Morse functions and excellent Morse functions

Depends on

• Critical points and critical values of a smooth function
• Nondegenerate critical points, nullity, index, and coindex

Used by

• A Morse function on a compact manifold has finitely many critical points Corollary
• A Morse gradient zero contributes (-1)^λ to the index Corollary
• Closed odd-dimensional manifolds have zero Euler characteristic Corollary
• The Euler number of the tangent bundle is the Euler characteristic Corollary
• The Morse critical-point sum is the Euler characteristic Corollary
• A Morse function can have two different critical points with the same critical value Counterexample
• Downward gradient-like vector fields for a Morse function Definition
• Morse function adapted to a cobordism Definition
• Morse numbers and the Morse polynomial Definition
• Negative-gradient trajectories of a Morse function Definition
• The mod-two Morse chain group Definition
• A standard embedded torus has generic height directions with four Morse critical points, but symmetry directions create degenerate or nongeneric behavior Example
• A standard torus height function has four critical points Example
• Continuation across a birth--death pair adds an acyclic summand Example
• The height function on the sphere is Morse and excellent Example
• The Morse complex of the circle Example
• The Morse complex of the two-sphere Example
• Two equal critical levels can be separated by adding disjoint bump perturbations near the corresponding critical points Example
• Two Morse functions on the circle have isomorphic Morse homology Example
• A closed smooth manifold has the homotopy type of a finite CW complex Lemma
• Critical values of disjoint trajectory closures can be interchanged Lemma
• Face-relative Morse decomposition into the ten sewing generators Lemma
• For a compact Morse function, disjoint local bump perturbations can separate finitely many equal critical values without changing the Hessians Lemma
• Gluing handle Morse models along collars Lemma
• On a compact manifold, a Morse function has finitely many critical points and a uniform Hessian gap on disjoint critical neighborhoods Lemma
• Regular sublevels are compact manifolds with boundary Lemma
• Separating critical values far from the boundary Lemma
• Standard handle admits an adapted Morse function Lemma
• Every Morse function admits a complete downward gradient-like field on a closed manifold Proposition
• Being Morse does not by itself force distinct critical values; excellence is a separate generic condition Remark
• Converse Poincare-Hopf for nowhere-zero fields Theorem
• Morse functions and handle decompositions correspond Theorem
• On a compact smooth manifold, the excellent Morse functions form an open dense subset of the C² and hence C^∞ topology Theorem
• Poincare-Hopf for closed manifolds Theorem
• Rearrangement of critical levels by index Theorem
• Self-indexing Morse functions exist Theorem
• Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric Theorem

Dependency tree · two levels

6 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

• Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)
• Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)