Transverse smooth maps - Imported Knowlpedia Content

Definition

Let $f:M\to P$ and $g:N\to P$ be smooth maps between finite-dimensional smooth manifolds without boundary. The maps $f$ and $g$ are transverse, written $f\pitchfork g$ , if for every pair $(x,y)\in M\times N$ with $f(x)=g(y)=z$ ,

df_x(T_xM)+dg_y(T_yN)=T_zP,

where $df_x$ and $dg_y$ are their differentials. The two differential images may overlap; only their sum must fill the target tangent space. If the images of $f$ and $g$ are disjoint, the condition holds vacuously.

Equivalent diagonal formulation

Define $(f,g):M\times N\to P\times P$ on the product manifold. Then $f\pitchfork g$ exactly when $(f,g)$ is transverse to the diagonal submanifold $\Delta_P\subset P\times P$ . The difference-map formulation in tangent spaces is the differential of this diagonal condition.

Fiber-product consequence

When $f\pitchfork g$ , the fiber product

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

is an embedded submanifold of $M\times N$ of dimension $\dim M+\dim N-\dim P$ . Its tangent space consists of pairs $(v,w)$ satisfying $df_x(v)=dg_y(w)$ Hirsch, Chapter 3.

Examples and non-examples

If $f$ is a submersion, then $f\pitchfork g$ for every smooth $g$ , because $df_x$ is surjective. Two constant maps with the same value in a positive-dimensional target are not transverse: both differential images are zero.

Conventions and scope

For manifolds with boundary or corners, extra hypotheses are needed to control the boundaries and corners of the fiber product. The boundaryless setting in the core avoids silently asserting those stronger conclusions.

References

M. W. Hirsch, Differential Topology, Springer, 1976. Springer DOI record. Relevant: Chapter 3, transversality and inverse images.
V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. AMS DOI record. Relevant: Chapter 2, transversality.