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 $\text{Con}(S \cup S') = \text{Con}(S) \vee \text{Con}(S')$.