Morse--Smale pairs

Depends on

• Downward gradient-like vector fields for a Morse function
• Stable and unstable sets of a critical point
• Global stable and unstable manifolds are immersed Euclidean spaces
• Transverse smooth maps

Used by

• Index-one trajectory moduli spaces are finite Corollary
• A noncompact continuation datum can lose its trajectories at infinity Counterexample
• The symmetric torus height flow is not Morse--Smale Counterexample
• A regular continuation datum between Morse--Smale pairs Definition
• A regular two-parameter continuation datum Definition
• Broken continuation trajectories and geometric convergence Definition
• Broken Morse trajectories Definition
• Canonical Morse homology of a closed manifold Definition
• Geometric convergence to a broken trajectory Definition
• Morse homology of a Morse--Smale pair Definition
• The continuation chain map Definition
• The mod-two Morse differential Definition
• The orientation line of a Morse critical point Definition
• A Morse--Smale flow on the circle Example
• A Morse--Smale height function on a tilted torus Example
• Broken trajectories in an index-two torus moduli space Example
• Continuation across a birth--death pair adds an acyclic summand Example
• The Morse complex of the circle Example
• The Morse complex of the two-sphere Example
• Two Morse functions on the circle have isomorphic Morse homology Example
• Arbitrary metric Morse--Smale end counts form finite Morse chain complexes Lemma
• Boundary orientation of the compactified one-dimensional Morse moduli space Lemma
• Breaking length is bounded by the index drop Lemma
• Broken Morse trajectories have strictly decreasing critical values and indices Lemma
• Cellular boundary coefficients are the signed trajectory counts Lemma
• Compactified unstable manifolds give the Morse--Smale CW decomposition Lemma
• Continuation solutions have critical limits and exponential decay Lemma
• Every broken trajectory is a limit of ordinary trajectories Lemma
• Finite flow matching gives local charts at metric-end broken trajectories Lemma
• Gluing once-broken index-two trajectories: collar ends Lemma
• Morse--Smale transversality and surjectivity of the linearized flow operator Lemma
• Orientation lines orient the continuation moduli spaces compatibly with gluing Lemma
• The continuation energy identity Lemma
• The continuation map of constant data is the identity Lemma
• Unstable orientations induce orientations of the trajectory moduli spaces Lemma
• A parametrized Morse trajectory space is a manifold Proposition
• The relative Morse complex of an adapted cobordism Proposition
• Ambient orientability is not required for Morse--Smale transversality Remark
• Compactness up to breaking needs closedness or a proper compactness package Remark
• Flow and compactness hypotheses for noncompact Morse homology Remark

…and 9 more results.

Dependency tree · two levels

11 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

• Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, §2.2 (standard reference, not scraped)