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Every compact smooth manifold admits an excellent Morse function
Statement
Assume the axiom of choice. Every compact smooth manifold admits an excellent Morse function.
Facts & Assumptions
Given: A compact smooth manifold and the axiom of choice (The Axiom of Choice).
Under countable choice, every smooth manifold embeds in some finite-dimensional Euclidean space (Every smooth manifold embeds in some finite-dimensional Euclidean space).
On a compact embedded manifold, a generic linear height function is Morse (For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions).
On a compact manifold, excellent Morse functions are dense among all smooth functions (On a compact smooth manifold, the excellent Morse functions form an open dense subset of the and hence topology). Its published density proof uses the relative jet-transversality construction, whose successive perturbation choices are covered here by the assumed axiom of choice.
Proof
Use the assumed axiom of choice for the countable selections in [L1], and choose a smooth embedding . Then [L2] gives a Morse height function on that embedding.
Use the assumed axiom of choice for the successive selections in the density proof underlying [L3], and apply [L3] to the Morse function . An excellent Morse function then lies arbitrarily close to and in particular exists on .
Therefore every compact smooth manifold admits an excellent Morse function.
Depends on
- The Axiom of Choice
- Every smooth manifold embeds in some finite-dimensional Euclidean space
- For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions
- On a compact smooth manifold, the excellent Morse functions form an open dense subset of the $C^2$ and hence $C^\infty$ topology
Used by
- Closed odd-dimensional manifolds have zero Euler characteristic Corollary
- The Euler number of the tangent bundle is the Euler characteristic Corollary
- Algebraic Lefschetz number via rational homology traces Definition
- Canonical Morse homology of a closed manifold Definition
- A closed smooth manifold has the homotopy type of a finite CW complex Lemma
- Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
- The orientation-twisted diagonal realizes the Lefschetz trace Lemma
- Finiteness and additivity of the Euler characteristic Proposition
- Converse Poincare-Hopf for nowhere-zero fields Theorem
- Poincare-Hopf for closed manifolds Theorem
Dependency tree · two levels
41 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, Part 11 (standard reference, not scraped)
- Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)