Conley Index

The Conley Index is a powerful tool in generalized Morse theory, extending the applicability of Morse theory to non-manifolds and degenerate gradients. Originating from the work of Conley, this index has found applications in various fields such as mathematical biology, rigorous numerics, and experimental time-series inference.

Definition

The Conley Index is defined for a complete locally compact metric space $X$ with a continuous flow $\phi: X \rightarrow X$. It is associated with a suitable compact subset $B \subseteq X$. The invariant set $S = \text{Inv}(B; \phi)$ is the set of all points $x \in B$ such that $\phi_t(x) \in B$ for all $t$. A compact set $B$ is an isolating block if $\text{Inv}(B; \phi)$ lies strictly in the interior of $B$, and no internal "tangencies" are permitted. The Conley Index of an invariant set $S$ with isolating block $B$ is denoted as $\text{Con}(S)$ and captures the topology of $B$ relative to how the flow exits $\partial B$.

Exit Set and Homological Conley Index

The exit set of $B$, denoted as $E$, is defined as $\{x \in \partial B: w(x) \cap B, \forall 0 < \phi \leq 1\}$. The Conley Index $\text{Con}(S)$ is the pointed homotopy type $h[B/E, \{E\}]$. It is the quotient space $B/E$ (up to homotopy) with $E$ remembered as an abstract basepoint. The homological Conley Index is defined as the relative singular homology $\text{Cone}(B) := H_*(B, E)$. The index is well-defined, meaning that any two isolating blocks with the same invariant set have equivalent Conley indices.

Examples and Special Cases

Morse Index Example

The Conley Index of a nondegenerate critical point of a Morse function with Morse index $\omega$ is the homotopy type of the sphere $S^\omega$ (with basepoint). In the special case where $\omega = 0$, the fixed point is a sink, and the exit set $E$ is the empty set. This results in a Conley Index of $S^0$, the disjoint union of two points.

Properties and Generalizations

The Conley Index has several remarkable properties. It generalizes the Morse index in that fixed points need not be nondegenerate, discrete, or stratified. Vector fields need not be smooth or gradient, and domains need not be manifolds or stratified spaces. However, it's worth noting that some flows, such as integrable Hamiltonian flows, admit few index pairs.

Theorems and Computational Aids

Conley Index Theorem Theorem. The Conley Index Theorem states that the Conley Index has the following properties: Invariance: The Conley Index depends only on $S$, not on the choice of index pair $(B, E)$ for $S$. Continuation: If $(B_t, E_t)$ is a continuous family of index pairs for a continuous family of flows, then $\text{Con}_t = \text{Con}(B_t, E_t)$ is constant. Additivity: If $(B, E)$ and $(B', E')$ are disjoint index pairs for $S$ and $S'$, then $\text{Con}(S \cup S') = \text{Con}(S) \vee \text{Con}(S')$.

See Also

• Morse Theory
• Homotopy Theory

(edit) Topics in Topological Data Analysis (📁) Background Point-set topology • Algebraic topology Manifolds Manifold • Configuration space • Differentiability • Vector space (bundle) • Tangent space (bundle) • Cotangent space (bundle) • Vector field (Flow, Critical points) • Braid • Transversality • Whitney embedding theorem • Stratified space Complexes CW Complex • Δ-Complex • Regular Complex (Simplicial, Cubical, Vietoris-Rips, Čech, Witness, Flag, Strategy • Decision Task) • Nerve • Phylogenetic tree • Discretized graph configuration space • State complex Euler Characteristic Euler characteristic • Curvature • Nonvanishing vector field • Fixed point index • Tame topology • Euler calculus • Target enumeration • Fubini theorem • Euler integral transform • Intrinsic volume • Gaussian random field Exact Sequences Chain homotopy • Snake lemma • Good pair • Mayer-Vietoris sequence • 5-lemma • Cellular homology • Coverage • Degree • Sperner's Lemma • Borsuk-Ulam theorem • Short exact sequence • Lefschetz index • Nash equilibrium • Hex • Barcode • Space of natural images • Zigzag persistence Homology Simplicial homology • Künneth theorem • Coefficient • Singular homology • Reduced homology • Čech homology • Relative homology • Excision Theorem • Local homology • Relation homology • Functoriality • Inverse kinematics • Winding number • Brouwer fixed point theorem Cohomology Duality • Cochain complex • Cohomology • Universal coefficient theorem • Poincaré duality • Alexander duality • Helly's Theorem • Numerical Euler integration • Forms • De Rham cohomology • Cup product • Helicity Theorem • Laplacian • Hodge Theorem • Cohomological coordination Morse Theory Morse homology • Excursion set • Definable functional • Euler-Fourier transform • Stratified Morse theory • Conley index • Forcing theory • Attractor-repeller pairs • Floer homology • Symplectic form • Hamiltonian field • Arnold's Conjecture • Lefschetz-Hopf theorem • Stratified Lefschetz-Morse Index • Discrete Morse theory • LS category • Unimodal decomposition Homotopy Fundamental group • Van-Kampen theorem • Covering space • Knot • Braid • Seifert conjecture • Higher homotopy groups • Biaxial nematics • Hurewicz Theorem • Whitehead Theorem • Eilenberg-MacLane space • Topological social choice • Fiber bundle • Section • Path space • Farber lemma • Sectional category • Topological complexity • Fibration • Homotopy extension property • Model category • Homotopy type theory Sheaves Cellular sheaf • Local cohomology • Fiber homology • Sheaf cohomology • Global sections theorem • Linear sheaf • Flow sheaf • Network coding sheaf • Information flow theorem • Čech cohomology • Preseaf • Stalk • Gluing axiom • PDE sheaf • Operations on sheaves • Vietoris mapping theorem • Sheaf-theoretic Nyquist-Shannon • Fourier-Sato transform • Cosheaf • Bézier curve • Barcodes • Interval Indecomposables • Étale Space • Leray spectral sequence • Spectral sheaf theory • Sheaf neural network Categorification Category • Groupoid • Monoid • Morphism • Equalizer • Functor • Kan complex • Nerve • Clustering • Natural transformation • Cone • Interleaving Limit • Terminal object • Product • Pullback • Homology Colimit • Initial object • Coproduct • Pushout • Cohomology Topological sheaf • Sheaf operations • Derived functors