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Transverse fibre products are embedded submanifolds

Statement

Let $F : M \to N$ and $G : P \to N$ be smooth and transverse. Then the fibre product

M \times_{N} P : = {(p, q) \in M \times P : F (p) = G (q)}

is an embedded submanifold of $M \times P$ .

Facts & Assumptions

Given: Smooth maps $F : M \to N$ and $G : P \to N$ with $F ⋔ G$ .

[F1]

Two smooth maps are transverse when their differential images span the target tangent space at every coincidence point (Transverse smooth maps).

[L1]

The diagonal $Δ_{N} \subseteq N \times N$ is an embedded submanifold (The diagonal is an embedded submanifold).

[L2]

Products of smooth maps are smooth, and the transverse preimage theorem applies to a map transverse to an embedded submanifold (Restrictions, corestrictions, and products of smooth maps are smooth, The transverse preimage theorem).

Proof

technique · direct

1.1

Define $H : M \times P \to N \times N$ by $H (p, q) = (F (p), G (q))$ . By [L2], $H$ is smooth, and $H^{- 1} (Δ_{N}) = {(p, q) : F (p) = G (q)} = M \times_{N} P .$

L1L2given

2.1

At a point $(p, q)$ with $F (p) = G (q) = y$ , the tangent space to $Δ_{N}$ is ${(u, u) : u \in T_{y} N}$ . Therefore transversality of $H$ to $Δ_{N}$ means that every pair $(a, b) \in T_{y} N \times T_{y} N$ can be written as $(a, b) = (d F_{p} u, d G_{q} v) + (w, w) .$ This is equivalent to $a - b \in d F_{p} (T_{p} M) + d G_{q} (T_{q} P)$ , which is exactly [F1].

F1step 1.1algebra

3.1

Hence $H ⋔ Δ_{N}$ , so [L2] applied with [L1] shows that $M \times_{N} P$ is an embedded submanifold of $M \times P$ .

L1L2step 2.1∎

Depends on

Used by

A fibre product of submersions Example

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)