Nondegenerate critical points, nullity, index, and coindex

Depends on

• The intrinsic Hessian of a smooth function at a critical point
• Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form

Used by

• A Morse gradient zero contributes (-1)^λ to the index Corollary
• Index and coindex swap under negation Corollary
• Morse homology recovers the Morse inequalities Corollary
• A degenerate critical set can be nonisolated Counterexample
• A noncompact continuation datum can lose its trajectories at infinity Counterexample
• Euler equality alone does not imply perfectness Counterexample
• A regular continuation datum between Morse--Smale pairs Definition
• Broken continuation trajectories and geometric convergence Definition
• Broken Morse trajectories Definition
• Morse functions and excellent Morse functions 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 mod-two Morse differential Definition
• The orientation line of a Morse critical point Definition
• A created cancelling pair contributes a (1+t)tᵏ term Example
• A Morse function on the torus is perfect over every field Example
• A standard torus height function has four critical points Example
• Continuation across a birth--death pair adds an acyclic summand Example
• Morse and cellular boundaries for a surface handle presentation Example
• Real projective space shows coefficient-dependent perfectness Example
• Relative Morse homology of a single-handle cobordism Example
• The height function on a sphere is perfect 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
• The standard quadratic form realizes every Morse index Example
• Two Morse functions on the circle have isomorphic Morse homology Example
• A smooth function is Morse if and only if its differential section is transverse to the zero section Lemma
• Breaking length is bounded by the index drop Lemma
• Cellular boundary coefficients are the signed trajectory counts Lemma
• Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
• Complex stratified Morse data, constructible transport and complex links Lemma
• Nondegenerate critical points are isolated Lemma
• Sylvester inertia makes the Morse index intrinsic Lemma
• The relative Morse complex of an adapted cobordism Proposition
• The zero-dimensional Morse convention Remark
• Continuation trajectories are compact up to breaking Theorem
• Handle duality from negating a Morse function Theorem

…and 5 more results.

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)
• Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)