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A Morse function on a compact manifold has finitely many critical points
Statement
If is a compact smooth manifold and is a Morse function, then has only finitely many critical points.
Facts & Assumptions
Given: A compact smooth manifold and a Morse function .
Every critical point of a Morse function is nondegenerate (Morse functions and excellent Morse functions).
Every nondegenerate critical point is isolated (Nondegenerate critical points are isolated).
Compactness means that every open cover has a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).
The vanishing of in a chart is equivalent to the vanishing of all coordinate partial derivatives (Coordinate formula for the differential of a function).
Proof
For each critical point , [F1] and [L1] give an open neighbourhood containing no critical point other than .
If is not critical, choose a chart around and write . By [L3], some partial derivative is nonzero at ; continuity keeps it nonzero on a smaller open neighbourhood , so contains no critical point.
The family of all together with all covers . By [L2], it has a finite subcover. Only finitely many sets of the form occur in that subcover, and each such set contains exactly one critical point by step 1.1. Therefore has finitely many critical points.
Hence every Morse function on a compact manifold has finitely many critical points.
Depends on
Used by
- Index-one trajectory moduli spaces are finite Corollary
- Morse Euler characteristic identity Corollary
- Morse homology recovers the Morse inequalities Corollary
- Geometric convergence to a broken trajectory Definition
- Morse homology of a Morse--Smale pair Definition
- Morse numbers and the Morse polynomial Definition
- The continuation chain map Definition
- The mod-two Morse chain group Definition
- The signed Morse differential over the integers Definition
- A closed smooth manifold has the homotopy type of a finite CW complex Lemma
- A negative-gradient trajectory on a compact Morse manifold has single critical alpha and omega limits Lemma
- Breaking length is bounded by the index drop Lemma
- Compact smooth manifolds have finite CW models under countable choice Lemma
- Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
- For a compact Morse function, disjoint local bump perturbations can separate finitely many equal critical values without changing the Hessians Lemma
- On a compact manifold, a Morse function has finitely many critical points and a uniform Hessian gap on disjoint critical neighborhoods Lemma
- Separating critical values far from the boundary Lemma
- Every Morse function admits a complete downward gradient-like field on a closed manifold Proposition
- Relative Morse inequalities for a cobordism Proposition
- The handle chain complex computes singular homology Proposition
- The relative Morse complex of an adapted cobordism Proposition
- Compactness up to breaking needs closedness or a proper compactness package Remark
- Adapted excellent Morse functions exist on compact cobordisms Theorem
- Compactness up to breaking of Morse trajectory spaces Theorem
- Continuation trajectories are compact up to breaking Theorem
- Morse functions and handle decompositions correspond Theorem
- Morse polynomial identity Theorem
- Rearrangement of critical levels by index Theorem
- Relative Morse--Smale perturbation of a gradient-like field Theorem
- The index-two compactification is a compact one-manifold with boundary Theorem
Dependency tree · two levels
15 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
- Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)
- Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)