Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory

TL;DR

The paper tackles computing the Conley complex and its connection matrix for a $P$-graded chain complex by weaving algebraic Morse theory with the homological perturbation lemma. A consistent $P$-graded splitting yields a contraction to a minimal Conley complex $(\mathcal{M}, d^{\mathcal{M}})$, whose differential is captured by a connection matrix $\Delta$ expressible as a sum over zigzag paths; this provides a rigorous, algebraic route to Conley data without relying on cell-graph matchings. The authors introduce a practical algorithm based on clearing optimisation to obtain the required splitting and perform column reductions to produce the connection matrix, with an $O(N^3)$ complexity bound. They further show how the Conley indices arise as relative homology pieces and discuss extensions to arbitrary coefficient rings and potential connections to spectral sequence formalisms, highlighting the method’s scalability and applicability in multiparameter settings. Overall, the work provides a principled, computable framework for extracting Conley-type invariants from $P$-graded chain complexes in a broad algebraic setting.

Abstract

Given a poset-graded chain complex of vector spaces, a Conley complex is the minimal chain-homotopic reduction of the initial complex that respects the poset grading. A connection matrix is a matrix representing the differential of the Conley complex. In this work, we give an algebraic derivation of the Conley complex and its connection matrix using homological perturbation theory and algebraic Morse theory. Under this framework, we use a graded splitting of relative chain groups to determine the connection matrix, rather than Forman's acyclic partial matching in the usual discrete Morse theory setting. This splitting is obtained by means of the clearing optimisation, a commonly used technique in persistent homology. Finally, we show how this algebraic perspective yields an algorithm for computing the connection matrix via column reductions on the differential of the initial complex.

Figures (2)

• Figure 1: Left: The simplicial complex $X = \Delta^2$ together with an example of a filtration function $f \colon X \to P$ for the linear poset $P = \{0,1,2,3\}$. Right: the vertices of $\Delta^2$ labelled by $v,u,w$.
• Figure 2: Left: the simplicial complex with the multivector field from Example 2.4. Right: the poset associated to the partition into Morse sets.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 2.1
• Example 2.2
• Remark 2.3
• Remark 2.4
• Example 2.5: Relative homology
• Definition 2.6
• Remark 2.7
• Example 2.8
• Example 2.9