without boundary. If $f$ and $g$ are transverse, their smooth fiber product is

M\times_PN=\{(x,y)\in M\times N:f(x)=g(y)\},

equipped with the unique smooth structure for which it is the corresponding embedded submanifold of the product manifold $M\times N$ . The coordinate projections restrict to smooth maps $\pi_M:M\times_PN\to M$ and $\pi_N:M\times_PN\to N$ , and satisfy $f\pi_M=g\pi_N$ .

Tangent space and dimension

At $(x,y)\in M\times_PN$ , with common image $z$ , its tangent space is the linear pullback

T_{(x,y)}(M\times_PN) =\{(v,w)\in T_xM\oplus T_yN:df_x(v)=dg_y(w)\}.

Transversality makes the difference map $(v,w)\mapsto df_x(v)-dg_y(w)$ surjective. The regular-level-set theorem therefore gives

\dim(M\times_PN)=\dim M+\dim N-\dim P.

Universal property

The square formed by the two projections is a categorical pullback in the category of smooth manifolds: if smooth maps $a:Q\to M$ and $b:Q\to N$ obey $fa=gb$ , there is a unique smooth map

q\longmapsto(a(q),b(q))

from $Q$ to $M\times_PN$ through which both maps factor. Thus the set-theoretic equality condition and the smooth universal property agree once the transverse submanifold structure exists.

Examples and scope

If $P$ is a point, the fiber product is the ordinary product $M\times N$ . If $g:N\hookrightarrow P$ is an embedded submanifold and $f$ is transverse to $N$ , then $M\times_PN$ identifies with the inverse-image submanifold $f^{-1}(N)$ .

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 6 on submanifolds, transversality, and fiber products.
Dominic Joyce, “On manifolds with corners,” in Advances in Geometric Analysis, Advanced Lectures in Mathematics 21, 2012, 225–258. Preprint record. Relevant: §6 on transverse fiber products in a category of manifolds with corners.