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The transverse preimage theorem
Statement
Let be smooth and let be an embedded submanifold of codimension . If , then is an embedded submanifold of of codimension . For each ,
Facts & Assumptions
Given: A smooth map transverse to an embedded submanifold .
Embedded submanifolds admit local defining submersions (Embedded submanifolds admit local defining submersions).
Transversality to is equivalent to surjectivity on the normal quotient (Transversality is equivalent to surjectivity on the normal quotient).
A regular level set is an embedded submanifold, and its tangent space is the kernel of the defining submersion differential (A regular level set is an embedded submanifold, The tangent space of a regular level set is the kernel).
Proof
Fix and put . By [L1], choose a neighbourhood of and a submersion such that .
At , the differential kills exactly . Therefore [L2] implies that is surjective. So is a regular value of near .
Since , [L3] shows that this set is an embedded codimension- submanifold near , with tangent space
Because was arbitrary, these local models glue to an embedded submanifold structure on with the stated tangent-space formula.
Depends on
Used by
- Transverse embedded submanifolds intersect in the expected codimension Corollary
- The intersection of coordinate spheres as a transverse level set Example
- A preimage need not be a submanifold without transversality False statement
- A co-oriented closed transversal detects nonvanishing rational homology of a compact leaf Lemma
- A collared Whitney disk can avoid an entire compact immersed image Lemma
- A relative map cell kills its class with the correct fundamental-group action Lemma
- A small regular homotopy removes triple images and preserves transverse branch pairs Lemma
- A taut foliation of a compact connected manifold has a single closed transversal Lemma
- Algebraic open inclusions surject on fundamental groups Lemma
- Belt-sphere complements in low handle levels preserve the fundamental group Lemma
- Compact smooth manifolds have finite CW models under countable choice Lemma
- Compact transverse complementary intersections are finite Lemma
- Existence of regular projections Lemma
- General position makes a Whitney disk embedded and interior-disjoint in the stable range Lemma
- General-position isotopies of links Lemma
- Metastable approximation of maps by embeddings Lemma
- Normal bundle of the zero locus of a transverse section Lemma
- Preimage orientation agrees with the local intersection sign Lemma
- The double point locus has the expected dimension 2m-n Lemma
- The group-ring modification lemma for embedded spheres Lemma
- Zero- and one-handles are eliminated in a simply connected h-cobordism Lemma
- Transverse preimages carry the pulled-back normal structure Proposition
- Whitney disjunction removes algebraically cancelling double points Proposition
- For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions Theorem
- Neat submanifolds have boundary-adapted slice charts Theorem
- Parametric transversality Theorem
- The Whitney trick in the codimension-two borderline case Theorem
- Transverse fibre products are embedded submanifolds Theorem
- Transverse preimages for maps from manifolds with boundary Theorem
Dependency tree · two levels
13 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
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
- Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)