Diagonal transversality criterion - Imported Knowlpedia Content

Statement

Let $f:M\to P$ and $g:N\to P$ be smooth maps between finite-dimensional smooth manifolds without boundary. Using the product manifolds, form the map

f\times g:M\times N\longrightarrow P\times P,\qquad (x,y)\longmapsto(f(x),g(y)).

Then $f$ and $g$ are transverse smooth maps if and only if $f\times g$ is transverse to the diagonal submanifold

\Delta_P=\{(z,z):z\in P\}.

Moreover, $(f\times g)^{-1}(\Delta_P)$ is the set-theoretic fiber product $M\times_PN$ .

Tangent-space calculation

At $(z,z)\in\Delta_P$ ,

T_{(z,z)}\Delta_P=\{(u,u):u\in T_zP\}.

The quotient of $T_zP\oplus T_zP$ by this diagonal subspace is identified with $T_zP$ by $(u,v)\mapsto u-v$ . Consequently, transversality of $f\times g$ to $\Delta_P$ is equivalent to surjectivity of

(v,w)\longmapsto df_x(v)-dg_y(w),

which is exactly the condition $df_x(T_xM)+dg_y(T_yN)=T_zP$ . Compare Guillemin and Pollack, Chapter 2.

Fiber-product consequence

When the equivalent conditions hold, the transverse preimage theorem makes $M\times_PN$ an embedded submanifold of $M\times N$ . Its tangent space is

\{(v,w):df_x(v)=dg_y(w)\},

and its dimension is $\dim M+\dim N-\dim P$ . This is the smooth fiber product construction.

Scope

The criterion is a reformulation, not an additional hypothesis. It is especially useful because coincidence equations become a single inverse-image problem. Versions for manifolds with boundary or corners require a notion of transversality compatible with the relevant strata.

References

Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. DOI record. Relevant: Chapter 2, diagonal formulation of transversality.
Morris W. Hirsch, Differential Topology, Springer, 1976. DOI record. Relevant: Chapter 3, transverse maps and inverse images.